molecular filaments and their assemblies underlie the function of a diverse range of materials , from intra- and inter - cellular matter to high - strength synthetic materials .condensed , parallel arrays , or bundles , of cytoskelatal filaments like f - actin and microtubules provide both mechanical reinforcement to cells and play crucial roles in vital cellular processes like adhesion , division and locomotion  .rope - like assemblies of extra - cellular protein filaments for example , collagen , cellulose or fibrin are also primary structural elements _ in - between _ cells , in plant and animal tissue  . as filaments are densely - packed and hence strongly interacting in bundles , the structure of intermolecular organization underlies the collective dynamic and mechanical properties of these assemblies , critical for their robust function in living systems . in the simplest models of this material prototype ,the ground state of bundles of straight filaments is universally a hexagonally - packed array of parallel filaments , which maximizes both the number of inter - filament contacts as well as the density within the bundle . as a consequence , filament bundles forming under the influence of cohesive inter - filament forces  as well as osmotic pressure ( depletion forces ) are observed to form hexagonal cross - sectional order .notably both the ( straight ) filament shape and the inter - filament spacing are uniform throughout a hexagonal bundle , which implies that both length and lateral size of self - assembled bundles is thermodynamically unlimited  .  in this article, we consider the more complex structure and thermodynamics of bundles formed by _ intrinsically - curved _ filaments , a distinction which we show frustrates homogeneous inter- and intra - filament geometry of straight , hexagonal bundles . for many common biofilaments f - actin or microtubules the filament backbone is straight in the absence of external stress or thermal fluctuations .there are , however , important exceptions , where the backbone filament geometry has a preferred , or natural , curvature , deriving from the shape and interactions between globular subunits  .most widely known among this class of intrinsically - curved filaments is the bacterial _ flagella _ , a corkscrew shaped protein filament helical radius ; and pitch ; diameter nm , shown in fig .[ figure1 ] ( b)that propels common microorganisms like _ e. coli _ and _ salmonella_. notably , bacteria swim , not by the actuation of single filaments , but by the screw - like rotation of coherent bundles of multiple flagella  , held together through a combination of contact and hydrodynamic interactions . more recently , new classes of curved - filament assemblies have been discovered as components of bacterial cytoskeleton "  .one such filament mreb , considered to be a prokaryotic analogue to f - actin , is observed to associate with inner - cell wall of bacteria in bundles that wind helically around the cell body , potentially shaping the growth of rod - like bacteria  . a second filament , crescentin , also forms bundles _ in vivo _  helical structure is believed to drive the curved and helical shapes of bacteria like _ caulabacter crescentus _ .unlike the case of straight - filament assembly , bundles of curved - filaments are subject to an unavoidable geometric conflict between inter - filament packing and the natural curvature of filaments .simply put , it is not possible to evenly space nearest - neighbor filaments throughout the ordered cross - section of the bundle while maintaining uniform shape of all filaments .this conflict is illustrated by the bundles of helical filaments in fig .[ figure1 ] ( c ) and ( d ) , which we denote , respectively , as _ isometric _ ( constant spacing ) and _ isomorphic _ ( constant shape ) packings .the perfect inter - filament packing constant center - to - center filament distance in the isometric bundle constrains the relative geometry between filaments so that the inter - filament separation , , between any filament pair is _ relatively parallel _ to the filament tangents  .this means that in isometric bundles all filament backbones are normal to the common plane of lattice packing , and does not _ twist _ around the backbones along the length of the bundle [ fig .[ figure1 ] ( c ) ] .importantly , this packing constrains the relationship between the curvature and torsion of a reference " filament and the shape of a filament at separation , whose respective curvature and torsion follow  where is the normal to reference filament .( [ eq : kaptau ] ) is the filamentary analogue of the curvature focussing " properties of evenly - spaced , layered materials ( i.e. smectics )  , embodying the fundamental frustration of the lateral assembly of filaments in densely - packed bundles . on one hand ,driving forces for dense - filament packing or cohesive filament generically penalize deviations from perfect lateral packing in a bundle .however , when these filaments are intrinsically curved , meaning that in their stress free configurations , perfect lateral packing requires the filament geometry to vary along normal directions ( ) in the bundle cross section , incurring a mechanical cost for change of filament shape .thus , in bundles of intrinsically - curved filaments , isometric packing is not compatible with constant filament shape , a property achieved in the isomorphic packing of fig .[ figure1 ] ( d ) . in the isomorphic state ,as we show below , the non - local nature of filament contact requires inter - filament spacing to vary _ along _ the bundle , straining either interactions that prefer constant spacing or thermodynamic forces favoring maximal density .the mutual frustration between isometric and isomorphic packing implies that the state of inter - filament packing in curved - filament bundle is determined not only by the lateral forces that drive filaments in to tightly - packed assemblies , but also by the mechanical costs of filament distortion . here, we show that interplay between geometry and mechanics in curved - filament bundles has a profound influence on the structure and dimensions of assembly , giving rise to thermodynamic behavior not exhibited by assemblies of unfrustrated , straight filaments .  in this paper , we study the frustration between constant spacing and constant shape in bundles of helically - shaped bundles , whose backbones have preferred values of curvature and torsion , and .helical filaments are prototypes of the frustration in curved - filament assemblies , whose degree of frustration can be tuned continuously from zero for straight filaments ( ) and the maximal frustration of ring filaments ( . in ref . , it was shown that the mechanical costs of filament shape change in isometric , helical filament bundles grow rapidly with lateral size of bundles , ultimately limiting the lateral radius of bundles that are unable to relax the constant - spacing constraints . in the present study, we develop a continuum theory to analyze the non - linear interplay between the distinct costs of inter - filament and intra - filament distortions in helical - filament bundles which relax both shape as well as the lateral packing .we study two models for the thermodynamic cost of inter - filament distortion : 1 ) an elastic model for disruption of inter - filament cohesion ; and 2 ) the osmotic excluded - volume cost of bundles formed in concentrated solutions of osmolytes excluded from the filament packing . analyzing a class of low - energy bundles that span continuously from the limiting isometric to isomorphic bundles ,we show that bundle packing evolves from perfect inter - filament spacing in the limit of vanishing bundle diameter to constant filament shape in the limit of large diameter .while inter - filament packing in cohesive bundles is distorted for any finite radius , we find in the limit of weak resistance to filament torsion , osmotically - condensed bundles remain isometric over finite range of small bundle size .finally , the consider the equilibrium size of bundles in the canonical ensemble , showing that the mechanical cost of shape change in small bundles stabilizes finite size bundles when the surface energy of bundles is below a threshold value , which is shown to increase with for both bundling mechanisms .  the remainder of this article is organized as follows . in sec .ii.a we introduce the geometry of helical filament bundles , as well as the mechanical cost for filament distortion and the thermodynamic costs for distortion of the packing in sec .ii.b . in sec . iii.a we analyze the variation of bundle packing with lateral size of bundles in both cohesive and osmotically - driven bundles , and in sec .iii.b we consider the equilibrium radius of bundles in solutions of freely - associating filaments above saturation .finally we conclude in sec .iv with a discussion of the implications on our study for bundles of flagellar filaments .we consider two models of helical - filament bundles , which differ in terms of the driving forces for assembly . in the first model ,assembly is driven by short - range , adhesive interactions between neighbor filaments in the bundle , while the second model considers densely packed bundles to form under the influence of osmotic pressure , that works against the total volume of the bundle interior . in both models ,an increase of inter - filament packing forces ( adhesion or pressure ) drives the bundles away from the isomorphic state preferred by unbundled filaments towards the isometric packing . to describe this transition, we consider a class of bundle configurations that may be continuously transformed from the isomorphic into the isometric state and _ vice versa_.   we consider bundle configurations described by a packing the cross - section around a central filament , whose center is described by , . the backbone of the central filament is described by a tangent , normal or bi - normal , the frenet frame , curvature and torsion , deriving from the relations , the curvature and torsion of this filament define the _ helical angle _ , shown in fig .[ figure1 ] ( a ) . in the bundle cross - section ,filaments are labeled by coordinates .the vector maps the position of the central filament to the filament at in the plane spanned by the basis vectors , and , we refer to the plane normal to as the _ packing plane _ , and the orthonormal frame as the _ packing frame_. we assume a regular , hexagonal lattice packing , i.e. and where and are integers labeling occupied lattice positions and lattice spacing .  