as indicated above , the present invention is based upon the present inventors &# 39 ; discovery that objects hidden in highly scattering turbid media can be imaged by a novel , fast and noise - resistant inverse method which comprises , among other things , transilluminating a highly scattering turbid medium using a plurality of intersecting source - detector pairs and using intensity measurements of the diffusive component of the detected light signals taken over a plurality of time windows ( or &# 34 ; time - slices &# 34 ;) in a novel inverse reconstruction algorithm . the following discussion provides the theoretical basis for the present invention : a scattered signal is , in general , a function of key parameters characterizing the medium . the forward problem is to build a mathematical model that allows computation of signals when a spatial distribution of key parameters is given . this mathematical model is clearly based on the fundamental physical principles governing the interaction . the inverse problem is to find the distribution of these key parameters that best fit the given signal data . for photon propagation in turbid media , the key parameters characterizing the interaction are the absorption length , the transport mean free path , the scattering mean free path , and the index of refraction of media . the parameters are in general functions of position ( r ) and wavelength ( λ ). in the case of optical diffusion tomography , the most commonly used key parameters are absorption and transport scattering coefficients , which are the inverse of absorption length and transport scattering mean free path , respectively . the mathematical model governing the photon propagation is the photonic transport theory or its approximations , such as the diffusion theory or the telegrapher &# 39 ; s equation . to show the basic principle of this technique , diffusion theory is applied here to model light propagation in turbid media . the diffusion equation in turbid media is ## equ3 ## where i ( r , t ) is the field of photon density , s ( r , t ) is the source of laser light , μ a is the absorption coefficient , d = 1 / 3 ( μ a +( 1 - g ) μ s )! is the diffusion coefficient , with μ s the scattering coefficient , g the scattering anisotropic factor and c the speed of light in the medium . we are interested in the following problem : given intensity measurements taken around the media either with objects located therein , i ( r , t ), or without objects located therein , i 0 ( r , t ), reconstruct a spatial map of the change in optical parameters ( absorption and diffusion coefficients ) due to the presence of hidden objects . the forward problem of scattering can be represented in a matrix form written as where y has m =( number of s - d pairs )×( number of time slices ) elements , which is the change in intensity profile and is a function of the position of source r s , the position of detector r d , and the time - slice in the profile t . x has n =( number of voxel ) elements wherein a voxel is a space grid element . x j = cδμ a ( r j ) or cδd ( r j ) is the ( time - independent ) change in the absorption coefficient or the diffusion coefficient in the voxel j due to the presence of hidden objects , and n is signal detector noise . w is an m × n matrix . its element is the weight function or &# 34 ; photon measurement density functions . &# 34 ; this weight function represents the change in signal at the detector placed at r d at time t from a source signal originating at r s associated with a unit change in absorption coefficient ( or diffusion coefficient ) at r j of the reference medium , as implemented by a diffusion process that carries the signal from source at r s to the detector at r d . its value is theoretically calculated according to the mathematical model , for example , the diffusion theory in the case shown below . using green &# 39 ; s function formula , for the case of absorption change , cδμ a , the element of w is given by ## equ4 ## where g 0 ( r j , r i , t ) is the green &# 39 ; s function of eq . ( 1 ) for a reference system ( the system with no hidden object ) which gives the normalized photon intensity at r j at time t as migrating photons originate from r i at time t = 0 . in eq . ( 3 ), the second green &# 39 ; s function in the integral , g 0 ( r j , r s , τ ) represents the photons that migrate from a source to a given voxel element . the first one , g 0 ( r d , r j , t - τ ), represents the photons that migrate from this voxel to a detector . δv is volume of a voxel . therefore , w in eq . ( 3 ) represents a normalized effect that photons originating from the source migrate through voxel j , then further migrate to the detector during a total time of t . in the case of diffusion or scattering change , both g 0 in the integral of eq . ( 3 ) are replaced by ∇ g 0 . ## equ5 ## according to certain previously - used inversion algorithms employed in image reconstruction , such as algebraic reconstruction technique ( art ), simultaneous algebraic reconstruction technique ( sart ) or conjugate gradients ( cgd ), the fractional signal change y =-( i - i 0 )/ i 0 , which corresponds to the first order perturbation , is used . such inversion reconstruction algorithms , however , generally requires a great number