time - delayed feedback is a common mechanism relevant in many biological system including excitable gene regulatory circuits and human balance .it has been investigated not only in biological systems , but also in a wider area : this includes but is not limited to semiconductor superlattices , chemical oscillators ( co oxidation on platinum ) and photosensitive oregonator models .these types of systems have been formulated mathematically in terms of recurrent models .recurrent connections introduce delayed feedback loops which may dramatically change the dynamic behavior of the system .it is known that delayed feedback presents an efficient method to control chaos or turbulence via stabilizing the unstable periodic orbits ( upos ) embedded in the chaotic attractor .feedback can be used to stabilize periodic orbits , or to control the coherence resonance ; the latter has also been studied experimentally for bistable system with delayed feedback .  over the past decades , neurobiologists found out that axons propagate the neuron s electrical signal to other neurons , and may sometimes feedback to the same neuron s dendrites  . these auto - synapses which establish a time - delayed feedback mechanism on a cellular level ,are called autapses and were described by van der loos and glaser in 1972 . since the discovery of such autapse in pyramidal cells in the cerebral neocortex , they were found in about 80% of all analyzed neurons including neurons of the human brain .however , what functional significance they offer in the neural systems is still not fully understood , as the autapses exhibit a broad range of delay time scales : these range from a few milliseconds to tenths of milliseconds .it was established that delayed feedback can induce bursting .its role for coding and processing of information in the brain has been evidenced experimentally for a variety of neural systems. however , the mechanisms leading to bursting are still open to debate ; in particular the presence of noise is expected to play a key role .  within this work we aim to contribute to this objective by considering the influence of intrinsic noise .it has been demonstrated , that noise leads to nontrivial effects in neuronal dynamics .some typical examples are stochastic resonance phenomena , coherence resonance and dynamical synchronization phenomena . in our situation thereoccurs an intrinsic source of noise which is due to the stochastic gating of ion channels ; i.e. so - called channel noise .the latter is inherently coupled to the properties of the axonal cell membrane and , _ a priori _ , can not be neglected .interestingly , this intrinsic noise effects the neuronal signaling at different levels : it was shown that it may control the occurrence of spontaneous action potentials , can improve the output of signal quality and may also account for the reliability of propagation , to name but a few .the manuscript is organized as follows : in sec .[ sec : model ] we present the biophysical model : it is given in terms of a stochastic hodgkin - huxley model containing a time - delayed feedback current . in sec .[ sec : detdyn ] we discuss the repetitive firing that occurs for the deterministic dynamics ( i.e. in absence of noise ) of this hodgkin - huxley model with delayed feedback .the stochastic dynamics are addressed in sect .[ sec : stochdyn ] . finally , we present our conclusions with sec . [sec : conclusion ] .in 1952 , hodgkin and huxley proposed an archetypical model for cell excitability and signal transmission along the axon of a neuronal cell .they postulated a set of continuous parallel pathways for the passage of the ionic and capacitive currents to represent the electrical properties of the neuronal cell membrane .the hodgkin - huxley model is widely regarded as a milestone achievement in biophysics and especially in electrophysiology .its applicability was extended to more complex systems beyond the originally studied excitability of the squid giant axon .experimental evidence of the channel noise has resulted in stochastic generalizations of the hodgkin - huxley model by considering the stochastic dynamics of the ion channel gating .these generalizations allow for microscopic modeling of the occurrence of spontaneous spiking and the phenomenon of biological stochastic resonance . in this context , it was shown recently , that channel noise improves the reliability for signal transmission along neuronal axons .we mathematically model the delayed feedback induced by an autapse ( sketched in fig . [fig : autapse ] ) by a feedback mechanism within a hodgkin - huxley model of the type put forward by pyragas . by an additional pathway in the dynamics of the membrane potential we consider the autapse leading to a delayed stimulus .particularly , we consider an excitable dynamics for the membrane potential , reading :    wherein the autaptic delayed stimulus \,,\end{aligned}\ ] ] is proportional to the difference of the membrane potential at time and that at an earlier time . here , corresponds to the coupling strength of the autapse mechanism and the finite delay time refers to the specific time delay caused by the autapse which is due to a finite signal propagation speed .the delay time is representing the elapsed time associated with the axonal propagation prior to the signal recurring onto the neuron ; is the membrane potential at the earlier time t- .the autaptic delayed stimulus in eq .[ curr ] results in an excitatory coupling mechanism in which spiking of a cell at an earlier time favors the initiation of a spiking event of the same cell at time .we note , that the ansatz for the delayed self - stimulus corresponds to electrotonic interaction , i.e. we consider an idealized situation wherein the autaptic delayed stimulus is proportional to the difference of the pre- and post - synaptic membrane potentials .moreover , as autapses are typically formed by chemical synapses , our modeling simulates the complex biophysical temporal evolution of the synaptic conductance by invoking a constant coupling strength .  in eq ., denotes the membrane capacitance , the time - dependent membrane potential , the leakage potential , and the reversal potentials for the potassium and sodium currents .the leakage conductance is given by , the potassium and sodium maximum conductances read and , respectively . the functions , and in eq . ([ eq : comp_mod ] ) denote the so - called stochastic gating variables ( see below ) , describing the mean fraction of open gates of the sodium and potassium channels at time .the typical values of parameters in our numerical simulation are those of the original hodgkin - huxley model ; i.e. , =1 , , , , , and .the stochastic dynamics of the gating variables , and depend on the voltage - dependent opening and closing rates and ( ) , which read :  [ eq : gating ]   the resulting stochastic dynamics is then modeled by a fokker - plank - type dynamics for the individual gates .specifically , this dynamics emerges within a large system size - expansion of the underlying markovian master - equation dynamics for the dynamics of the number of open gates .this so obtained langevin - dynamics is then interpreted in the it sense , reading explicitly :   with . here , ( ) denote independent gaussian white noise sources with vanishing mean and auto - correlation functions :  [ eq : noisecor ]   note , that the noise strengths are determined by the number of potassium and sodium channels and . throughout this workwe assume constant ion channel densities ; i.e. following the original hodgkin - huxley model we use for potassium and for the sodium channels per  .an integration step was used in the simulations and for the generation of the gaussian distributed random numbers , the box - muller algorithm was used .in the absence of both , the delayed feedback , i.e. , and zero noise strength , with the latter being formally achieved by and , the original hodgkin - huxley dynamics are recovered . these exhibit a single stable fixed point , which is the _ rest state _ with the rest potential .any temporary disturbance , e.g. caused by an applied external current stimulus , decays and the system relaxes back to the rest state . note that upon a proper choice of the initial condition , a single action potential may occur , before the system relaxes to the rest state .the phase diagram for repetitive firing within the hodgkin huxley model containing the delay coupling strength is depicted _ vs. _ the delay time .the two black solid lines indicate the boundaries of critical coupling strength versus the delay time at which the behavior changes from the rest state to a repetitive firing .the gray areas give the parameter regime for which repetitive firing is observed ., title="fig : " ] +  in presence of the autaptic delay coupling , the fixed point solution remains stable , but upon increasing the delay time parameter , there emerges now a stable , oscillatory solution , presenting repetitive ( tonic ) spiking . to investigate this repetitive spiking behavior in more detail , we systematically varied the two parameters ( and ) of the delay coupling scheme .in doing so , we determined the first occurrence of repetitive firing after a spike was created .the resulting phase diagram with the positive and negative critical coupling strength and , respectively , is depicted in fig .