due to the constraints of packing, the geometry of filaments varies through the cross - section of a bundle .given our construction , we describe the variation in terms of the functions , and characterizing respective values of backbone orientation , curvature and torsion for filaments at .we focus on the limit of narrow bundle cross - sections , such that and .the shape of the filament at derives from both the shape of as well as the tilt and rotation of the frame with respect to the geometry of the central curve at .the rotation of filament positions in the packing plane along the arc length of the bundle is described by the transformation , where here , is the tilt angle of the packing plane , the _ packing angle _ , with respect to the central tangent , i.e. .thus , and constitute two independent parameters specifying the geometry of the packing .we consider bundles that are homogeneous along their length so that both and are constant .the relationship between geometry ( i.e. orientation , curvature and torsion ) of the central filament to the geometry of a filament at derives from rotation of the frame along the bundle , which follows straightforwardly from eqs .( [ uv ] ) and ( [ upvp ] ) , \label{rot}\end{aligned}\ ] ] where from eq .( [ rot ] ) we note that the net rotation of filament positions around the center of the cross section , is the simply the sum rotation of and around , , and the rotation of and relative the to that coordinate system , . in our analysis , we focus on the set of low - energy states that span the two - limiting packings , isometric and isomorphic . we show in the appendix that vanishing torsion of filament positions in the packing plane ( i.e. ) is a necessary condition for both constant filament shape and constant filament spacing , and we therefore , expect this condition to also describe the low - energy states intermediate to these extreme cases . that is , we restrict our analysis to so that there is _ no rotation _ of the filament position in the packing plane along the bundle .the variation of filament geometry throughout the packing plane derives from a calculation of the frenet geometry based on the construction of eqs .( [ uv])-([omegas ] ) ( see appendix ) . for backbone orientation ( tangents )we find , first order ( in ) correction  which is manifestly perpendicular to .the variation of backbone curvature and torsion are described by where and , \label{dtau}\end{aligned}\ ] ] where the common root of expressions occurs for packing angle , notably , shape change of filaments ( to lowest order ) is distributed anisotropically in the packing plane : curvature and torsion are maximally distorted for filaments along , while no shape change is required for filaments at .this construction encompasses two extremal limits of the packing geometry :  _ isomorphic packing _ : - inspection of eqs .( [ dtperp ] ) - ( [ dtau ] ) immediately reveals these conditions achieve uniform orientation and shape the packing plane , .solution of eq .( [ omegas ] ) gives the inclination and rotation rate , , the isomorphic packing , in this packing , the packing plane is fixed ( say , the plane in the lab frame ) and remains normal to the pitch axis of the helices ( direction ) . though filaments are parallel in the packing plane , inter - filament distances in this plane are not the true distances between the helical curves .that is , and are not _ relatively parallel _ to the backbone , .below we show that this implies variation of inter - filament spacing the pitch axis of the helices in isomorphic bundles.  _ isometric packing _ : - in the isometric packing , filaments are parallel in the packing packing plane , and there is no twist of the packing around . because and are relatively parallel to , distances in the packing plane are true inter - filament distances , and hexagonal packing in this plane implies uniform neighbor spacing along the length and throughout the cross section of the bundle . while spacing is constant , shape varies throughout the packing plane according to which derive from eqs .( [ dkappa ] ) and ( [ dtau ] ) for and , consistent with the shape variation of eq .( [ eq : kaptau ] ) .   in this section, we describe the mechanical cost of distortions of filaments away from a preferred helical shape in bundles , characterized by constant values of backbone curvature and torsion .we adopt the kirchoff - love theory of filament mechanics , where distortions are captured the rotation of a material frame along the backbone  . for each filament , labeled by position , this frameis described by the triad where is along the backbone tangent and describe the position of the filament cross section , and represents an internal , rotational degree of freedom of the filament .the geometry of filament backbone is locally described by three rotation rates , where for simplicity we have suppressed the dependence of each quantity .the first two rates , and describe the bending of the filament backbone projected onto the two material directions in the filament cross section , while the last rate describes the rotation , or torsion , of cross - section around the filament backbone , . in this framework ,a helical filament is described by a preferred constant value of twist , curvature , and a preferred direction of bending , which we choose to be .  for simplicity ,we focus our attention on the limit of easy twist " filament mechanics  , which can be justified strictly for the case where the bending modulus of the filament is much larger than the twist modulus of the filament . in this limit ,for any modulations of and the torsional state of the filament adjusts to maintain alignment of the normal to , the direction of preferred curvature .explicitly , and the mechanical cost of shape change for a filament at becomes simply , ^ 2 + c \big [ \tau({{\mathbf x } } ) - \tau_0\big]^2 \big\}.\ ] ] to describe a filament bundle , we consider uniform areal density of filaments and integrate over circular bundle cross section of radius .( [ kaptau ] ) we find the total mechanical energy for a bundle filaments where is the filament contour length . in deriving eq .( [ emech ] ) , we have assumed that shape of the central filament remains close the preferred shape for sufficiently narrow bundles , specifically , , which follows from minimization of eq .( [ emech ] ) with respect to and . in the remainder of the article, we drop the distinction drop the distinct between the preferred helical shape and shape of the central filament , and take and . in fig .[ packing ] we show graphically the spatial variation of bending energy density in bundles ranging from bundles ranging isometric to isomorphic packing .we consider two models of the thermodynamic cost of inhomogeneous filament packing : 1 ) a continuum elastic model of cohesive filament bundles ; and 2 ) the osmotic cost of excluded volume in densely packed bundles . in both models , free energy costs arise as the consequence of the non - local nature of inter - filament contact .consider two filaments with backbones centered at and in the packing plane .assuming that , filament backbones may be approximated locally as straight segments of orientation , for nearby points . rather than the in - plane separation ,interactions between homogeneous filaments are natural functions of which is the _ distance of closest approach _ between the filament centers  . since when filaments are locally tilted into the packing plane , inter - filament spacing is necessarily reduced relative to the packing plane distance . from eq .( [ dtperp ] ) , the in - plane components are \\t_{v } ( { { \mathbf x } } ) & = & \cos ( \omega s ) \sin \theta  \big [ 1 + \cos \theta  \omega_p ( { { { \bf n}}}\cdot { { \mathbf x } } )  \big ]  , \label{tin}\end{aligned}\ ] ] which shows that for in - plane tilt ( in the packing frame ) varies along bundle , as well as within the bundle cross - section ( for ) . the interfilament strain generated by tilt in the packing plane is depicted in fig .[ packing ] graphically for bundles ranging isometric to isomorphic .below we find that inhomogeneous patterns of tilt , lead directly to thermodynamic costs associated with distortions away from the isometric state .   in this sectionwe derive a model for filaments condensed by the finite - range attractive interactions , where the energetic cost of small changes in local inter - filament spacing are described by a non - linear continuum elastic theory of columnar materials  .the role of non - linear coupling between strains and tilts has been previously discussed and analyzed extensively in the context of inhomogeneous packing in chiral ( twisted ) filament bundles  .the elastic energy has the form , , \label{eelas}\ ] ] where and are the lam constants corresponding to the isotropic elastic response of a hexagonally - ordered material  . here, the 2-component strain , describes changes in inter - filament spacing of nearby filaments , , relative to initial separation , where is the two - component in - plane separation between filaments in the reference state .we consider as a reference state , filaments initially oriented normally to the packing plane , so that indices and sum over and directions and .the form of the elastic strain follows , which is sensitive only to changes in inter - filament spacing perpendicular to backbone orientation  proportional because in - plane rotations are small  ] .importantly , eq .( [ uij ] ) shows that inter - filament spacing , and consequently , inter - filament cohesive energy changes both with shifts of relative in - plane filament positions , as described by and with deformations that tilt backbones into the packing plane , as described by .note that while these latter terms vary along the bundles according to eq .( [ tin ] ) , _ in the packing frame _ the relative positions of filaments are _ fixed _ along , according to eq .( [ rv ] ) .this implies that the net effect of in - plane tilt , when averaged over the length of the bundle , is to generate an _isotropic _ compression , which in turn , generates a tensile response in the packing . to offset the costs of tilt - induced compression, we consider an isotropic , in - plane strain by where and are coefficients parameterizing the isotropic dilation of the packing at a radius from the central filament .inserting into eq .( [ eelas ] ) and minimizing over and we find , \label{eelastic}\ ] ] where again we retain terms to lowest order in and where is the in - plane density of filaments .   in this section, we derive the cost associated with volume change of filaments bundled under the influence of an osmotic pressure , .this model is relevant to the case of filaments in a solution of high - molecular weight polymers , whose size excludes them from the interstitial space between densely - packed filaments in condensed bundles .the osmotic free - energy cost of a bundle is simply , where is volume within of the bundle . here, we consider a hard tube " model of filament packing , which requires that the local distance of closest approach between any filament and its neighbor in the packing satisfies , where is the filament diameter .as described above , local tilt of filaments into the packing plane frustrates the uniform spacing , hexagonal close - packing , requiring a distortion of the bundle cross - section . to avoid overlap along nearest neighbor direction ( unit vector in packing plane )we assume that the lattice spacing , , in the packing plane _ expands _ to maintain , ^ 2 \big ) .\ ] ] since the bond direction is fixed in the packing - plane while in - plane components of rotate along the bundle according to eq .