of iterations in order for a clear image map of hidden objects to be obtained . such iterations frequently can take an amount of calculation time that exceeds the practical time limit for clinical applications . a more accurate non - perturbative approximation based on a cumulant expansion has been developed by us , which fits the experimental data by y =- ln ( i / i 0 ). this treatment , to some extent , automatically includes higher order non - linear contributions . the inversion is a reconstruction of optical parameters from signals . in order to make a fast inversion for clinical applications , we have developed the following novel inverse reconstruction procedure based on a modification of the iteration formula described in shaw , &# 34 ; improvement of the numerical resolution of an instrument by numerical solution of the integral equation ,&# 34 ; j . math . analys . and applic ., 37 : 83 - 112 ( 1972 ), which is incorporated herein by reference : ## equ6 ## where y , (- ln ( i / i 0 )), is the input data for inversion . x . sup . ( k ) and x . sup . ( k + 1 ) are k th and ( k + 1 ) th iteration solutions , respectively , for the change of parameters . the initial value of x is set to zero in the absence of prior information about hidden objects . t represents the transpose matrix . the matrix λ , which regularizes or converts an ill - posed problem to a well - posed problem , is chosen to be diagonal , λ ij = λ j δ ij . λ j =& lt ; nn & gt ;/& lt ; δx j δx j & gt ;, with & lt ; . . . & gt ; the statistical average , and δx j = x j -& lt ; x j & gt ;. thus , λ j represents the ratio of mean square of noise to the expected mean square deviation in the change of parameters ( structural noise ). since w does not relate to the position and optical parameters of hidden objects , having w for a given reference system ( either homogeneous or inhomogeneous ), the inverse matrix w t w + λ ! - 1 can be precalculated and stored as a database , which can be used in clinical cases to find hidden objects . this strategy separates time - consuming work in inversion processes into two parts . pre - computation is time - consuming and may require the use of a supercomputer . however , it greatly reduces the burden of performing inverse processes in daily clinical cases . one of the main difficulties in inversion is that most inverse problems are ill - posed . if no noise existed , an inverse image could be reconstructed from eq . ( 2 ) via : x t = y t w w t w ! - 1 . the existence of a small amount of noise should not have much effect on the results obtained . however , the matrix w t w is generally ill - posed or near - singular . in other words , this matrix has some near - zero eigen values . this occurs when the number of measurements m is less than the number of unknown parameters n or when the measured signals are not fully independent of each other . in this case , the effect of small noise will be magnified in the inverse process , and the solution of inversion becomes unstable . one of the methods to overcome the ill - posed problems is regularization , which adds a λ to the matrix w t w . λ is relatively small compared to the matrix w t w ; therefore , regularization makes the matrix well - posed or well - defined , but does not change the characteristics of the image map . the value of λ is set according to the experimental noise level and structural noise present in the change of optical parameters . if || λw t w ||& lt ; 1 , one may use the following expansion to correct the error introduced by regularization : if noise n exists , y = wx + n . one wants to find an inverse operator , x = ly , which gives the estimated value x of x , that , in the sense of statistical average , has minimum error . shaw has derived the following solution : with & lt ; . . . & gt ; statistical average , δx = x -& lt ; x & gt ;. we see that eq . ( 6 ) is a well - posed expression . this approach is the so - called &# 34 ; well - posed stochastic extension of ill - posed linear processes &# 34 ; or &# 34 ; statistical regularization . &# 34 ; comparing eq . ( 4 ) and eq . ( 6 ), λ in eq . ( 4 ) is related to & lt ; nn & gt ;/& lt ; δxδx & gt ;. thus , λ j represents the ratio of mean square of detector noise to the expected mean square deviation in the change of parameters ( structural noise ). the key point is that since w is only reference - related , and does not relate to the position and optical parameters of hidden objects , the inverse matrix w t w + λ ! - 1 of a given pattern of the reference system , which can be either homogenous or inhomogeneous , can be pre - calculated and pre - stored as a database for various patients for clinical use . our tests show that this novel inverse procedure converges after only 2 - 4 iterations , which is much faster than the art or sart procedures currently used . it takes about one minute to form an image in our reconstruction tests on a silicon graphic instruments ( sgi ) workstation computer . to demonstrate the present technique , the following experiment was performed . schematic drawings of the experimental arrangement are shown in fig1 ( a ) and 1 ( b ). ultrashort pulses of 100 fs pulse duration at a wavelength of 625 nm were coupled into a rectangular shape random medium by an optical fiber . this