[ fig:1 ] .we point out the resulting asymmetry between positive and negative coupling .  for sub - threshold coupling strength , i.e. ( in the white regions of fig .[ fig:1 ] ) , excitations die out and the system relaxes to the rest state . for supra - threshold coupling , i.e. or a repetitive firing is observed . if the delay time becomes shorter than the _ refractory time interval _ , being ca . , which is the time the system needs for spiking and returning to the rest state , the modulus of the critical coupling strength of the bifurcation increases drastically .this is due to the fact , that in the so - called _ undershoot phase _ ( i.e. the part of the course of the action potential when the membrane potential is smaller than the membrane potential at rest ) the neuron is rather insensitive to any stimuli .hence , a stronger coupling strength is needed in order to excite the system from the _ undershoot phase _ and to obtain repetitive firing . in case of largerdelay the value of the critical coupling saturates . for longer delay timesmore than one spike may fit into time interval given by .doublets , triplets and multiplets may appear .below , we shall present examples where we have chosen and used a positive coupling strength .the critical value for repetitive firing corresponds to . for larger coupling action potential induces repetitive firing . in section [ sec : stochdyn ] we refer to this regime as _ supra - threshold delay coupling regime_. in case of lower coupling strength ;i.e. the state of a repetitive firing is excitable and noise is needed to support the repetition of an action potential . in the next section [ sec : stochdyn ]this regime will be addressed as _ sub - threshold delay coupling regime_. remarkably , for negative coupling one finds an overall similar similar behavior that differs , however , quantitatively ( not shown ) .+  +  in fig .[ fig : spiketrain ] the simulated membrane potential is shown for an exemplarily chosen noise level and for two cases , namely in panel ( a ) without delay coupling ( ) and in panel ( b ) with a finite delay coupling ) . in absence of delay couplingthe occurrence of spontaneous , i.e. noise induced actions potentials occur irregular while a characteristic bursting pattern can be detected in presence of finite delay - coupling .  in order to explain the spiking dynamics quantitatively , we determined the time intervals between two spike events , see in fig .[ fig : spiketrain](a ) , as obtained from simulations of the stochastic hodgkin - huxley model with delayed feedback coupling , cf .note that in order to detect the occurrences of action potentials from the simulated membrane potential dynamics , we did define a specific threshold barrier for detection .in particular , whenever the membrane potential exceeds the value of , the occurrence of an action potential is assigned .in fact , the so determined spike occurrences depend only weakly on the actual choice of the detection barrier .  these observed stochastically emerging interspike intervals the render the ( normalized ) interspike interval histograms ( isih ) and the mean interspike interval  where denotes the number of spikes obtained in the individual numerical simulation .the bin width for the histograms has been set at .both these quantities form the basis of our analysis of spike trains .we begin this study with the dynamics of zero delay coupling for the neuron , i.e. with .the distribution of the interspike intervals at a fixed channel noise intensity is depicted in fig .[ fig : isihdifferente ] . for large time intervals ,the distribution decays exponentially .small time intervals are suppressed because of the neuron s refractory state . for isih exhibits one broad maximum located around the internal time scale .this time scale systematically diminishes with increasing noise level .the minimal time scale for undistorted spikes is around the refractory time of .  in presence of noise and finite feedback coupling ,the neuron still can return to its fix - point with nonvanishing probability .this is why the peak at the intrinsic time scale and a broad exponential decay is still detected in the isih in fig .[ fig : isihdifferente ] .however , the isihs become multimodal in case of finite , nonvanishing feedback .several maxima occur at larger time intervals and , in between , a number of deep dips are observed .such forms of the distributions of the isih are indicators of an enhanced coherence . whereas in the sub - threshold regime (dashed line ) the additional extrema merely show up , they become increasingly dominant in the case of supra - threshold coupling ( note the logarithmic scale of the ordinate in fig .[ fig : isihdifferente ] ) .( color online ) interspike interval histograms ( isih ) versus the interspike interval of the stochastic hodgkin - huxley model with a delay coupling mechanism at different coupling strengths , as indicated in the figure in units of .the bin width for has been chosen throughout this work at .the chosen number of potassium ion channels is , while the number of sodium ion channels is set at .the delay time is set at .,title="fig : " ] +  in the case of sub - threshold coupling the repetitive oscillatory spiking is noise supported , as a non - vanishing delayed stimulus shifts the stable fix - point towards the threshold for excitation .consequently , the generation of spikes becomes more likely .however , the dynamics is still excitable ; i.e. a threshold still exists and noise is needed to induce its excitation .consequently , a sub - threshold delayed stimulus favors some characteristic interspike interval time .as far as the system remains excitable , an activation time after the delayed stimulus did set in is necessary in order to create the next noise - induced spike .therefore , the time is not equal to the delay time , but equals the sum of the delay time and an activation time , being around in this considered case .( color online ) interspike interval histogram ( isih ) _ vs. _ interspike duration for different noise levels corresponding to different number of sodium and potassium channels as indicated .the chosen delay time is : ; the coupling strength is set at and .,title="fig : " ] +  in addition , there occurs a suppression of certain time scales when the delay coupling is applied to the neuronal dynamics , cf .[ fig : isihdifferente ] . for positive the system favors frequencies associated to the time and suppresses frequencies slightly larger than those .the feedback hinders the generation of spikes by noise if the feedback current approaches values around the polarized refractory period .  in the sub - threshold regimethe intrinsic time scale and correspondingly the height and the position of the associated peak distinctly depends on the noise level . in fig .[ fig : isihnoise ] we depict results from simulations for different noise levels , i.e. different numbers of embedded ion channels . with increasing noise level the generation of spontaneous spikes by noise gains influence substantially .as a result , the intrinsic time scale becomes changed and the larger time intervals reflect the broad , exponentially decaying isih .the former sharply peaked maximum at switches towards a noise supported broader peak at smaller times , and the maximum shifts with growing noise towards the minimal time for a interspike interval located at the refractory time .interspike intervals around this minimal time are promoted mainly by the presence of noise .they do not necessarily relate to the delay mechanism .in contrast , the time scale induced by the delayed feedback is only slightly affected by the noise level .this is a somewhat striking feature recalling the fact that in the case of an excitable dynamics these spikes in fact require the presence of noise to become activated .the robustness of the positions of the time scales as well as their separation from the noisy time scale in terms of the resulting bimodal shape of the isih thus indicates a noisy synchronization phenomenon .( color online ) interspike interval histogram ( isih ) for different values of the supra - threshold coupling strength : the distribution of the interspike intervals is depicted for fixed delay time , the numbers of ion channels are and .the values of the coupling strengths indicated in the plot are in units of . ]we next consider the case with coupling strengths beyond the critical strength , i.e. , . here , the spikes repeat deterministically .however , noise - induced skipping of spikes leads to a transition from the oscillatory state with repetitive firing to the excitable state . in turn ,the _ back - transitions _ from the excitable to the oscillatory state are caused by noise - induced spontaneous spikes , cf .[ fig : spiketrain ] . the spikes repeated deterministicallylead to sharp peaks in the isih , while the noise - induced _ back - transitions _ from excitable to the oscillatory state result in a broad distribution with exponentially decaying tail . in fig .[ fig : isihe ] , the resulting isih is depicted for various supra - threshold coupling strengths . upon the chosen parameter values for the noise level ,i.e. the number of sodium and potassium ion channels , and the coupling parameters and , the isih therefore exhibits sharp peaks and more or less pronounced broad background : for stronger coupling or weaker noise the broad background diminishes .contrarily , the peak height at shows a strong dependence on the noise level , indicating the competitive interplay between channel noise and the delayed feedback mechanism : with increasing noise level the height of the peak at decreases , while the intrinsic time scale acquires increasing influence ( depicted for the sub - threshold coupling regime in fig .