( [ tin ] ) , this condition must hold at for _ all neighbor directions _ , including the in - plane tilt direction .neglecting the correction to this gives us the packing - plane lattice dimension , with the areal density of filaments in the packing plane of , where .the cross - sectional area of the packing plane is simply , the total volume excluded by the core of bundle is equal to the volume swept out by translating the cross - sectional area of the packing plane along the contour of the central filament .since the packing plane is inclined by an angle with respect to the tangent of , this excluded volume is simply , and the osmotic cost forming the bundle becomes notably , the volume - expansion cost for osmotically condensed filaments grows more quickly at small than for elastic distortion model .we now analyze the structural transition of helical filament bundles of fixed radius in terms of the two models that penalize distortions of inter - filament packing , described by and derived in the previous section .we determine the equilibrium variation of packing angle , which serves as an order parameter , with bundle size , filament shape and respective costs of inter- and intra - filament distortion .specifically , we show that in the limit of strong packing forces , bundles approach the isometric state as radius vanishes , while in the opposite limit of large mechanical costs for intra - filament distortion , bundles approach the isomorphic state , .first , we analyze the state of packing in both models for a given , and , and then consider the equilibrium size of freely - associating filament - bundles in solution .the free energy cost of filament distortion and inter - filament forces in cohesive bundles is described by where the forms of these functions are given be eqs .( [ emech ] ) and ( [ eelastic ] ) , respectively . defining the dimensionless measures of curvature and torsion variation , and grouping the -dependent terms into ( free energy per filament ) we note the state of packing ( as described by ) is determined by following three dimensionless parameters for cohesive bundles the first of these , , is a simply measure of central filament shape , while the second , , is simply an intrinsic property of the mechanics of the filaments .the last of these parameters , , is a dimensionless measure of bundle size that characterizes the relative costs inter- vs. intra - filament distortions .hence , this parameter plays a role similar to the fppl - von krmn number for problems involving confined elastic membranes . for small bundles , , the inter - filament elastic energies are large in comparison to the cost of filament shape change , while the inverse is true for large bundles , where .  to understand the size - dependence of packing we consider two cases for .first , we analyze the extreme easy twist " limit , where , and the minimization of with respect to is analytically tractable , resulting in as shown in fig .[ the_el ] ( a ) this behavior demonstrates that cohesive bundles adopt the extremal isometric or isomorphic packings only in the ( single filament ) and ( bulk assembly ) limits where the respective costs of inter- and intra - filament distortion dominate .  extending the analysis to the case of a finite twist elastic cost , , we consider the small limit of eq .( [ fcoh ] ) , finding optimal packing for narrow bundles , which indicates a much more rapid change in packing angle with size for small bundles , , due to finite torsional stiffness .the change in power - law dependence derives from the fact that in the small angle limit the , whereas .hence , the torque " acting on packing angle generated by bending vanishes in the isometric state ( ) , while the torque generated by torsional cost remains finite for finite .the singular , power - law evolution of at small bundle size is also strongly dependent on filament geometry .as filament geometry approaches shape , , the small- evolution from isometric to isometric bundles becomes infinitely sharp , indicative of the negligible disruption of inter - filament packing in this limit . in the opposite limit of large tilt angles cost of filament torsion in the isometric state becomes negligible , and the dependence of packing angle on bundle size ( for narrow bundles ) approaches the linear evolution of eq .( [ thecob0 ] ) . the numerical solution for shown in fig .[ the_el ] ( b ) for several filament shapes , ranging from the rapid jump between isometric to isomorphic bundles for nearly straight filaments ( ) to the more gradual evolution of packing in highly curved filaments ( ) .the fre- energy cost of distortion of filament shape and inter - filament forces in osmotically - condensed bundles is described by where the forms of these functions are given be eqs .( [ emech ] ) and ( [ fosm ] ) , respectively . following a similar analysis to the above for cohesive bundles we group the -dependent terms into  for pressure - induced bundles ,the packing is determined by following three dimensionless parameters the first of these two parameters , reflecting intrinsic filament properties , are appear identically to the analysis of cohesive bundles eq .( [ parcoh ] ) , where as the dimensionless measure of bundle size , , captures the relative cost of bundle volume change to filament distortion , including the dependence on osmotic pressure .again , we first analyze equilibrium packing in the limit .the minimization of is further simplified by noting which allows us to approximate . minimizing with respect to in this limit we find , which is shown in fig . [ the_osm ] ( a ) .thus , in comparison to cohesive bundles , we find a more abrupt change a packing with bundle size for osmotic bundles .notably we find a _second - order transition _ from the isometric state ( ) occurring at a critical size .near to this critical value , for , we find indicating a high sensitivity of packing near the critical size .as in the case of elastic restoring forces , we find that large bundles asymptotically approach the isomorphic packing .  following the analysis of finite torsional stiffness ( ) above for cohesive bundles , we analyze the small limit of eq .( [ fosm ] ) to determine the small diameter evolution of packing , hence , for finite torsional stiffness , bundles are distorted from the isometric packing for any non - zero bundle radius , deriving again from the non - zero torque exerted on the isometric bundle by torsional mechanics of filaments .[ the_osm ] ( b ) shows the evolution of packing angle with bundle radius for .notably , as we found for cohesive bundles , when the helical angle approaches , effects of torsional stiffness at small packing angles vanish , and we find that bundles remain isometric for a finite range of small bundles , exhibiting a second - order transition to at a critical size .   having analyzed the size - variation of optimal packing , we now consider the equilibrium size of bundles formed in concentrated filament solutions . in a previous study of helical filament bundles  , which assumedthe packing remains locked to the isometric limit , it was shown that the thermodynamically optimal size of bundles remains finite over a broad range of cohesive energy due to the prohibitive cost of filament shape change in large bundles .presently , we generalize this analysis by considering the relaxation of filament distortion through adjustment of bundle from the constant - spacing towards constant - shape packing .we consider solutions in the canonical ensemble , where the filament number and volume are constant .we further assume that all but a negligible number of filaments belong to bundles ( i.e. far above the saturation point ) and consider a size distribution that is sharply peaked around radius . in this case , an additional - free energy cost enters , associated with the surface energy , , and the exposed bundle surface area , where we assume such that the contribution from the bundle ends is negligible .  to analyze the evolution of bundle size distribution with , we assume and remains sufficiently small that the central filament locks into the preferred filament geometry , and .for the purposes of simplifying the analysis we focus on to the limit of easy twist where , though it is straightforward to extend the more general analysis of . the free energy per filament of bundles of size may be written as , where represents the bulk costs associated with packing distortion ( optimized over packing angle , ) .  in both models , the cost associated with the competition between inter- and intra - filament packing are monotonically increasing functions of ( see fig .[ fvsr ] ) . for small radius for cohesive bundles and for osmotic bundles bundles are packed isometrically and the mechanical cost associated with filament distortion grows as . in the opposite large bundle limit , bulk cost of packing in both bundles ( per filament ) saturates as the bundle approaches the isomorphic state , where and , favors small bundles , acting against lateral bundle growth .  as shown in fig .[ fvsr ] , below a critical value of surface energy , , the minimal free energy state is characterized by a finite bundle size .the critical surface energy derives from the solution to where and is finite - size of bundles at the point of equilibrium with the bulk assembly ( ) . for cohesive bundles this yields for osmotic bundles , it gives when the optimal bundle remains isometric at the critical surface energy , which holds only for . for larger helix angles ,the packing and size of osmotic bundles must be determined numerically .we may estimate the thermodynamic properties in this limit , however , by approximating equilibrium bundle size to be at the critical value , .since this overestimates the free energy of the finite - size bundle , this approximation gives a lower bound on the critical surface energy , .\label{sigc_os_large}\ ] ]  in figure [ phase ] we plot the phase diagrams for models of cohesion- and pressure - driven condensation of helical filaments , where a critical surface energy separates a phase of finite - size bundles from bulk assembly ( in the isomorphic packing ) . from eqs .( [ sigc_el ] ) and ( [ sigc_os ] ) it is easy to show that for small helix angles that critical surface energy increases as power law for both models : and .this gives the intuitive result that as the preferred filament approaches a straight geometry , thermodynamic costs associated with packing frustration must vanish .minimization of the surface energy ( for any value of ) for bundles of rod - like filaments results in aggregates of unlimited lateral size , the result of standard arguments for 2d aggregation . in the large helix angle limit ,the costs of inter - filament packing of the isomorphic , limit increase in both models .consequently , the critical surface energy needed to offset the bulk costs of isomorphic state also increases indicating a broader thermodynamic stability of finite - sized bundles .hence for cohesive bundles we find , and a maximal stability for bundles at , or the limit . we find a profoundly different result for the limit of osmotically - driven bundles , as diverges due to divergent cost of isomorphic packing of the ring - filament " limit .in this paper , we studied a continuum model for the formation of bundles of intrinsically - curved , helical filaments . a key aspect of this problem is a fundamental frustration between the curved shape of individual filaments and the tendency for even filament spacing , or maximal density , in the cross section . as implied by eq .