[ fig : isihnoise ] ) . note that we have selected the delay time ( ) so that the separation between the broad background which is peaked at the intrinsic time scale and the delayed induced sharp peaks at and multiples thereof is educible and visible with the isihs , cf .[ fig : isihnoise ] and fig .[ fig : isihe ] .  strikingly , with increasing coupling strength the multimodal structure collapses to a unimodal one , cf fig .[ fig : isihe ] .the most probable interspike interval now centers at one value . due to this supra - threshold driving , each spontaneous spiking eventis repeated periodically and even the noise dominated scale now collapses towards a narrow peak .note also that the distribution gains in sharpness as the coupling strength increases , cf .[ fig : isihe ] .   as function of the delay time : the mean interspike interval , cf .eq . , is plotted versus the delay time for the following parameters : the used ion channel numbers are , and the coupling strengths are set at ( black dashed line ) , ( red dotted line ) , and ( blue solid line ) . ]the inverse of the number of spikes that fit into the delay - time interval is plotted as function of the delay time for , and a delay coupling strength of . ]  let us focus on the dependence of the interspike intervals on the delay time .for this purpose we consider the mean interspike interval , cf .this quantity inherits the characteristic dependence on since the isih shape is unimodal .larger deviations from the mean interspike interval are found during transitions between different synchronized states where the histogram becomes multimodal .  in fig .[ fig : mean ] the mean interspike interval is depicted for a fixed channel noise strength . if the delay coupling mechanism does not dominate the spiking , i.e. for small supra - threshold coupling strengths , the mean interspike interval exhibits a smooth dependence on the delay time .however for larger supra - threshold driving strengths , for which the isih exhibits a unimodal structure consisting of a sharp peak , the mean interspike time varies with the delay time in an almost piecewise linear fashion , displaying sharp , triangle - like textures , cf .[ fig : mean ] . at these sharp peak locations ,the number of spikes that fit according to the intrinsic time into a full time length given by the delay time , just increases by unity with increasing delay time .the mean interspike interval is henceforth proportional to the ratio of the delay time and the number of spikes fitting into this very delay - time interval ; yielding in fig . [ fig : kappa ] the behavior of _ vs. _ the delay time is depicted . at multiples of the noise - dependent intrinsic time scale , such characteristic steps do occur indeed .we find that stronger noise intensity mimics a decrease of the coupling strength .nearby those typical steps noise is able to induce newly generated spikes at the end of the intervals reducing thereby the ratio .alternatively , the role of noise decreases the number of spikes towards the left boundary of a new synchronized region .therefore the corresponding results at increased noise strength attain a form that is similar to the case of lower delay coupling strength .a comparable scenario has been reported in other systems such as for a noisy van der pol dynamics , see in refs . near the hopf bifurcation , for semiconductor superlattices and stochastic excitable dynamics .we find that noise can induce oscillations with a well - defined time scale , and the simultaneous application of a delay of duration is able to stabilize this orbit .we have seen that the neuron behaves almost like an oscillatory system whenever the delay coupling rules the dynamics .this brings to mind a description in terms of a phase oscillator model which we develop next .let us assign an intrinsic frequency by .as pointed out above , the autaptic feedback leads to a stabilization of the internal rhythmicity as determined by the model parameters .this in turn results in synchronized patterns , cf .[ fig : kappa ] .therefore we investigate the question whether the observed synchronization in this studied neuronal model with autapse can be characterized by a kuramoto - type model of a phase oscillator including delayed feedback ? put differently , upon neglecting the details of the shape of the spike trains the essential information can be reduced to the phase of the oscillatory neuron .let us next compare our findings with the results obtained from a pure phase dynamics .accordingly , the proposed dynamics for the underlying phase is modeled as :  \ , , \end{aligned}\ ] ]  where denotes the angular intrinsic frequency of the oscillator and is the delay - coupling strength .searching for solutions with fixed frequency , we make the ansatz : the resulting , self - consistent equation reads : \ , , \end{aligned}\ ] ] which was solved numerically . comparing this phase oscillator dynamics with our oscillatory stochastic neuron dynamics exhibiting an autapse , we make the following parameter identification , i.e. , and obtain from an expansion in small delay time the relation . in fig .[ fig : comp ] we present the comparison between the hodgin - huxley modeling and the the phase oscillator modeling : even though the quantitative behavior is different , a good qualitative agreement is detected .the positions of the spikes where the autapse model jumps between bursts of different number of multiplets are nicely reproduced by this simplified phase oscillator modeling .a further improvement of the modeling would require a more optimal determination of the -dependent parameters of this kuramoto model with feedback . +  , [ eq : gating ] , [ eq : correlator - a ] , [ eq : noisecor ] ) with the dynamics of the feedback assisted kuramoto - type phase oscillator model , eq .( [ eq : phase - oscillator ] ) . the mean interspike interval ( red dashed lines ) is the result of the numerical integration of the stochastic hodgkin - huxley model with delayed feedback ; there is quantitative good agreement between period of oscillations as obtained from a phase oscillator dynamics of the kuramoto - type delayed feedback model ( solid black line ) .regions with multiple solutions in the phase model correspond to discontinuous jumps between different types of multiplets in the hodgkin - huxley model .the parameters for the neuronal modeling are chosen as , and ; those of the phase oscillator are given by oscillator frequency and feedback strength . ]with this work we have investigated the effect of delayed feedback on the dynamics of a _ stochastic _hodgkin - huxley neuron . using the original hodgkin - huxley parameters , we simulated numerically the stochastic hodgkin - huxley model taking into account the effects of intrinsic channel noise . a pyragas - like delayed feedback mechanism is employed to model the autapse phenomena , in which a neuron s dendrite back - couples to itself . the two basic parameters used in our study are the strength of the autapse - coupling and the time - delay resulting from the finite length of the self - connecting dendrite .  we have found that the neuron dynamics exhibits intriguing new times scale that stem from the autaptic connection .since the rest state of the neuron is always stable , noise or an initial spike is necessary to create activity , i.e. the spiking dynamics . for a small number of and channels ,the noise becomes sizable and the excitory dynamics remains practically unaffected by the delay .in contrast , smaller noise levels and stronger coupling strengths induce a different synchronization phenomena between the delay time and the intrinsic ( also noise dependent ) time scales . the delay time andthese intrinsic time scales determine how many spikes will be created and become subsequently locked during one delay epoch .we further underline that our exemplary study can be mimicked qualitatively in terms of a reduced description given by a kuramoto phase dynamics with built - 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Abstract: the effect of intrinsic channel noise is investigated for the dynamic response of a neuronal cell with a delayed feedback loop . the loop is based on the so - called autapse phenomenon in which dendrites establish not only connections to neighboring cells but as well to its own axon . the biophysical modeling is achieved in terms of a stochastic hodgkin - huxley model containing such a built in delayed feedback . the fluctuations stem from intrinsic channel noise , being caused by the stochastic nature of the gating dynamics of ion channels . the influence of the delayed stimulus is systematically analyzed with respect to the coupling parameter and the delay time in terms of the interspike interval histograms and the average interspike interval . the delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution , exhibiting a delay - induced reduction of the spontaneous spiking activity at characteristic frequencies . moreover , a specific frequency - locking mechanism is detected for the mean interspike interval .


recently the numerical computation of gravitational waves from coalescing black holes has received considerable attention and effort for closely related reasons .( i ) the problem is a testbed for numerical relativity and has been the center of a collaboration of eight universities .( ii ) the results are necessary to understand details , and perhaps even gross features of coalescing black holes , presently viewed by many as the most promising source of gravitational waves detectable by earth based instruments .( iii ) the problem is interesting in its own regard as the first solution of einstein s equations for strong field interactions without simplifying symmetry .the job of the numerical codes under development will be to evolve forward in time from an initial value solution on a starting hypersurface .there are many ways that initial data can be chosen to represent the starting state of two black holes .if the black holes are far apart on the initial hypersurface , then any reasonable choice of hypersurface data will suffice .unfortunately the instabilities in evolution codes mean that the codes can only run for a relatively short time , and initial data will have to be specified on a hypersurface as late as possible .  for the most part , the initial data used to the present has been that put forth by york and bowen , and coworkers .this scheme starts by requiring that the initial geometry be conformally flat and that the extrinsic curvature have a property called `` longitudinal , '' equivalent to the condition that it be derivable from a vector potential .the solution is then made inversion symmetric , so that topological throats representing the black holes connect two isometric asymptotically flat universes .( this step of making the hypersurface data inversion symmetric turns out to destroy `` longitudinality'' . )the advantages of this approach to initial data include the following : ( i ) since it is widely adopted in the numerical relativity community , comparison of results of different groups is facilitated .( ii ) it is specific and definitive and the numerical methods of solving for these initial data are highly developed .( iii ) the inversion symmetry gave a simple inner boundary condition for evolution codes .( iv ) an argument could be made that the initial data was free of , or had a minimal content of , radiation encoded ( in some sense ) in the initial data themselves , so that the outgoing radiation found in a spacetime evolution should represent only the radiation due to the dynamical interaction of the black holes .the bowen - york ( hereafter by ) scheme served very well in the role intended for it .one should ask , however , whether it is the appropriate scheme for computations supporting the search for gravitational waves .there are strong reasons for concluding that the answer is `` no . ''one immediate reason is that the kerr geometry itself can not be put in this scheme ; a constant time slice of the kerr geometry is not conformally flat .even a single spinning by hole will therefore radiate gravitational waves as it settles down into its kerr final state .a second rather direct reason comes from recent calculations we carried out for radiation from a particle falling radially into a schwarzschild hole , hereafter paper i and paper ii . in these calculations ,the `` particle '' was taken as a perturbation of the background black hole spacetime and computations could be done relatively easily for a variety of choices of data on an initial hypersurface . furthermore , evolved data could be extracted on later hypersurfaces and compared with various prescriptions ( by etc . ) for hypersurface data . in the approachused , based on the perturbation formalism of moncrief , the initial and evolved data was represented by a spacetime function that is totally gauge invariant , so that the interpretation of the physics was not tainted by questions of coordinate choices .the calculations were used to study a set of hypersurface data that somewhat generalize the by data .those sets considered were all conformally flat and longitudinal ( cfl ) . in practicethis gave , for each multipole of the perturbation , a complete set of differential equations that had to be satisfied by the hypersurface geometry and extrinsic curvature ( or , in practice , and its time derivative ) .the only remaining freedom is the choice of boundary conditions ( consistent with asymptotic flatness ) .these choices are equivalent to the choice of conditions deep inside the throats representing the holes , i.e. , in the asymptotically flat universes at the end of the throats away from `` our '' asymptotically flat universe .we focused on two choices of these boundary conditions . once choice was the choice of the by data itselfthe second was the choice that the solution at the horizon be `` frozen '' or `` matched '' to its initial value .this approach was based on the fact that very near the background horizon a simple analytic solution could be given for the evolution of the data on the initial hypersurface to subsequent hypersurfaces .the solution for was simply that it was `` frozen '' to retain its initial value at the horizon .the solution for reflected more detail , and was said to be `` matched '' to the initial solution at the horizon .we could also choose to have `` frozen , '' but we could choose the boundary conditions for to be the by conditions , and so forth .  for a particle falling from rest at a large initial distance , we found that the solution that evolved , and the cfl data on later hypersurfaces were in serious disagreement . if evolved data were replaced by cfl data on a late hypersurface the resulting waveforms and radiated energies were drastically altered .we found furthermore : ( i ) in the class of cfl hypersurface data the best choice is the `` frozen - matched '' choice , but that the difference between this and other cfl choices is small compared to the disagreement with the evolved solution .( ii ) the disagreement between evolved hypersurface data and cfl data does not lie , in any obvious way , in the `` radiation content '' of the data , but rather in the details of the data near the location of the particle .( iii ) the disagreement between the true and cfl data was much more severe in than in .in the present paper we look again at the problem of head on collision of two nonspinning holes in the particle limit , and ask whether there is a more successful alternative prescription for hypersurface data . more specifically, let us consider a configuration of black holes at late times that evolved from a precursor configuration at large separation , and let us suppose that we know the positions and momenta of the holes on the late time hypersurface. is there a prescription for the data on the late hypersurface which is in good agreement with the `` true '' hypersurface data ( i.e. , the data evolved from the large - separation precursor ) , and which gives an accurate prediction of the radiated energy and waveform ?the answer to this question seems to be that there is , and the prescription , at least in the particle limit , is a very simple one : that the data be `` convective , '' in a sense to be defined below .  in sec .iia we sketch the mathematics of the moncrief formalism we use and of our notation and conventions , and we present some results of evolved data vs. cfl data ; this sketch will be quite brief ; a reader interested in the details will find them in paper ii .the proposal for convective data is discussed in general in sec .iib , and is implemented in a particular manner , for radial infall , in sec.iic .numerical results from this implementation are given in sec.iid .a discussion of the implications and possible extensions of this work is presented in sec .the numerical results we will present below will be for a particle infalling radially into a schwarzschild black hole . due to the axial symmetry of our problem , and the absence of rotation , the perturbations are pure even parity .to describe the perturbations , we use the regge - wheeler notation , but not the regge - wheeler gauge . in this notation the even parity metric perturbations are decomposed into scalar , vector and tensor spherical harmonics . with coefficient functions , depending on the standard radial and time coordinates of the schwarzschild background . for simplicity, we drop the multipole index on these perturbation functions .the perturbation coefficients describe only the perturbed 3-geometry , and are independent of the choice of shift and lapse . in terms of these , moncrief defines the totally gauge invariant function + \frac{2}{r}(r-2m)\left(r^2\partial g/\partial r-2h_1 \right)\ , \ ] ] where we have used zerilli s normalization for and his notation   the basic wave equation for an infalling particle is given in paper i as here is the regge - wheeler `` tortoise '' coordinate and is the zerilli potential the right hand side of eq .( [ rtzerilli ] ) is a source term constructed from the particle stress energy . for a point particle of proper mass ,the stress energy is given by \delta^2[\omega]/r^2\ , \ ] ] where is the particle 4-velocity , and gives the radial position of the particle as a function of coordinate time , starting from at   the total energy radiated in the multipole , after a hypersurface at is    an even parity multipole perturbation of the 3-geometry has the four functional ( of ) degrees of freedom contained in , but these must satisfy the ( perturbed , multipole ) hamiltonian constraint , a single second order differential equation containing the particle source . the choice of a conformally flat 3-geometry reduces the problem to a single functional degree of freedom ( since , in an appropriate coordinate system , and ) satisfying the single hamiltonian constraint .the only freedom is the choice of the constants specifying the homogeneous solution of the hamiltonian constraint .one of those constants is fixed by the condition that the solution be well behaved at spatial infinity .the remaining constant can be specified to `` freeze '' the horizon value of at its initial value , to make the solution inversion symmetric , or in other ways .if the requirement of conformal flatness is dropped , then any sufficiently smooth function can be added to a particular solution that solves the hamiltonian constraint .  for thingsare a bit more complicated . for an even parity perturbative extrinsic curvaturethere are four functions of , analogous to the four functions specifying the 3-geometry perturbations .there are two second order equations , equivalent to and .if we require that the extrinsic curvature be longitudinal , then the extrinsic curvature must be derivable from a vector potential , and for a given even parity multipole the functional degrees of freedom are and .since these functions must satisfy two second order differential equations , we have no remaining functional freedom .we have only to choose the four constants that specify how much our solution contains of the four homogeneous solutions of the momentum constraints .two of these constants are fixed by the requirement that the extrinsic geometry be well behaved at spatial infinity .the remaining two constants can be chosen to make the solution of the by type , to make the solution `` match '' the initial solution at the horizon , etc .if the requirement of longitudinality is dropped , then any sufficiently smooth function can be added to any particular solution of the momentum constraints .   