( [ eq : kaptau ] ) , isometric packing requires filament distortions , whose costs increase rapidly with cross - section thickness . in comparison ,the thermodynamic costs of imperfect packing in both models is relatively insensitive to bundle radius . while the ( per filament ) cost of filament distortion in isometric bundles grows as , according to eqs .( [ eelastic ] ) and ( [ fosm ] ) the inter - filament packing cost in isomorphic ( per filament ) is independent of radius .this difference in size sensitivity underlies the generic evolution from isometric packings for small bundles ( ) , where mechanical distortion is negligible , to isomorphic packings for large bundles ( ) , where filament distortion becomes prohibitively expensive .  despite the common tendency towards constant shape packings in larger bungles, we find a markedly different sensitivity of inter - filament packing to lateral bundle radius for cohesive bundles and osmotically - condensed bundles , as evidenced by the -dependence of optimal packing angles shown in figs .[ the_el ] and [ the_osm ] . in cohesive bundles, packing evolves smoothly with for all helix angles and mechanical properties , exhibiting a non - isometric packing ( i.e. ) for any finite radius .in contrast , when the cost of filament torsion is negligible ( either or ) , in osmotically - condensed bundles , we find that bundles lock - in " to the isometric packing over finite range of small radii , , exhibiting a continuous , second - order transition to the non - isometric packing beyond this critical size .the difference in -dependence of the packing in the two models derive ultimately from a different stiffness " to interfilament distortion from the isomorphic state , . in the elastic modelthis leads directly to a packing angle dependence , , indicating a soft " -dependence of packing cost that is generically weaker than the -dependence of driving bundles toward the isomorphic state .this can be contrasted with the stiffer " -dependance of packing free energy in osmotic bundles , , a comparable dependence to the bending costs , which implies a finite size range over which the isometric packing is stable .the evolution from isometric to isomorphic packing has important consequences for the equilibrium size of self - assembled bundles of helical filaments . in ref . , it was shown that constraining bundles of helical filament to the isometric packing leads to self - limited lateral radii of equilibrium bundles .hence , this system belongs to an unusual class of associating molecular systems along with self - twisted bundles of chiral filaments  , where the equilibrium dimensions of the condensed state are finite , but mesoscopic much larger that filament diameter despite the finite range of the interactions .importantly , the present study demonstrates that even when inter - filament packing is free to relax from the isometric packing , the self - limiting behavior of helical filament bundles is retained over a broad range of cohesive conditions . from fig .[ phase ] , we note again a significant difference between the stable range of self - limited bundles in cohesive and osmotic bundles , characterized the large- limit of the , the surface energy where finite - size bundles are in equilibrium with bulk ( infinite- ) assembly . while finite - diameter cohesive bundles reach a maximal range of stability in ring - filament limit , , diverges in this limit , indicating that the stable range of finite bundles becomes infinite in the limit , where the costs of packing frustration are maximal .we conclude with a brief discussion of relevant physical parameters for a model system of helical filaments , the bacterial flagella . owing the precise control of helical geometry offered through the multiple polymorphic shapes of flagella  ,the condensed states of reconstituted flagella provide an ideal system for exploring the thermodynamics of curved - filament assembly .recent experiments have exploited the intrinsic shape of reconstituted flagella in colloidal suspensions as a means to study study the unusual liquid crystalline mesophases of spiral - shaped filaments  . here , we consider the properties of _ bundles _ of flagella that may be formed in highly - concentrated solutions based on our analysis . for simplicity , we focus on the case of bundles of flagella osmotically - condensed in the presence of a osmolytes sufficiently large to be excluded from the interstitial regions of the dense bundle core . of particular importance is the size scale , in eq .( [ parosm ] ) , which determines both the range over which bundles are isometrically packed ( for ) as well as size range of finite - size bundles in equilibrium according to eq .( [ sigc_os ] ) .we recast the definition of this length as , where constitutes a pressure scale determined by the deformation cost of filaments ( we have dropped numerical factors of order unity ) .force - extension measurements of flagella under flow suggest a bending modulus ,  . using the preferred curvature of the _ normal _ and_ coiled _ polymorphs , and , we find predictions for the characteristic osmotic pressures , and .compared to the typical scale of osmotic pressure of order  for few percent of 8000 mw peg the modest pressure scale set by flagellar bending cost suggests that the characteristic bundle size is _ mesoscopic _ , orders of magnitude larger filament diameter , . since for helical angles of these flagellar polymorphs  and our theory suggests that packings remain isomorphic for _ bundle radii up to the range of microns_.  turning now to the range of self - limited lateral assembly , we consider the surface energy of bundle under the influence of osmotic pressure , , where is the effective size of that depletion zone " ( roughly , the osmolyte radius ) that excludes osmolytes near the bundle surface  . from the critical surface energy of eq .( [ sigc_os ] ) , we expect finite diameter bundles _ above _ a critical pressure , , again suggesting the threshold osmotic pressures are nominal , in the range the pa .thus , we expect reasonable experimental conditions to be in the range of self - limited bundle sizes , for which the balance between surface energy and filament distortion in isometric bundles determines the equilibrium dependence of bundle size on osmotic pressure , which is of order 10 for .these estimates suggest a strong ( measurable ) pressure dependence self - limited diameter of flagellar bundles in the range of weak osmotic pressures .i would like to thank l. cajamarca for numerous discussions and a careful reading of the manuscript .i would also like to acknowledge p. ziherl and the hospitality of the jozef stefan institute at the university of ljubljana where some of this work was completed .i am grateful to z. dogic and s. yardimci for providing the florescence image of flagellar bundling .this work was supported by the nsf through grant cmmi 10 - 68852 .here , we derive the shape variation of filaments in the family of bundles described by eqs .( [ rv])-([omegas ] ) . the unit tangent, , of filaments at follows straightforwardly from eq .( [ rv ] ) where and ( { { { \bf n}}}\cdot { { \mathbf x } } )  .\label{dt}\ ] ] reflects mapping of arc - length elements at to the central filament arc - length .recall in this notation objects that appear as explicit functions of denote geometry quantities at position in the packing plane , and objects not written as functions of refer to geometric quantities related to the central filament at . note also that we are working in the limit of narrow bundles where it is sufficient to consider first - order in correction to the geometry ( relative to the central filament ) .the filament curvature derives from the magnitude of from which we derive where using the fact that is perpendicular to we find , ( { { { \bf n}}}\cdot { { \mathbf x } } ) , \ ] ] which when substituted into eq .( [ kappaa ] ) yields eq .( [ dkap ] ) .  -2 \big [ ( \omega+ \omega_0)\sin \theta - \cos \theta \omega_p\big ]  , \label{dkappaa}\end{gathered}\ ] ] where .the variation of torsion follows from where and are the first order variation to normal and bi - normal , respectively , relative to the central filament .the normal correction follows from , where we used in the second line .similarly , defining we have the correction to the bi - normal , the component of the derivative of eq .( [ db ] ) can be written as from eq .( [ dt ] ) we have ( { { { \bf n}}}\cdot { { \mathbf x}}),\ ] ] and differentiating eq .( [ dsdt ] ) we find  \\-\omega_p \omega ( \omega+\omega_0)\sin \theta  -\sin \theta  \omega_p\omega^2  -\omega_p^2\tau \big\ } ( { { { \bf n}}}\cdot { { \mathbf x } } ) .\end{gathered}\ ] ] substituting these into eq .( [ taueq ] ) we find   -\kappa^{-2}\big\ { - \tau \big [ ( \omega+ \omega_0)^2 + \omega_p^2 \big ] \\ + \big [ \kappa^2 + ( \omega+ \omega_0)^2 \big ] \big [ \cos \theta ( \omega+\omega_0)+\sin \theta \omega_p\big ]  \\ - \omega_p\big[\sin \theta ( \omega+\omega_0 ) \omega + \sin \theta \omega^2 + \omega_p \tau \big ] \big\ }  , \label{dtaua}\end{gathered}\ ] ] where .

Abstract: densely - packed bundles of biological filaments ( filamentous proteins ) are common and critical structural elements in range of biological materials . while most bundles form from intrinsically straight filaments , there are notable examples of protein filaments possessing a natural , or intrinsic , curvature , such as the helical bacterial flagellum . we study the non - linear interplay between thermodynamic preference for dense and regular inter - filament packing and the mechanical preference for uniform filament shape in bundles of helically - curved filaments . geometric constraints in bundles make perfect inter - filament ( constant spacing , or _ isometric _ ) packing incompatible with perfect intra - filament ( constant shape , or _ isomorphic _ ) packing . as a consequence , we predict that bundle packing exhibits a strong sensitivity to bundle size , evolving from the isometric packing at small radii to an isomorphic packing at large radii . the nature of the transition between these extremal states depends on thermodynamic costs of packing distortion , with packing in elastically - constrained bundles evolving smoothly with size , while packing in osmotically - compressed bundles may exhibit a singular transition from the isometric packing at a finite bundle radius . we consider the equilibrium assembly of bundles in a saturated solution of filaments and show that mechanical cost of isomorphic packing leads to self - limited equilibrium bundle diameters , whose size and range of thermodynamic stability depend both on condensation mechanism , as well as the helical geometry of filaments .