herewe briefly review the numerical results found for the cfl prescription . in the results presented here a particle started its infall from schwarzschild radial coordinate , where is the mass of the spacetime ( in the usual units ) . at subsequent times ,when the radial location of the particle is smaller , and the time label of a constant hypersurface is larger , we can extract the evolved , replace it with a prescribed cfl etc .figure [ fig : reven ] shows the result that is most important . at various times( parameterized by the particle location at these times ) the numerical evolution was stopped , and the evolved quadrupole and were replaced by the cfl prescribed data representing the particle position and momentum on this hypersurface .evolution forward from this hypersurface is then done numerically .the resulting energy is then numerically computed .the true energy for infall from is . if we evolve the prescribed data from a later surface we see that we get reasonably good agreement with the true energy for larger than around , corresponding to around . for smaller the use of prescribed cfl databadly overestimates the radiated energy .comparisons of the true ( i.e. , evolved ) quadrupolar and the prescribed quadrupolar are shown in the next two figures .both are for a particle starting from rest at initial separation at . for both ,the comparison is made at .this corresponds to the particle being at , and having momentum .a comparison of is given in fig .[ fig : revpsi ] in which the solid curve shows the evolved and the dashed curve is the cf for a particle of mass , at location , with momentum . the corresponding comparison for is shown in fig .  [ fig : revpsidot ] . from these comparisonsit appears that the cf solutions give a reasonably good approximation to near the particle , but in the strong field region of the background , the cfl form of has a very different shape near the particle than that of the evolved .we note here that the evolved will , in general , have a  ] .( note that the resulting time - dependent function ) ] .if on some hypersurface , one chooses initial data then , in principle , numerical evolution would give a purely periodic solution .for corresponding to circular motion , one could , on the other hand , give other hypersurface data , such as cfl .the result would be a burst of initial radiation as the spacetime `` settles down '' to the periodic solution . if the circular orbit is meant to represent a late stage in a binary coalescence which is adiabatically decaying due to radiation reaction, then the periodic solution is obviously the appropriate one , and an initial burst of radiation is an anomaly due to the wrong choice of initial conditions .we now apply the idea of convective to a particle of mass falling radially into a schwarzschild hole . as the appropriate form of on a hypersurfacewe choose a conformally flat solution , with `` frozen '' horizon conditions .that is , on any hypersurface , the limit of is the same as on the hypersurface .the meaning of this solution can be nicely described with the notation developed in paper ii . for even parity -pole perturbations of the 3-geometry , there are only two gauge invariant functions of .one is the moncrief defined by eq .( [ psidef ] ) .the second is , a gauge invariant measure of deviation of the 3-geometry from conformal flatness .( in the regge - wheeler gauge . ) for all cfl choices is zero on the hypersurface .this choice exhausts the functional freedom on the hypersurface , and the 3-geometry is then fixed by the hamiltonian constraint . in practicethis means that satisfies a second - order differential equation in the variable ; the frozen horizon condition fixes the solution to that equation .we are then choosing the same hypersurface as a cfl choice .what is crucial is the difference in our choice of . hereour convective choice is tantamount to taking  ] soon after the initial hypersurface .the new prescription agrees very well with the evolved data , both in its general shape near the particle , and in its numerical values ; by contrast the by data disagrees sharply with the evolved data .this disagreement grows at later times , and at the by and evolved curves have little similarity . on the other hand ,the new prescription gives a curve that agrees qualitatively with the true data , and agrees in size within a factor of two .the same comparison is made in figs .[ fig : psidot15 ] for a particle falling from rest at at . herewe see that both the new prescription and by data agree with the evolved data reasonably well until the particle approaches the strong field region near the peak of the zerilli potential . at smaller radiithere are significant quantitative differences between the data of the new prescription and the evolved data , but these differences are much milder than the wide divergence of the by data from the evolved data .a comparison of outgoing waveforms appears in figs .[ fig : wvfrm ] .the solid curves in both ( _ a _ ) and ( _ b _ ) show the true waveform , seen by an observer at , for the quadrupolar part of , for a particle falling from rest , at time from position ( equivalent to $ ] ) .the dotted curve in ( _ a _ ) shows the outgoing waveform that results if conformally flat , frozen - horizon , convective data is substituted for true data on the hypersurface .similarly the dashed curve in ( _ a _ ) shows the waveform if the substitution is made at .the three curves show remarkable agreement after the start of quasinormal ringing . only at early timesis there a noticeable difference : the long shallow dip in the true waveform is reduced or missing from the waveforms generated by the data of the new prescription .the dotted curve in ( _ b _ ) shows a waveform comparison at a yet later time .although the new waveform has roughly the same magnitude of quasinormal ringing as the true waveform , it has a large phase shift relative to the true curve .this can be understood with the idea that quasinormal ringing is generated near the peak of the curvature potential .for the dotted curve in ( _ b _ ) , the particle is placed at a smaller radius than the potential peak , and has no history of having moved inward through the potential peak .the quasinormal ringing excited by the prescribed data , then , must start at the time of the hypersurface on which prescribed data is defined .for the true curve , the quasinormal ringing starts earlier , at the time the particle passed going inward .numerical experiments confirm with reasonable accuracy that the phase shift depends on particle position in this manner .  in the work of abrahams and cook , and baker and li ,the `` close limit '' was used to evolve prescribed data for the head on collision of equal mass holes .the close limit approximation applies to a hypersurface late enough that a single horizon surrounds both colliding holes , and only the large radius fields of the colliding holes are of importance .we have seen in paper ii , however , that in the particle limit this close limit does not work at _ any _ separation of the holes , if by prescribed data is used .the method fails at large separation , because the close limit conditions fail ; the method fails at small separation because the by data is a bad representation of the true data . in fig .[ fig : clap ] we see that the close limit does work at sufficiently small separations if the new prescription ( conformally flat , horizon frozen , convective ) is used .we have seen that the convective prescription for initial data for a black hole coalescence gives a much better description of the way in which hypersurface data evolves than does the cfl prescription .the demonstration , however , has been confined to an example of a coalescence which ( i ) was a head on collision ( ii ) used a conformally flat 3-geometry , and ( iii ) involved the particle limit of nonrotating holes .it will be relatively simple to study another particle - limit example that is free of the first two of these limitations . as already mentioned in sec .iib , the steady state solution for a particle in circular orbit around a hole is an example of a solution that is convective for any time slice .it is straightforward to find this convective periodic solution in the form of a fourier series , and this solution will not be conformally flat .this opens the possibility for several interesting comparisons .it will be reasonably straightforward to start with cfl data for a particle which at one instant has the correct position and momentum for a circular orbit .similarly , it is simple to find initial conditions corresponding to a convective , initially conformally flat data .both these solutions could be evolved forward in time and the anomalous radiation could be found that is emitted as the field settles into its steady state solution .the amount of radiation energy in this anomalous initial burst is an indicator of how sensitive conclusions will be to the details of the initial data prescribed .if the energy is significant compared to the steady state emission during a single orbit , it is a sign that cfl initial data will give misleading results for black hole coalescence , at least for coalescing holes of very different mass .a considerably more difficult issue is the particle - limit context of our development .how are these ideas to be applied to coalescence of holes of comparable mass ? in a very general way one might try to use the idea of a time sequence of 3-geometries in order to fix the initial extrinsic curvature .there are many caveats that go with such a proposal , along with the question of just how one implements such a scheme ? one could , for example , add time dependence to the conformal factor with the convective description , by making the parameters ( location and momentum of the holes ) time dependent .but we do not know at the outset what the correct dynamical sequence is for strong field motion .there is a more serious difficulty .such a scheme would give a definitive prescription for finding the time derivatives of the 3-geometries , and ( with some physically motivated choice of lapse and shift ) would give us the initial extrinsic curvature .this extrinsic curvature should in general give a result that does not satisfy the momentum constraint .a more promising approach might be to try to generalize the condition .one might , for example , use the cotton - york tensor , which vanishes if and only if the 3-geometry is conformally flat . by setting the time derivative of this tensor to zeroone would capture , in the nonlinear theory , the same condition as in perturbation theory . a scheme like this ,however , is based on conformal flatness , and this raises the question of whether conformal flatness itself should be retained in an astrophysically motivated specification of initial data .somewhat related to the question of a convective prescription , but somewhat independent , is the question of the frozen horizon condition . for representing initial data that evolved from an earlier astrophysical configuration , the frozen horizon condition is `` obviously '' a correct constraint on data on a late hypersurface .whether or not the convective prescription is used in connection with numerical relativity , the frozen horizon condition should be used if one wants a representation of an astrophysical problem .

Abstract: numerical relativity codes now being developed will evolve initial data representing colliding black holes at a relatively late stage in the collision . the choice of initial data used for code development has been made on the basis of mathematical definitiveness and usefulness for computational implementation . by using the `` particle limit '' ( the limit of an extreme ratio of masses of colliding holes ) we recently showed that the standard choice is not a good representation of astrophysically generated initial data . here we show that , for the particle limit , there is a very simple alternative choice that appears to give excellent results . that choice , `` convective '' initial data is , roughly speaking , equivalent to the start of a time sequence of parameterized solutions of the hamiltonian constraint ; for a particle in circular orbit , it is the initial data of the steady state solution on any hypersurface . the implementation of related schemes for equal mass holes is discussed .


photoacoustic tomography ( pat ) is a novel coupled - physics ( hybrid ) modality for non - invasive biomedical imaging that combines the high contrast of optical tomography with the high spatial resolution of acoustic imaging .its principle consists in illuminating a sample by an electromagnetic pulse that , due to the photoacoustic effect , generates pressure waves inside of the object ; see figure [ fig : pat ] .the generated pressure waves ( the acoustic signals ) then propagate through the sample and beyond , and the pressure is recorded outside of the sample .finally , mathematical algorithms are used to reconstruct an image of the interior ( see , for example , ) .  in this paperwe work with the standard model of pat , where the acoustic pressure solves the standard wave equation ( see ) .the goal of pat is to recover the initial pressure distribution ( at time ) , given by a function , from measurements of the pressure that is recorded on the hyperplane .here , denotes the half space .that is , we assume that we know on ( the acquisition surface ) , and our goal is to reconstruct from ( possibly approximate ) knowledge of .this is an inverse problem , which amounts to an ( approximate ) inversion of the operator . we will refer to that problem as the inverse problem of pat with a planar acquisition geometry .note that similar approaches can be considered for other acquisition geometries as well .a semitransparent sample is illuminated with a short optical pulse . due to optical absorption and subsequent thermal expansion an acoustic pressure waveis induced within the sample .the pressure waves are measured outside of the sample and used to reconstruct an image of the interior . ]  in the case of exact data , several approaches have been derived for solving the inverse problem of pat considering different acquisition geometries .this includes time reversal ( see ) , fourier domain algorithms ( see ) , analytic reconstruction formulas of back - projection type ( see ) , as well as iterative approaches . for the case of noisy data , it is well known that iterative methods ( including variational methods and sparse reconstructions ) tend to be more accurate than analytic methods .  among those iterative techniques , sparse regularization approacheshave gained a lot of attention during the last years as they have proven to perform well for noisy data as well as for incomplete data problems .one of their main advantages consists in their ability to combine efficient regularization with good feature preservation and to ( to some extent ) to compensate for the missing data .however , these advantages of sparse regularization methods come with the cost of typically significantly longer reconstruction times than fbp - type approaches .this is because the forward and adjoint operators have to be evaluated repeatedly .due to that reason , fbp type methods ( or other direct approaches ) are often preferred over the more elaborate sparse regularization techniques .  in this paperwe develop numerically efficient reconstruction method for pat with planar geometry that effectively deals with noisy data , where regularization is achieved by enforcing sparsity constraints in the reconstruction with respect to wavelet coefficients of .more precisely , we derive a direct method for calculating a minimizer the -tikhonov functional where denotes an orthonormal wavelet basis and are weights . to achieve that , as one of our main results ,we construct a wavelet - vaguelette decomposition for the operator .that is , given a wavelet basis , we construct a system in that , in the case of exact data , gives rise to an inversion formula of the form .given noisy data , we show that a combination of this formula with soft - thresholding , namely provides a minimizer of the functional . additionally , we derive an efficient algorithm for the vaguelette transform and provide an implementation for the wvd reconstruction . moreover , we show order optimality of our method in the case of deterministic noise as well as is the case of gaussian random noise .we also consider generalizations of our method and , in particular , we show how wvd reconstruction can be combined with an additional tv - prior .sparse regularization has been widely used as a reconstruction method for general inverse problems and there is a vast literature on that topic ( see , for example , ) . in the statistical setting ,-tikhonov regularization is known as lasso or basis pursuit . in most cases ,the reconstructions are computed by employing iterative algorithms ( such as iterative soft - thresholding ) to minimize the -tikhonov functional . as mentioned above, those methods have the disadvantage to be slower than fbp type methods , as the forward and adjoint problem have to be solved repeatedly .  wavelet - vaguelette decompositions and generalizations like biorthogonal curvelet decompositions and shearlet decompositionshave been derived for the classical radon transform in 2d , see . to the best of our knowledge, this paper is the first to provide a wvd for photoacoustic tomography as well as an efficient direct implementation of sparse regularization using wavelets for that case .[ [ organization - of - the - paper ] ] organization of the paper + + + + + + + + + + + + + + + + + + + + + + + + +  in section [ sec : pat ] we review the mathematical principles of pat with a flat observation surface and collect results required for our further analysis . in section [ sec : wvd ] we derive the wvd for the forward operator .the wvd is then used to define the soft - thresholding estimator in section [ sec : noisy ] . in that sectionwe also discuss the equivalence to variational estimation such as -tikhonov regularization .the efficient implementation of the estimator requires and efficient implementation of the vaguelette transform .such an algorithm is derived in section [ sec : num ] , where we also present results of our numerical simulationlet denote the space of compactly supported functions that are supported in the half space , where .we write and consider the initial value problem with .we assume that the pressure data is observed on the hyperplane and define the corresponding pat forward operator as where denotes the unique solution of .our aim is to recover from exact or approximate knowledge of .we are particularly interested in the cases and , as they are of practical relevance in pat ( see ) .nevertheless , in what follows , we consider the case of general dimension since this does not introduce additional difficulties .the following isometry property for the wave equation is central in the analysis we derive below . for odd dimensions, it has been obtained in  , and for even dimensions in  ; see also .[ lem : isometry ] for any we have   see for odd and  for even .  from lemma [ lem : isometry ]it follows that extends to an isometry on the space with respect to the scalar product . in view of the isometry property and the desired wavelet - vaguelette decomposition , instead of the operator , it is more convenient to work with the modified operator it is not hard to see that is an isometry with respect to the inner product we have the following result .[ lem : aiso ] let be defined as in . then, the following assertions hold :  1 .[ lem : aiso1 ] for all we have .2 .  [ lem : aiso2 ] the operator uniquely extends to an isometry .  [ lem : aiso1 ] according to lemma [ lem : isometry ] for every we have   [ lem : aiso2 ] the set is dense in and , according to [ lem : aiso1 ] , the operator is an isometry on .consequently , it follows from the general hilbert space theory that the operator can be extended in a unique manner to an isometry .finally , from the construction of it is clear that .   for the following considerations it will be convenient to apply the operator to functions that are defined on rather than on the half space .that is , we need to extend the operator in a meaningful way to an operator .one possibility to do this would be to consider the wave equation with initial data and then to proceed as above . however , any function that is odd in the last variable would be in the kernel of the resulting operator .therefore , we use a different extension that leads to an isometric operator on .  to that end, we define the operator by .then is an isometric isomorphism with and , where is defined in an obvious way .we are now able to define the announced extension of .[ def : afull ] we define the operator by here and below denotes the orthogonal projection onto a closed subspace of .[ thm : isofull ] the operator is an isometry .  any function can be written in the form and satisfies . from definition [ def : afull ] and lemma [ lem : aiso ] we then conclude that for every .  in what follows , we will also consider the operator as a densely defined operator on .here , is the set of all ( equivalence classes of ) functions such that .in this section we will state explicit expressions for the operator and its dual .for that purpose , we consider the spherical radon transform , which is defined as follows : where denotes the unit sphere in and is its surface measure .a simple calculation ( application of fubini s theorem ) shows that the dual of the operator is given by the operator is called _ spherical backprojection operator _ , because integrates the function over all spheres that pass through the point .we will also consider the ( fractional ) differentiation operators the formal adjoints of those operators are given by for and for , where   we are now able to provide explicit expressions for the operator and its dual .we have   the identity follows from the well known explicit expression for the solution of the wave equation ( see and ) .the identity follows from by applying calculation rules for the adjoint .in this section , for a given wavelet basis of , we construct the wvd of the operators and that we defined in the previous section and prove inversion formulae for the case of exact data . to that end , we particularly will make use of the isometry relation that we proved in section [ sec : pat ] .  the basic idea of the wvd is to start with an orthogonal wavelet basis and to construct a possibly non - orthogonal basis system of the image space in such a way that the operator and the prior information are simultaneously ( nearly ) diagonalized . for readers convenience , we summarized some basic facts about wavelets in appendix [ app : wavelets ] .let be a linear , not necessarily bounded , operator and let be an orthonormal wavelet basis of .the construction of a wavelet - vaguelette decomposition for the operator amounts to finding families , in satisfying the following properties :  1 .  quasi - singular relations ( with ) : 2 .biorthogonal relations : .near - orthogonality relations : here , means that there are constants such that .  