recent progress in the growth of chemical vapour deposition ( cvd ) diamonds is challenging readout electronics developers . because of its large bandgap of 5.5 ev and very low boron and nitrogen impurity concentrations - 5 ppb  cvd diamond has a negligible intrinsic noise at room temperature .in fact , the mean leakage current of a typical device is less than 1 pa . combining this with a very high carrier mobility of about /(v/ m ) , allowing for a complete charge collection in few ns ( 8 ns for 500 m thick device biased at 1 v/ m ) , results in an intrinsic noise charge collected within the duration of the signal of the order of .05 electrons . except for very slow , cryogenic , charge sensitive amplifiers , able to reduce the rms noise to a few electrons ,such precision is out of reach of modern fast amplifiers .indeed , the best broadband amplifiers have input referred noise rms of electrons corresponding to the input referred noise of 10 ( 50 input impedance ) in 8 ns signal . even conventional charge amplifiers with 50 - 300 decay time have more than 200 electrons rms noise . therefore , for diamond detector applicationsthe dominant ( by three - five orders of magnitude ) source of noise is the readout electronics .  in the present note we present a series of tests on various fast amplifiers aimed to identify the best solution for nuclear and particle physics applications .these applications demand a measurement of both the energy released in the bulk of the diamond by ionizing particles and the time .these tests were performed using various radioactive sources .we selected a 4.7 x 4.7mm2 , 500 um thick single crystal cvd diamond detector capacitance can be estimated according to the plain  we selected 4.7.7 mm , 500 m thick single crystal cvd diamond detector produced by diamond detectors ltd  .the crystal has two electrodes deposited on its upper and lower major surfaces , made of few nm of dlc followed by 100 nm of gold .each contact is bonded via a gold microwire onto pcb and then to its sma connector in such a way that both contacts can be read out independently .detector capacitance can be estimated according to the plane capacitor equation ( neglecting eventual fringe field contribution ) : this result is in good agreement with the value measured by hp 4280a capacimeter .the diamond detector was polarized at 1 v/ m by means of ortec 710 bias supply .this value remains below detector breakdown voltage ( at 1.4 v/ m discharges , likely to be attributed to crystal defects or contact disuniformity , occur ) allowing for complete charge collection  .  in the present work , we used two types of readout : one - side readout , shown in fig .[ fig : setup_single ] and two - side readout , shown in fig .[ fig : setup_double ] .the one - side readout was used to characterize energy resolution of of various amplifiers . in this casethe diamond was polarized by applying 1 v/ m bias voltage from the readout side and the other contact was connected to ground .schematic drawing of the experimental setup in case of the one - side readout . ]schematic drawing of the experimental setup in case of the two - side readout . ]the two - side readout was used to study ideal timing resolution of the cvd diamond detector .indeed , the major contribution to both energy and timing resolutions come from the readout electronics and in particular from the first amplification stage . using the same diamond detector signalread out independently from opposite sides by two separate amplifiers allows to study the electronic contribution to the timing resolution using realistic signals .  for single amplifier characterization measurementsthe diamond detector was connected to the amplifier input through a 2.4 cm huber & suhner sma `` i '' connector , while the second sma connector of the detector was shorted to ground by sma tap .the overall capacitance of the system in front of the amplifier ( `` i '' connector , two sma connectors of detector case and sma tap ) was found to be 6.6 pf .the amplifier output was connected to the digitizer via 1.5 m huber & suhner sucoflex 104 cable .the amplified signal was read out by sis3305 digitizer with analog bandwidth of 2.2 ghz .the digitizer was operated in four - channel interleaved 5 gs / s mode .for timing response measurements with small signals an additional amplifier philips scientific 744 with bandwidth of 1.8 ghz and gain was inserted between primary amplifier output and digitizer input .the data measured by the digitizer was acquired through the vme bus by concurrent technologies vx 813/091 single board computer ( sbc ) which incorporates a tsi148 vme controller .a simple daq program was developed using dma transfer of digitizer memory to the sbc memory .the data were saved on the local compact flash disk and transferred to workstation for the off - line analysis .in order to measure coincidence events and to reduce accessible energy threshold the standard sis3305 digitizer firmware rev.1c.0b had to be modified .sis3305 digitizer features three xilinx fpgas on - board , two virtex 5 fpgas serve as interfaces for the two 2.5 gs / s adc cores , and one virtex 4 fpga manages the board vme interface .partially precompiled firmware of adc fpgas was acquired from the manufacturer for our purposes .this firmware version 1c.0b included only simple internal triggers : data acquisition is started when one adc sample goes above or below imposed threshold in one of digitizer channels .it also includes a schmidt trigger , which fires when one adc sample goes above ( below ) a first threshold and turns off when after another single adc sample is below ( above ) a second threshold .the trigger was modified using the xilinx ise design suite version 13.2 .first of all a trigger that fires when a configurable number of samples is above ( below ) a given threshold was implemented . for the initialization of this parameter for each channel we used bits 30 - 26 of the corresponding sis3305_trigger_gate_gt_thresholds_adc1 - 8 and sis3305_trigger_gate_lt_thresholds_adc1 - 8 ( 0x2020 - 0x203c and 0x3020 - 0x303c ) registers  .this allows to select the number of consecutive samples above / below threshold in selected channel in the range from 1 to 32 ( 5-bit word ) corresponding to time interval from 0.8 ns to 25.6 ns .the 4.8 ns long channel s trigger - valid signal goes on when the number of consecutive samples above / below threshold in given channel reaches the configured value . in case of the 2 or 4 channel interleaved modes ( 2.5 gs / s and 5 gs / s ,respectively ) an asynchronous and of the corresponding channel trigger - valid signals is taken and then synchronized with 250 mhz system clock to give the final adc core internal trigger .the acquisition mode is determined by the first three bits of sis3305_event_config_adc1_4 ( 0x2000 ) or sis3305_event_config_adc5_8 ( 0x3000 ) register . in this waythe real number of samples above / below threshold in 2 or 4 channel interleaved modes is equal to the configured value multiplied by 2 or 4 , respectively .the adc core internal trigger signal can be extended from its natural length of 4 ns up to 128 ns by configuring bits 14 - 10 of sis3305_trigger_gate_gt_thresholds_adc1 ( 0x2020 ) register .this 5-bit word allows to modify the coincidence time window with precision of 4 ns .it has to be mentioned that since many firmware blocks were provided in a precompiled form we had to follow its existing design .in particular , the data from each adc input channel acquired at 1.25 gs / s rate are split in adc - fpga in six parallel 208.3 mhz flows .this architecture limits the trigger selectivity in the 2 or 4 channel interleaved modes .indeed , the length of trigger - valid signal from each adc channel is 4.8 ns , therefore imposing only one sample above / below threshold in one of interleaved modes ( 2 or 4 channels ) there is a probability to trigger on accidental coincidence of background hits ( whose rate ) in selected channels within ns time window of ) . in order to prevent this possible background the continuity of trigger condition among interleaved channelscan be activated by an additional configuration flag .setting bit-15 of sis3305_trigger_gate_gt_thresholds_adc1 ( 0x2020 ) register the difference between indexes of trigger samples within 6-sample data blocks from interleaved adc channels is checked to be not larger than unity .the coincidence between two adc cores ( channels 1 - 4 and 5 - 8 ) is implemented in the vme - fpga by an asynchronous and between two adc core internal trigger signals , then synchronized with 125 mhz system clock and extended to 32 ns length . setting bit-20 of sis3305_trigger_out_select_reg ( 0x40 ) register allows to route the coincidence signal onto a lemo trigger out connector . thus connecting the latter to the lemo trigger in connector and enabling the external lemo trigger in bit in sis3305_control_status ( 0x0 )register and external trigger bit in sis3305_event_config_adc1_4/5_8 ( 0x2000 and 0x3000 ) register allows to measure coincidence events only .the coincidence interval can be configured with 4 ns steps up to 128 ns as explained above .similarly , bit-21 of sis3305_trigger_out_select_reg ( 0x40 ) register allows to trigger on or of two adc core internal triggers .this feature permits to acquire both adcs when one of two adc cores had an internal trigger .it has to be noticed that in this modified firmware the schmidt trigger has been removed and therefore trigger - off setting in bits 25 - 16 of sis3305_trigger_gate_gt_thresholds_adc1 - 8 and sis3305_trigger_gate_lt_thresholds_adc1 - 8 registers ( 0x2020 - 0x203c and 0x3020 - 0x303c ) are not more significant .moreover , the internal trigger logic in the 2 channel interleaved mode ( 2.5 gs / s ) assumes that two pairs of channels are : 1 and 2 , 3 and 4 .this modified firmware is available at  .the noise level referred to the input was measured simultaneously with signal acquisition by comparing the voltages in the samples before the signal peak to the baseline voltage .this was done for the full digitizer sampling rate of 5 gs / s and summing four interleaved channel amplitudes to restrict the sampling rate to 1.25 gs / s ( sis3305 digitizer analog bandwidth is 2.2 ghz ) .the resulted distributions are shown in fig .[ fig : noise_distr ] and their rms values are indicated in fig .[ fig : noise_rms ] .noise distributions as referred to input for different amplifiers measured by sis3305 2.2 ghz bandwidth digitizer in 5 gs / s mode ( left ) and 1.25 gs / s mode ( right).,title="fig : " ]  noise distributions as referred to input for different amplifiers measured by sis3305 2.2 ghz bandwidth digitizer in 5 gs / s mode ( left ) and 1.25 gs / s mode ( right).,title="fig : " ]   rms of noise as referred to input for different amplifiers .upper points are for 5 gs / s mode , while lower points are for 1.25 gs / s mode . ].[table : noise_rms]noise rms values as referred to input for different amplifiers measured with 2.2 ghz bandwidth digitizer in 5 gs / s mode compared to values declared by the manufacturer . [ cols="^,^,^ " , ]in many experiments the amplifier can not be installed close to the detector .insertion of a cable between detector and the amplifier introduces noise and signal distortions , in particular when the impedance of the cable and amplifier is not well matched .this is particularly important for charge and transimpedance amplifiers .standard broadband amplifiers featuring 50 input impedance are almost insensitive to the insertion of the input cable . in our applicationthe presence of a .5 m long cable is mandatory .hence we tested transimpedance and charge amplifiers cividec c6 and cividec cx with four different cables : 150 cm of 50 /87 pf / m sf105 ( indicated as rg58 ) , 150 cm of 75 /67 pf / m rg59 , 166 cm of 93 /44 pf / m rg62 , 616 cm of 185 /22.3 pf / m rg114 .the comparison of signals produced by cividec c6 and cividec cx amplifiers connected to the detector via different cables is shown in fig .[ fig : cable_distortion ] .distortion of the signal from diamond detector introduced by insertion of various cables between detector and cividec c6 transimpedance amplifier ( left ) and cividec cx charge amplifier ( right).,title="fig : " ]  distortion of the signal from diamond detector introduced by insertion of various cables between detector and cividec c6 transimpedance amplifier ( left ) and cividec cx charge amplifier ( right).,title="fig : " ]  using the cividec c6 amplifier , the insertion of a cable leads to strong signal reflections , reducing for higher impedance cables and almost absent for 185 rg114 cable . from thiswe deduce that the amplifier input impedance is about 200 .using cividec cx amplifier cable insertion leads to signal suppression proportional to the overall capacitance of the cable .indeed , the rg62 cable with 44 pf / m capacitance shows smallest suppression , while rg114 with 22.3 pf / m capacitance exhibits larger suppression because its 4 times larger length . this can be better quantified by measuring peak amplitude of the output signals as shown in fig .