such a decomposition ( if it exists )is called a wvd for the operator .given such wvd for an operator , one can always obtain an explicit inversion formula for the operator of the form note the analogy between and the svd decomposition .the numbers depend here only on the scale parameter and have the same meaning as singular values in the svd .thus , are referred to as quasi singular values .similarly to the svd , the decay of the quasi singular values reflects the the ill - posedness of the inverse problem .a wvd decomposition has been constructed for the classical computed tomography modeled by the two dimensional radon transform , where ; see . in the case of the two dimensional radon transform ,a generalization of the wvd , a so - called biorthogonal curvelet decomposition was constructed in . in , the authors derived biorthogonal shearlet decompositions for two and three dimensional radon transforms .  in the case of photoacoustic tomography , there are no such decompositions available so far . in the next subsection , we establish a wvd for the operators and , which serve as forward operators for pat with a planar observation surface , and so automatically obtain an inversion formula for exact data .let be a wavelet basis of .the desired wavelet - vaguelette decomposition of and is in fact a direct consequence of the isometry relation of theorem [ thm : isofull ] .[ thm : wvd ] for every define and let .then , the family is a wvd for the operator .moreover , we have the inversion formulae   we start with an arbitrary function and express this function in terms of a wavelet expansion with respect to . then according to the isometry property is an orthonormal basis of and further .in particular , this amounts to a wvd decomposition with and .further , is a consequence of the wvd .next , let be an element in . then consequently , applying yields .note that the identity in theorem [ thm : wvd ] is an explicit inversion formula for the operator in the spirit of a wvd . instead of an orthonormal wavelet basis it uses the family , which is non - orthogonal with respect to the inner product . restricted to functions in that vanish outside , where is any compact subset , the operator is an isomorphism .then , allows to characterize the besov norm of any function that is supported in .figure [ fig : wv ] shows a vertical , a horizontal and a diagonal daubechies 10 wavelet and the corresponding vaguelettes obtained by application of the operator .in what follows , we assume to be an orthonormal compactly supported wavelet basis of .further , by we denote the corresponding vaguelette basis of that satisfies .if exact data are available , then the wvd decomposition  provides an exact reconstruction formula for the unknown .however , in practical applications the data ( or ) are only known up to some errors ( e.g. noise ) .we therefore assume that we are given erroneous ( noisy ) data where is the error term and the exact unknown .we consider both the deterministic and the stochastic case , in which different models for are assumed :  * in the deterministic case , we assume that a bound is available . * in the stochastic case , we assume that , where is a white noise process .  in the deterministic situation the constant is referred to as the noise level ; in the stochastic case it is the noise standard deviation .the goal in both situations is to estimate the unknown from data in .   in section [ sec : wvd ]we have shown that the reproducing formula holds in the case of exact data .if the data is corrupted by noise , i.e. , , the inner products can not be computed exactly .instead , they are estimated by first evaluating and then applying the soft - thresholding operation \\ y-  { w } & \text { if } y  >  { w}\\ \end{cases}\ ] ] with appropriate thresholds .[ def : wvd - soft ] for any nonnegative sequence , the wvd soft thresholding - estimator for the solution of is defined by with the nonlinear soft - thresholding function defined in .  in the deterministic casewe assume that the error term is known to satisfy . in this case , due the isometry property of the operator , one obtains a stable reconstruction by applying the adjoint , as the adjoint operator satisfies , i.e. , corresponds to the wvd - thresholding estimator with , it follows that also the class of wvd - thresholding estimators yields the optimal error estimate . without additional knowledge about the noisethere is no need to apply the wvd soft - thresholding estimator with , even if the function is known to belong to some smoothness class , such as a sobolev or besov ball .however , if a portion of the noise energy is known to correspond to high frequency components , then using non - zero thresholds may significantly outperform given that belongs to some smoothness class .for example , this happens in the case of stochastic white noise , where the energy - spectrum is uniformly distributed .suppose , where is a white noise process .in the case of random noise , it is common to measure the performance of an estimator in terms of the worst - case risk  of a subset , further define the nonlinear minimax risk , the minimax risk using the wvd estimator  and the linear minimax risk , respectively by from the definition it is clear that no reconstruction method can have worst - case risk smaller than the non - linear minimax risk .we are in particular interested in asymptotic behavior for the case and is a ball in a besov space .[ thm : random ] suppose that and let be a ball in the besov - norm having the form then , as , the following hold  1 .[ thm : randomn ] , 2 .[ thm : randomw ] for some constant , 3 .[ thm : randoml ] with .  follows from ( * ? ? ?* theorem 4 ) for the special case .theorem [ thm : random ] implies that , despite its simplicity , the wvd - thresholding estimator is order optimal on any besov - ball and the rate can not be improved by any other estimator ( up to some constant factors ) . on the other hand , if , then the exponent in the linear minimax rate is strictly smaller than the exponent in the non - linear minimax rate , .therefore , no linear estimator can give the optimal convergence order .in particular , this implies that the wvd - thresholding estimator outperforms filter based regularization methods including the truncated svd or quadratic tikhonov regularization .the wvd - based soft thresholding estimation can be characterized via various variational minimization schemes that , as we shall discuss later , in turn offer several extension of the wvd - estimators .let [thm : var ] be a sequence of thresholds and let .then the following assertions are equivalent :  1 .  [ thm : var1 ] [ thm : var2 ] ; 3 .[ thm : var3 ] is the unique solution of the constraint optimization problem   : let denote the minimizer of . because is an orthonormal basis of we have .further , , which shows that is the unique minimizer of the latter functional is minimized by minimizing every summand independently in the first argument .the minimizer is given by one - dimensional soft thresholding which gives and therefore . : let denote the solution of . using that is an orthogonal basis and , shows that can be equivalently characterized as the minimizer of the minimization problem can again be solved separately for every component which is again given by the one - dimensional soft thresholding . as above thisimplies .the variational characterizations of theorem [ thm : var ] have several important implications .first , they provide an explicit minimizer for the -tikhonov functional which in general has to be minimized by an iterative algorithm , such as the iterative soft thresholding algorithm and its variants . for the analysis of -tikhonov regularization for inverse problemssee .another important consequence is that the wvd - soft thresholding estimator can be generalized in various directions .in particular , one can get a generalization of by replacing the -norm in by an appropriate regularization functional ( for example , can be chosen as the total variation norm , see section [ sec : implementation ] ) .that is , instead of solving the problem , one aims at solving the problem this generalization constitutes a hybrid version that combines the wvd estimator with more general regularization functionals .such hybrid approaches have been introduced independently in ( see also ) .it is also related to the dantzig and multiscale estimators of ., not all features of can be stably recovered .only singularities can be reconstructed for which intersects .,scaledwidth=70.0% ]in this section , we provide algorithms for the calculation of the vaguelette transform and the corresponding wvd estimator . moreover , we consider a hybrid version of the wvd estimator that combines wavelets and tv regularization and discuss its implementation .we also present some numerical examples . throughout the following an orthonormal wavelet basis of and is the corresponding orthogonal vaguelette basis of defined by ( see section [ sec : wvd ] ) .the considered noisy data model assumes that the data can be collected on the whole hyperplane . however , this is not feasible in practice since the data can be collected only on a finite subset . in this subsectionwe discuss the effects of partial ( or limited view ) data and discretization .first , we address the limited data issue . in the considered imaging setup ,the data is collected on a subset  ] denotes the characteristic function of  ] , the actual sampled data are given by \coloneqq   g (  n \delta_n , m \delta_m )   + { z}[n , m ] \ , ,   \quad ( n , m ) \in { \left\{1 , \dots , n\right\}}^{d-1 } \times { \left\{1 , \dots , m\right\ } }   \,.\ ] ] here and are natural numbers , and are the sampling step sizes and  ]  compute the discrete wavelet transform   assuming discrete data of the form = g (  n \delta_n , m \delta_m)$ ] , the discrete vaguelette transform can efficiently be computed by algorithm [ alg : v ] .both the steps in this algorithm are well known and numerically efficient . for the first step we use a numerical approximation of by numerically implementing  with a filtered backprojection ( fbp ) algorithm ( see ) . for evaluating the second step we use the implementation of the fast discrete wavelet transform provided by the matlab function wavedec2 .   for the numerical experiments , that will be presented in the next section , we implement the wvd soft - thresholding estimator defined by as well as a hybrid vaguelette - tv approach defined by with being the tv norm of .in both cases we choose as a half of the so called universal threshold that can be derived from extreme value theory  .the implementation of the wvd soft - thresholding estimator is summarized in algorithm [ alg : soft ] .it is based on the discrete vaguelette transform that was presented in algorithm [ alg : v ] .compute vaguelette coefficients by applying algorithm [ alg : v ]  apply soft thresholding to using threshold   apply the inverse wavelet transform  given discrete data , the hybrid vaguelette - tv approach can be written in the form where denotes the discrete fbp operator .one recognizes that can be implemented by applying a hybrid wavelet - tv denoising algorithm to . following  , we rewrite the optimization problem  in the form and introduce the associated augmented lagrangian operator then , can be solved by the alternating direction method of multipliers ( admm ) , introduced in , which alternatingly performs minimization steps with respect to and and maximization steps with respect to . the resulting admm algorithm for solvingis summarized in algorithm [ alg : admm - tv ] .        for performing the -update in algorithm [ alg : admm - tv ] we have to solve the unconstraint total variation minimization problem . for this purposewe use chambolle s dual projection algorithm .   in this sectionwe present a numerical example testing the vaguelette soft - thresholding and the hybrid approach .for that purpose we consider a simple phantom that consists of the superposition of three uniformly absorbing spheres as illustrated in the top image in figure [ fig : fg ] .the data are computed numerically using an implementation according to . to simulate data errors , we added i.i.d gaussian noise with standard deviation . the relative -error in the noisy data is .results of our reconstruction from noisy data are shown figure [ fig : rec ] .the top image shows the non - regularized reconstruction , the middle image the vaguelette - thresholding reconstruction and the bottom image the reconstruction with the hybrid vaguelette - tv method .figure [ fig : err ] shows corresponding difference images between the reconstructions and the true phantom .one clearly observes that the vaguelette estimators significantly reduce the error in the reconstruction. this can be quantified by the relative -errors , which are equal to for the unregularized reconstruction , for the vaguelette thresholding estimator for the hybrid estimator .the relative -error for the hybrid reconstruction is slightly larger than for the vaguelette - thresholding estimator .this is reasonable since the vaguelette - thresholding estimator minimizes the -norm ( see ) among all potential reconstructions that are compatible with the data , whereas the hybrid estimator minimizes the total variation .however , in some more appropriate error measure the hybrid reconstruction may outperform the thresholding estimator .further note that all reconstruction methods contain some limited data artifacts , which can not removed completely by wavelet methods .in this paper we developed a regularization framework using wavelet sparsity in photoacoustic tomography .for that purpose we derived wavelet - vaguelette ( wvd ) decompositions ( see theorem [ thm : wvd ] ) and an easy but efficient implementation of the corresponding vaguelette transform ( see algorithm [ alg : v ] ) . using the wvd we derived an explicit formula for minimizing the sparse tikhonov functional that can be implemented without any iterative reconstruction procedure ( see algorithm [ alg : soft ] ) .the considered regularization approach has been shown to provide optimal error estimates in the deterministic as well as in the stochastic setting ( see theorem [ thm : random ] ) . in order to account for wavelet artifacts we also developed hybride regularization methods combining wavelet sparsity with total variation ( see algorithm [ alg : admm - tv ] ) .numerical results demonstrate the feasibility and efficiency of our reconstruction approaches .future work will be done to extend our approach to more general measurements geometries in pat and also to different radon type inverse problems .we recall some basic facts about orthogonal wavelets as we need them for our purpose ( in particular in section [ sec : wvd ] ) . for task of function estimation, wavelets are known to sparsely represent many signals and , hence , they can be used to effectively encode prior information .another useful property of wavelets consists in the ability to characterize several classical smoothness measures ( eg .sobolev and besov norms ) in terms of the decay properties of wavelet coefficients .we will also use that property in section [ sec : noisy ] . for a detailed introduction to waveletswe refer to .the construction of a wavelet onb is based on the concept of a multiresolution analysis ( msa ) , which is given by a sequence subspaces in that satisfy the following requirements ( see ) :  * for all it holds that . *the union is dense in . * . * for every , we have .* there is a function such that the translates constitute an onb of the scaling space .the function is called _ scaling function _ and the spaces are called _ scaling ( or approximation ) spaces at scale j_. to each scaling space , one can associate a _ wavelet ( or detail ) space _ , that are defined to be the orthogonal complements of in , i.e. , .because of the above properties it holds that , for each and , the functions constitute an onb for the scaling space . one can also show that from the existence of the scaling function it follows that there exists a so - called _ generating wavelet ( or mother wavelet ) _ such that , for each and , the functions constitute an onb for the spaces .hence , we have that , for every , the following mappings are bijections :   the above constructions provide the following decompositions of the signal space into the sum of the scaling spaces and the wavelet spaces , or into a sum of only wavelet spaces : from these decomposition we immediately get the following decompositions of signals : the coefficients of the above decomposition are called the _ wavelet and scaling coefficients of _ , respectively , and the corresponding mapping that maps to those coefficients is called the _ wavelet transform of . a detailed construction of orthogonal ( and biorthogonal ) wavelet systems together with many interesting details may be found , for example , in  .   wavelet bases in higher dimensionscan be defined by taking tensor products of the one - dimensional wavelet and scaling functions . as a concrete example consider and let be the generating wavelet and scaling functions ,respectively . then , an orthogonal wavelet basis of is defined by translates and scaled versions of the ( tensor product ) functions the corresponding scaling functions at scale of two variables are defined as translated and scaled versions of   analogously to the above construction in two dimensions , one can define a wavelet onb in by a tensor product construction . in this casewe get different generating wavelets and one generating scaling function . therefore , a wavelet onb in is system in , where the index consists of three parameters , the scale index , the location index , and the orientation index .the wavelet basis elements are again defined as dilates and translates of the generating wavelets : if we also add the index set and consider the scaling functions defined by the we get similar decompositions to the one - dimensional case :     orthonormal wavelet bases can be used to characterize smoothness of functions in terms of the decay properties of wavelet coefficients . in particular , wavelets provide a convenient and a numerically efficient way to characterize besov spaces and calculate besov smoothness of functions , see . skipping the details , we here only mention that the besov spaces can be defined by the condition , that the besov norm {\sum_{j\in{\mathbb{n } } } 2^{jsq } \ , { \left\lvert { \mathcal w}{f}{\left(j,{\,\cdot\,}\right)}\right\rvert}_{\ell^p}^q } \qquad \text { with } \quad s = r +  \frac{d}{2 } - \frac{d}{p }  \,,\ ] ] is well defined and finite .we note that , the given definition of is an equivalent norm to the classical definition of besov norms .the above characterization holds as long as the generating wavelet has vanishing moments and is -times continuously differentiable .  in section [ sec : noisy ] , we use the above characterization of besov norms in terms of wavelet coefficients in order to evaluate the performance of our method for functions that lie in balls of besov spaces for some given norm parameters , and smoothness parameter .m. agranovsky , p. kuchment , and l. kunyansky . on reconstruction formulas and algorithms for the thermoacoustic tomography . in l. v. wang , editor , _ photoacoustic imaging and spectroscopy _ , chapter 8 , pages 89101 .crc press , 2009 .m. m. betcke , b. t. cox , n. huynh , e. z. zhang , p. c. beard , and s. r. arridge .acoustic wave field reconstruction from compressed measurements with application in photoacoustic tomography , 2016 .arxiv preprint arxiv:1609.02763 .g. paltauf , r. nuster , m. haltmeier , and p. burgholzer .photoacoustic tomography with integrating area and line detectors . in _photoacoustic imaging and spectroscopy _ , chapter 20 , pages 251263 .crc press , 2009 .j. poudel , t. p. matthews , l. li , m. a. anastasio , and l. v. wang .mitigation of artifacts due to isolated acoustic heterogeneities in photoacoustic computed tomography using a variable data truncation - based reconstruction method ., 22(4):041018 , 2017 .k. wang , r. w. schoonover , r. su , a. oraevsky , and m. a. anastasio .discrete imaging models for three - dimensional optoacoustic tomography using radially symmetric expansion functions . , 33(5):11801193 , 2014

Abstract: in this paper we consider the reconstruction problem of photoacoustic tomography ( pat ) with a flat observation surface . we develop a direct reconstruction method that employs regularization with wavelet sparsity constraints . to that end , we derive a wavelet - vaguelette decomposition ( wvd ) for the pat forward operator and a corresponding explicit reconstruction formula in the case of exact data . in the case of noisy data , we combine the wvd reconstruction formula with soft - thresholding which yields a spatially adaptive estimation method . we demonstrate that our method is statistically optimal for white random noise if the unknown function is assumed to lie in any besov - ball . we present generalizations of this approach and , in particular , we discuss the combination of vaguelette soft - thresholding with a tv prior . we also provide an efficient implementation of the vaguelette transform that leads to fast image reconstruction algorithms supported by numerical results .   * key words : * photoacoustic tomography , image reconstruction , wavelet - vaguelette decomposition , variational regularization , sparsity constraints , wavelet - tv regularization .   * ams subject classification : * 49n45 , 65n21 , 92c55 .