[ fig : cable_reduction ] .reduction of the output signal amplitude introduced by the insertion of various cables between the detector and cividec c6 transimpedance amplifier ( left ) and cividec cx charge amplifier ( right).,title="fig : " ]  reduction of the output signal amplitude introduced by the insertion of various cables between the detector and cividec c6 transimpedance amplifier ( left ) and cividec cx charge amplifier ( right).,title="fig : " ]  the energy resolution variations due to insertion of different cables are different for two amplifiers as shown in fig .[ fig : cable_resolution ] .cividec c6 amplifier energy resolution changes by about 10% with rg114 and rg62 cables , while no effect is observed for other cables . for cividec cx amplifier insertion of any cable leads to 30 - 40% resolution loss .deposited energy resolution variations introduced by insertion of various cables between detector and cividec c6 transimpedance amplifier ( left ) and cividec cx charge amplifier ( right).,title="fig : " ]  deposited energy resolution variations introduced by insertion of various cables between detector and cividec c6 transimpedance amplifier ( left ) and cividec cx charge amplifier ( right).,title="fig : " ]a number of modern commercial amplifiers for diamond detectors were characterized for typical nuclear and particle physics applications . these include four broadband amplifiers one transimpedance amplifier and one charge amplifier .we compared their energy and timing resolutions in order to select the best candidate for measuring 200 kev deposited energy signals . among these amplifiersonly cividec c6 and cividec cx amplifiers are able to reach energy thresholds as low as 100 kev. however , the latter one is charge amplifier featuring very long signals , not suitable for for timing application . the obtained timing resolution at 200 kev of the only amplifier which met our requirements , the cividec c6 , was found to be 1.2 ns ( resolution of two cividec c6 amplifiers in coincidence ) , which was factor of 6 larger than our target resolution of 200 ps .this resolution could be further reduced by a factor by differential readout of the detector , but would remain factor of 4 above the requirement .element six ltd , http://www.e6cvd.com/cvd struck innovative systeme , http://www.struck.de/sis3305.html h. kagan and w. trischuk , cvd diamond for electronic devices and sensors , wiley , p.207 ( 2009 ) .diamond detector ltd , http://www.diamonddetectors.com http://www.ge.infn.it/~osipenko/sis3305/sis3305_rev2c.zip p. moritz , e. berdermann , k. blasche , h. stelzer and f. zeytouni , diamond detectors for beam diagnostics in heavy ion accelerators , frascati , 153 - 155 ( 1997 ) .caen s.p.a ., http://www.caentechnologies.com cividec instrumentation gmbh , http://www.cividec.at wisnam s.r.l . , http://www.wisnam.com

Abstract: the development of chemical vapour deposition ( cvd ) diamond detectors requests for novel signal amplifiers , capable to match the superb signal - to - noise ratio and timing response of these detectors . existing amplifiers are still far away from this goal and are the dominant contributors to the overall system noise and the main source of degradation of the energy and timing resolution . we tested a number of commercial amplifiers designed for diamond detector readout to identify the best solution for a particular application . this application required a deposited energy threshold below 100 kev and timing resolution of the order of 200 ps at 200 kev . none of tested amplifiers satisfies these requirements . the best solution to such application found to be the cividec c6 amplifier , which allows 100 kev minimal threshold , but its coincidence timing resolution at 200 kev is as large as 1.2 ns .


the next generation of large - scale radio interferometers , such as askap  , mwa  , paper  , and ska  , promise incredible scientific reward but also incredible data analysis challenges .the tremendous volume of data , high dynamic range , wide bandwidth , large amounts of radio interference , and significant foregrounds present serious instrumentation and analysis difficulties  .image deconvolution the process of removing the effects of signal mode coupling due to the primary beam , noise mode coupling due to sampling , incomplete fourier mode sampling , and noise is an important first step in scientific analysis from these instruments  .the ideal algorithm for performing image deconvolution suitable for these upcoming surveys would be : ( 1 ) robust to changes in the input signal and instrument configuration , ( 2 ) parametrized by as few tunable inputs as possible , ( 3 ) as automated as possible , ( 4 ) driven by the _ data _ rather than user - selected guesses as to the source signal , ( 5 ) as informative as possible given the difficulty of processing the data even a single time , ( 6 ) scalable to the extreme sizes of future data sets , ( 7 ) as fast as possible , and ( 8) able to easily incorporate modeling of foregrounds or other systematics .unfortunately , the most commonly used algorithm for interferometric image reconstruction , clean  , satisfies few if any of these criteria .originally developed to remove point source foregrounds from a smooth background source image , clean works by iteratively removing the effects of high - intensity point sources in the fourier - transformed -plane .while the original algorithm excels in a limited number of cases , it is not appropriate for general reconstruction problems .modern implementations of clean that appear in standard radio astronomy packages ( e.g. , casa  ) incorporate several advanced features such as simultaneous deconvolution at multiple scales  and time - varying analysis  .  however , even these more advanced versions of clean require significant fine - tuning and supervision during the deconvolution process .users must select thresholds for deciding which pixels contain `` point sources '' and the amount by which to remove the point source effects in -space .the optimal choices for these thresholds are not known in advance for a given observation .users may also select specific regions of the image to apply more or fewer cleaning iterations .clean has no final end state : users must decide when an image has been deconvolved `` enough '' without introducing unwanted artefacts .finally , clean gives no information on the uncertainties in the reconstruction .realizing the shortcomings of traditional and more sophisticated clean - based algorithms , methods based on a regularized likelihood , such as the maximum entropy method ( mem ;  ) generated significant interest .mem produces an image estimate which minimizes the difference between the estimate and the data given the level of noise and some chosen metric .while mem requires less fine - tuning and supervision than clean , the user must still choose the metric ( usually expressed as an entropy functional ) , and the optimal choice of the metric is not known in advance  .the final reconstructed image is thus only optimal in regards to that metric . similarly to clean, mem gives no uncertainty information and in general tends to underestimate the source image intensity  .  as  and  pointed out, optimal reconstruction techniques must be spatially adaptive and operate at multiple scales simultaneously .many authors have proposed relatively new alternative methods based on compressed sensing techniques , bayesian processes  , and separation of smooth and point - like components  .  in this work ,we present a general purpose , bayesian reconstruction algorithm to infer the source image from realistic interferometric radio data .bayesian analysis starts with building a generative probabilistic forward model of the data .given this model choice , the posterior probability density function ( the _ posterior distribution _ ) quantifies what is known about the source image once the data have been obtained  .the purpose of this paper is to demonstrate that a surprisingly simple choice of model for the source image , that of an _ isotropic gaussian process _ , performs very well in realistically simulated examples for a test suite of widely diverse images .the key to practical bayesian image analysis is to be able to navigate efficiently through the very high dimensional parameter space , since every pixel value is an independent parameter .for example , in this paper we will explore posterior distributions in -dimensional parameter spaces .gibbs sampling is a powerful technique that has been used successfully for sky maps with more than pixels in the context of cosmic microwave background signal reconstruction and power spectrum estimation  .conceptually , gibbs sampling iterates between samples of the signal and its power spectrum in a way that respects the joint posterior distribution of signal and power spectrum given the image .this separation allows for significant speedups compared to grid based evaluations of the posterior : the algorithm scales as , where is the total number of pixels , in the ideal pregridded flat - sky limit discussed here .our assumptions for our mock observations allow us to pregrid the intensities before the iterative solution step . more general curved - sky analysis would necessarily be more expensive , either with gridding - regridding steps during the analysis in an aw - projection method or with the spherical harmonic transform operation in a full skythis is similar to the computational scalability of traditional clean .however , clean typically completes in order steps , whereas our method usually requires iterations . as we will see below, the primary computational cost in each iteration comes from solving a matrix - vector equation .fortunately , this is a common problem in computational science and there are many fast , scalable solutions available  .while the gibbs sampling framework itself is independent of prior ( see , for example ,  for an implementation based on fluxon models ) we choose an isotropic gaussian process prior .gaussian processes are surprisingly flexible in describing a variety of images  . because gibbs sampling can be understood as a non - linear generalization of the least - squares optimal signal reconstruction provided by the wiener filter ( see section [ sec : method ] ) without requiring a choice of the signal covariance _ a priori _ ,successive samples are always constrained by the data in regions of high signal - to - noise . in regions of low signal - to - noise , the gaussian process is the least informative completion to a full probabilistic model . in this regimethe method still maps out the signal likelihood taking into account all modeled signals and imperfections in the data at the two - point correlation level .in addition to the speed gains mentioned above , the method of gibbs sampling offers several advantages over traditional deconvolution techniques : ( 1 ) it explores the full posterior shape , giving _statistical information on the resulting image ; ( 2 ) it _ automatically _ takes into account signal mode coupling from the primary beam ; ( 3 ) the sampled representation produced by the method makes marginalization _ trivial _ ; ( 4 ) it provides optimal reconstruction ( i.e. , wiener filtering ) _ without _ assuming a signal covariance ; ( 5 ) since we have already specified the signal prior , the method has _ no _ tunable parameters and operates completely unsupervised ; and ( 6 ) it offers a _ sparse _ reconstruction technique with a full bayesian motivation .section [ sec : sims ] below presents our mock observations and interferometer setup used to assess our new method .next , in section [ sec : method ] we outline the method of gibbs sampling as applied to radio interferometers . in section [ sec : comparison ]we compare our method to a proxy for traditional clean in terms of reconstructed image fidelity , residuals , and statistics such as rms error and residual signal - to - noise levels .finally , we conclude in section [ sec : conclusion ] with a discussion of planned extensions and the potential role of this method in future observations .we model the visibility data obtained from an interferometric observation as where is a vector containing a discretization of the input sky , is a primary beam pattern , is a fourier transform operator that converts from pixel space to the -plane , is an interferometer pattern in the -plane , and is a gaussian realization of the noise .we discretize the signal , data , and noise with elements .we generate all input visibilities within a  degree square patch discretized to pixels per side , similar to the approach of  and  .while this size of patch violates the strict flat - sky approximation , it allows us to explore the validity of our technique at higher resolutions and probe the range of scales accessible to realistic interferometers . because it is easier in interferometry to perform calculations in visibility space than in image space , we do not establish an observation wavelength for these mock observations , since our results and conclusions are independent of wavelength .we model the primary beam pattern as a gaussian with peak value of unity and standard deviation deg . with these parametersthe primary beam decreases to a value of halfway to the edge of the box .this allows us to include all fourier modes up to the nyquist frequency in our analysis and ensures that the periodic boundary conditions inherent in the fourier transform do not cause unwanted edge - effects .we prevent the primary beam from reaching values below .this value reflects a balance between faithfully representing the suppression of signals far from the point center ( so that fourier transforms have correct periodic boundaries ) and the need to preserve numerical stability in the conjugate - gradient algorithm utilized in the method below .we assemble the interferometer array in a simple way by randomly placing 12 antennas and selecting all baseline pairs within the -plane .we then allow the assembly to rotate uniformly for 6 hours while observing the same sky patch at the north celestial pole .this choice of antenna arrangement and integration time roughly corresponds to existing instruments , such as an extended configuration of alma  or vla  , although our fiducial setup uses fewer elements to highlight the performance of our method in less - than - optimal observing regimes .the interferometer pattern is discretized to the same resolution as our input images ( pixels on a side ; described below ) .we construct the interferometer pattern by placing a value of one wherever a baseline length intersects a pixel during its rotation and zeros elsewhere .we show the resulting -plane coverage in figure [ fig : pattern ] .this configuration covers roughly of the -plane , although the coverage varies significantly for each -bin , where is the radius of a given annulus in the -plane .some bins , especially at very low and very high , have zero coverage due to the lack of baselines at that distance .however , even if these bins had adequate coverage , we expect statistics here to be relatively poor due to the reduced number of modes in these regions .most bins have at least coverage and several bins have complete coverage .we determine the noise per pixel by summing the integration time spent in that pixel by all baselines .we do not adopt a noise model for a particular instrument ; rather , we set the noise variance to be where is the observation time in pixel .we then set an overall signal - to - noise ratio of by multiplying all noise variances by a constant value to maintain .this provides a scaling of the noise that would normally be caused by instrument effects such as the effective area of the apertures and the system temperature in a realistic observation .we use a gaussian realization of this noise variance to generate in equation ( [ eq : data ] ) .we then multiply this noise by the interferometer pattern to maintain consistency with the signal .figure [ fig : noiseanddata ] shows a particular noise realization for one of our test images and the resulting data .radio interferometers are used to study a wide variety of interesting astrophysical and cosmological phenomena , such as supernova remnants  , molecular gas clouds  , cluster radio halos  , magnetic fields in dwarf galaxies  , the interstellar medium  , the galactic center  , and the cosmic microwave background  . to assess the ability of our method to cope with this variety of targets we select eight input images drawn from the casa  user guide , which we present in figure [ fig : inputs ] .the test images represent a diverse variety of realistic and a few unrealistic imaging scenarios , such as a protoplanetary disk , a face - on spiral galaxy , a cluster , an agn jet and lobe , and the face of einstein .these images were provided in a mix of resolutions and dynamic ranges . to simplify our analysis ( but without loss of generality ) , we remapped all images to a uniform grid pixels on a side and renormalized all intensities to a peak of unity .some test images had artificial artefacts built - in , and the remapping procedure introduced some additional glitches in the image .we left these artefacts intact to test the ability of our isotropic gaussian process prior to recover highly anisotropic portions of the data .previous works have extensively discussed gibbs samples ( e.g. , * ? ? ?* ; * ? ? ?* ) , so we only briefly introduce the relevant equations as applied to interferometric observations here .we begin with some initial guess of the angular power spectrum and progressively iterate samples from the conditional distributions where is the least squares estimate of the signal given the data ( i.e. , , where represents a full signal - to - data operator ) .the samples converge to samples from the joint distribution after a sufficient number of iterations .given an angular power spectrum sample , we generate a new signal sample by drawing from a multivariate gaussian with mean and variance . here and are the signal and noise covariance , respectively .we do this by solving the set of equations where we define the matrix operator as in the above equations is the primary beam transpose and is the inverse fourier transform .the first term in the right - hand side of the above equation provides the solution for the wiener - filtered map , while the second and third terms of eq .( [ eq : sky ] ) provide random fluctuations with the required variance .the vectors and are of length with elements drawn from a standard normal distribution . as an illustration ,figure [ fig : iteration ] shows an example wiener - filtered map ( i.e. , from just solving the first term in the right - hand side ) and full signal sample at a single iteration for the _ einstein _ test case .we see that the fluctuation terms in equation ( [ eq : sky ] ) fill in regions weakly constrained by the data with a guess that mimics the known portions of the signal given the level of noise .the signal covariance matrix is diagonal in the -plane for isotropic signals ( which we assume as part of our gaussian process prior ) , so , where , with being the radial distance in the -plane .here and throughout we assume the flat - sky approximation that makes this identity valid . by construction, gibbs sampling explores the exact posterior and therefore treats the couplings introduced by partial sky coverage optimally  .the algorithm `` knows '' about the couplings since they are contained in the data model ( eq . [ eq : data ] ) which underlies the analysis .however , to a very good approximationthe noise covariance matrix is diagonal , and thus we will assume this for simplicity .we assign to the matrix entries equal to , where is the noise variance for the pixel in the -plane .the construction provides a pseudo - inverse of , so that any locations in the -plane with no antenna coverage do not yield infinities when taking the inverse .we solve numerically the above matrix - vector equation using a preconditioned conjugate - gradient scheme  .the preconditioner approximates the diagonal components of and is where is the fourier transform of the primary beam pattern .we implemented the code to solve the above equations with the open - source petsc library  and the mpi - parallelized version of fftw  .given the latest signal sample , , we generate a new angular power spectrum sample from eq .( [ eq : iterations ] ) by computing the variance in annuli of constant on the fourier - transformed signal .we then use this variance to draw from the probability density , which follows an inverse gamma distribution , by creating a vector of length ( assuming a jeffreys ignorance prior ) and unit gaussian random elements . here, is the number of pixels in the bin .the next power spectrum sample is then simply   in the above , we assume to be constant across the width of each annulus in the -plane .the width of each annulus can be set as desired . for the test cases which we work with in this paper we chose the width to be , where is the longest baseline of the interferometer .this is four times the -space resolution .this choice limits correlations between angular power spectrum bins which develop as a consequence of partial sky coverage .all -bins have uniform width except for the first , which we restrict to cover only the central zone where we enforce , since our analysis can not constrain the dc mode .we wish to capture as much power spectrum information as possible , so we correspondingly widen the width of the second bin to close this gap .  to determine convergence so that our iterative samples from the conditional distributions ( eq . [ eq : iterations ] ) are indeed samples from the joint distribution , we employ multiple chains with different random number seeds .our convergence criterion is the gelman - rubin ( g - r ) statistic , which compares the variance among chains to the variance within each chain .the g - r statistic asymptotes to unity , so convergence is said to be achieved when this statistic is below a given tolerance level for each -bin  . for our test images we stopped iterating when the g - r statistic reached less than , which took from 500 to 1500 steps , depending on the image .  after convergence ,we take the mean of the signal samples as our reconstructed image and simply calculate the variance for each pixel to assess the uncertainty in the reconstruction .figure [ fig : variance ] shows the variance for the _ einstein _ test image .we naturally see insignificant variance in the center of the primary beam where the data support is strongest , with steadily increasing variance away from the pointing center .we may use the variance to build a mask of the final reconstructed image in which we only accept pixels below a given variance threshold . while this certainly is nt necessary for the method to work , it provides an easy way to combine the posterior mean and variance information in a single plot .first , we compute the asymptotic variance , or the mean of the variance along the outer edge of image plane .we then reject any pixel whose variance is greater than _ and _ whose signal - to - noise ratio .this last criterion prevents us from masking low - intensity , but still well - constrained , pixels .the values chosen to create the mask are arbitrary , and we choose them solely on aesthetic merit .we repeat : this mask is not necessary to perform the deconvolution and produce a reconstructed signal .we only use it to neatly incorporate the measured uncertainty information in the plotted images .as we see in figure [ fig : variance ] , the acceptance mask generally follows the primary beam pattern .the detailed shape near the mask edge is influenced by the relative signal - to - noise levels and highlights the non - obvious nature of the regions of reliable reconstruction .we also notice an asymmetry in the acceptance mask : this is in response to the asymmetric nature of the fiducial interferometer setup ( figure [ fig : pattern ] ) and provides strong evidence that the symmetric nature of our gaussian process prior does not greatly influence our inference .we compare images recovered by gibbs sampling , as described in the previous section , to those recovered by a proxy for the point source - based clean algorithm .clean is implemented in various radio interferometric imaging packages ( such as casa ) .however , it is not straightforward to use these packages to reconstruct images from simulated visibilities already defined on gridded coordinates ; these packages are instead tailored to analyse observations made by real interferometric telescopes , with data in a specific format .consequently , we compare to reconstructions made with a proxy for the point source - based clean algorithm . while there are more sophisticated implementations of clean that would undoubtedly perform better with our test images ,this gives us a simple standard of comparison allowing us to demonstrate the viability of our method .we will include further comparisons in future work .it was shown by that reconstruction with the dirac basis ( i.e. pixel basis ) results in very similar reconstruction quality to clean ( see * ? ? ?* figure 1 ) .this is to be expected since it is known already that clean is closely related to reconstruction with the dirac basis .we thus take a similar approach to that taken by recent studies ( mcewen & wiaux 2011 ; carrillo et al . 2012 ; wolz et al .2013 ; carrillo et al . 2013 ) and use reconstruction with a dirac basis as a proxy for the clean algorithm . the reconstructed image that serves as a clean proxyis therefore given by the solution of the optimisation problem : where denotes the norm and is related to a residual noise level estimator ( see , e.g. , * ? ? ?* ; * ? ? ?we solve this problem using the sparse optimisation ( sopt ) package  using the douglas - rachford splitting algorithm .note that the sopt package is a versatile code , capable of solving much more sophisticated optimisation problems than the straight - forward minimisation performed here ( for examples of more extensive use see carrillo et al .2012 , 2013b ; wolz et al .2013 ) .figures [ fig : gallery1 ] and [ fig : gallery2 ] are galleries of the input images ( without noise ) , our posterior mean reconstruction using gibbs sampling , and a standard reconstruction using our implementation of clean .for all these images we have zoomed in to the inner 10 degrees where the primary beam selects the most prominent signal .we have applied our variance - based acceptance mask for our gibbs sampling reconstructions .we see that gibbs sampling is able to recover the complete range of source images to very high fidelity .we even faithfully recover glitches in the input image , such as the asymmetric protoplanetary disk in the _ swolf _ image and discontinuous lobe structure in the _3c288 _ image .this emphasizes the power of the wiener filter : in regions of strong data support , our samples are driven to the data regardless of choice of prior .we also recover low - intensity portions of the image both in the center of the image and towards the edge of the primary beam , such as the regions between the spiral arms in the _m51 ha _ image . however , there are other low - intensity regions , such as the lower portion of the mustache in the _ einstein _ image , where the signal is too low to distinguish it from the noise , and our mask rejects those pixels .a comparison to the clean images especially highlights the abilities of our gibbs sampling reconstruction technique . in all images ,we recover a broader range of fluxes over a wider extent than clean ( as in , for example , the _ cluster _ reconstruction ) . also , gibbs sampling is able to recover portions of the image further into the edges of the primary beam , as can be easily seen in the _ 3c288 _ and _ hco+4 - 360 _ images .finally , in images with glitches , such as _ swolf _ , clean tends to exaggerate the asymmetries .figure [ fig : residuals ] shows the residuals ( i.e. , difference maps ) between the mean posterior signal generated by gibbs sampling and the input test image . in all casesthe variance - based acceptance mask has been applied .we see that the maximum residual occurs in the center of the _ einstein _ test image , where the gibbs method slightly underestimates the source intensity .this is probably due to the method attempting to average the very high - contrast features adjacent to the center .the _ einstein _image turns out to be the most difficult : the residuals for all other images are typically an order of magnitude smaller . while the residuals are somewhat correlated with the distributions of the source image , like the _ einstein _ test image it appears that our method performs poorest in regions of high intensity contrast . however , these differences are very mild : typically on the order of 1% of the input source intensity .we may further quantify the differences between the input signals and their reconstructions with clean and our gibbs sampling method by binning the intensities , as we do in figure [ fig : histograms ] .bounding the histograms of the posterior mean are 2 error bars measured from the variance in the generated samples .this is another example of the kind of information unavailable in traditional reconstruction techniques . for almost all test images and intensity bins , our posterior meanreconstruction is within two standard deviations of the input signal .this is unsurprising : our bayesian method automatically discovers the local variance because that variance is directly related to the relative level of signal and noise in a given bin . in a few bins , such as the high - intensity bin of the _ einstein _ test image , our method tends to underestimate the true intensity .this is due to the high pixel - by - pixel contrast in the central image region discussed above .while the clean image intensities largely fall within the uncertainty ranges of the gibbs reconstruction , there is a systematic steepening of the distributions : clean tends to have too many low - intensity pixels and correspondingly too many high - intensity pixels .this validates the discussion above which noted that clean does not fully reproduce the observed range of input fluxes .finally , we may further simplify the comparison by reducing our measurement error to a single scalar .two error metrics are commonly used : the root - mean - square ( rms ) of the residual map , and the signal - to - noise ratio ( snr ) .this last quantity itself has several variations ; we take that of  : where is the standard deviation of image x , with denoting the original image and denoting the reconstructed image . for both measureswe first apply our variance - based acceptance mask before calculating the error metrics .figure [ fig : error ] shows the rms and snr for each of our test images for both gibbs sampling and traditional clean .as expected from our residuals , gibbs sampling performed most poorly in terms of rms with the _ einstein _ test image due to its highly complex structure . with the exception of that image and _ g41.1 - 0.3.b _ , all rms errors are below .the snr for all reconstructions with gibbs sampling fall between 10 and 25 . in both rms and snr measuresgibbs sampling outperforms clean for all test images .the largest differences occur in images with high degrees of symmetry ( e.g. , _m51 ha _ , _ swolf _ ) since in these cases our isotropic gaussian process prior performs best in marginal signal - to - noise portions of the image .we have presented an innovative method for deconvolving radio interferometric images using the bayesian method of gibbs sampling .our method naturally offers several advantages over traditional deconvolution approaches .it fully accounts for signal and noise mode coupling and incomplete -plane coverage in an automatic and well - motivated fashion , requires no fine - tuning or supervision as the method progresses , and provides an informative description of the uncertainty information in the signal reconstruction .we choose an isotropic gaussian process image prior , though we do nt specify the signal covariance in advance .we have tested our method with a realistic interferometric observing scenario of a wide variety of source images .these images represent typical targets , including protoplanetary disks and active galactic nuclei jet - and - lobe systems .note that our choice of prior is wrong for _ all test cases _ , but the iterative application of the wiener filter allows us to discover the source images within the noise and incomplete -plane coverage .we find that our method is quite robust : regardless of the structure of the source image we are able to recover the intensity distribution to a very high fidelity .our method outperforms traditional point source - based clean in terms of intensity distributions , rms error , and reconstruction signal - to - noise ratio .as expected given our isotropic gaussian process prior , we perform best on images with large amount of symmetry , though the wiener filter provides a route for reliable reconstructions of asymmetric images in regions of strong data support regardless of source structure .  using our methodwe can also easily incorporate uncertainty information in the reconstructed image . as discussed in earlier works  , the error model generated from the isotropic gaussian processprior tends to underestimate the true errors , but this situation is far preferable to no error model at all . with our error modelwe can construct acceptance masks based on the local ( i.e. , pixel - by - pixel ) signal - to - variance ratios .the desired threshold can be adjusted based on the desired level of confidence in the deconvolution .since the sample variance is directly tied to the relative level of noise by way of the wiener filter , our method naturally and self - consistently determines this mask .while the test cases we have presented in this work use an interferometric setup with realistic noise levels , primary beam shape , and antenna array , they do represent a relatively simplistic observing scenario .a fully implemented method would include simultaneous solutions of multiple frequencies , mosaicked images , and wide bandwidth observations .also , we have specified an overall snr of 10 for our fiducial observations , and we must examine the performance of this method in different regimes .  additionally , future large - scale interferometers will deliver incredibly high volumes of data .image deconvolution from even a single pointing at a single frequency will tax most computing systems . to accommodate future data setswe have implemented our algorithm using the mpi - parallelized petsc  toolkit , so our approach automatically grows with the size of the data without loss of scalability .algorithmically , the most expensive portion of our approach is the solution to equation ( [ eq : sky ] ) , which scales as in the ideal pregridded flat - sky approximation we have presented here .while we have not discussed foreground removal in this work , this method easily accommodates modeling in two fashions .first , partial signal or foreground information , if known in advance , can enter as an additional prior .alternatively , foreground models can be added as an additional sampling step within the algorithm  .the resulting posterior mean signal will thus _ automatically _ include marginalizations over the unconstrained parameters of the model .the same approach can be taken with the noise : if , for example , the noise spectrum is known but the absolute amplitude is not , we can sample over that amplitude .this flexibility , in addition to the other strengths discussed above , suggest that gibbs sampling is a promising response to the challenges of contemporary and future radio interferometric observations .m. sutter and b. d. wandelt acknowledge support from nsf grant ast-0908902 .b. d. wandelt also acknowledges funding from an anr chaire dexcellence , the upmc chaire internationale in theoretical cosmology , and nsf ast-0708849 .mcewen is supported in part by a newton international fellowship from the royal society and the british academy .l. zhang and p. timbie acknowledge support from nsf grant ast-0908900 .e. f. bunn acknowledges support from nsf grant ast-0908900 .g. s. tucker and a. karakci acknowledge support from nsf grant ast- 0908844 .

Abstract: we present a novel , general - purpose method for deconvolving and denoising images from gridded radio interferometric visibilities using bayesian inference based on a gaussian process model . the method automatically takes into account incomplete coverage of the -plane , signal mode coupling due to the primary beam , and noise mode coupling due to sampling . our method uses gibbs sampling to efficiently explore the full posterior distribution of the underlying signal image given the data . we use a set of widely diverse mock images with a realistic interferometer setup and level of noise to assess the method . compared to results from a proxy for point source - based clean method we find that in terms of rms error and signal - to - noise ratio our approach performs better than traditional deconvolution techniques , regardless of the structure of the source image in our test suite . our implementation scales as , provides full statistical and uncertainty information of the reconstructed image , requires no supervision , and provides a robust , consistent framework for incorporating noise and parameter marginalizations and foreground removal .   = -0.8 in  [ firstpage ]  instrumentation : interferometers , methods : data analysis , methods : statistical