geometry was chosen because it is easy to construct an analytical expression of green &# 39 ; s function g 0 in a finite sized 3d volume , which is suitable for testing the physical modeling and the inversion algorithm . the scattered pulses around the medium were collected by 7 optical fibers and time resolved by a streak camera . the size of the sample was 60 × 60 mm and the height was 90 mm . the fibers were located at mid - plane . the source fiber was at the center of the boundary wall . two fibers each were placed on the opposite and the side boundary walls of the sample , and one fiber detector was placed on the incident wall . the scattered pulses collected by the 7 detection fibers and a reference pulse were coupled onto the input slit of the streak camera for multi - channel detection . the medium was intralipid solution with a transport mean free path of 2 . 5 mm and absorption length of about 500 mm . the hidden object was a black painted aluminum rod of diameter of 4 . 8 mm . temporal profiles of the scattered pulses around the medium were measured . in this experimental setting , the structure was z - independent , so the reconstructed image is presented as a 2 - d image . however , the present method could also be used as a 3 - d imaging model . as an example , a typical set of temporal profiles measured for one s - d pair is shown in fig2 . in this case , the detector is located straight across the medium in the line of incidence . measurements of the uniform medium without the object and with the object placed in three locations 90 degrees apart are also shown . it is clearly shown that the intensity is reduced by introducing the absorbing object , and a larger reduction was observed when the object was placed in the line of the source detector pair . the measurement of 3 object positions is equivalent to 3 measurements by rotating the s - d pairs by 90 degrees while fixing the object location . in the image reconstruction , an area in the sample was divided into 28 × 28 = 784 voxels , which corresponds to an actual voxel size of 2 × 2 mm . measurements of 7 s - d pairs with the above mentioned object locations were used . for each temporal intensity profile , intensity data at 40 time slices uniformly distributed from 602 ps to 2045 ps were used . thus , the total number of input data points was 21 × 40 = 840 . this is one of the important features of our method -- the use of hundreds of time slices from the temporal profile of one s - d pair for image reconstruction gives much more information than data obtained in the frequency domain with much higher speed resolution in sub - mm to mm range to detect a defect . the robustness of the method was demonstrated with a small number of fibers . in our programs , y and w are magnified by a factor of 100 . fig3 ( a ) through 3 ( e ) show images obtained for x . sup . ( 1 ) in eq . ( 4 ) with different regularization parameters . using large λ 0 , the image of the hidden object has already appeared after the first iteration . with a decreasing of λ 0 down to 10 - 2 , the image is gradually broken . fig4 ( a ) through 4 ( h ) show the results of different numbers of iterations , when the small regularization λ 0 is used . we see that after m = 50 iterations , the image of the hidden object begins to appear . after m = 500 , the image becomes stable as no visible change in the shape of the object occurs between m = 500 to m = 2000 . fig5 ( a ) and 5 ( b ) show an image obtained where ( a ) λ 0 = 1000 , without iteration , by running 1 minute on a silicon graphic incorp . ( sgi ) indy computer where w t w + λ ! - 1 is stored ; and ( b ) λ 0 = 10 - 2 , with 2000 iterations , by running 3 hours on the same computer , respectively . as can be seen , the two images are similar , but the latter has better resolution . noting that the difference of regularization parameters for these two images is up to 5 orders of magnitude , the stability of the imaging process for the present technique is confirmed . in both fig5 ( a ) and 5 ( b ), the presence of an absorber can be clearly seen . the location of the maximum change is about 7 mm away from the center which is about the actual location of the object ( 10 mm ). the full width of half maximum of the absorber is 8 mm which is comparable to the diameter of the object ( 4 . 8 mm ). the fluctuation at other spatial locations are less than one third of the peak value . using different regularization parameters , the same location of the absorber was found , but the maximum peak value and the full width of half maximum were different . one possible reason that the center of the image deviates from that of the experimental set is that the hidden object is not a pure transparent scattering absorber , and the possible effect of reflection on the surface of the rod was not included in our physical model . for comparison , the result using the sart inversion algorithm , commonly used by others , after 100 iterations ( taking x = 0 as initial input ), which runs 3 hours on the same sgi workstation , is shown in fig6 . there is no clear image of the absorber , noticing that the maximum absorption coefficient is only 0 . 0025 which is about 40 times smaller than that obtained by our new algorithm . we have also tested our technique on simulated temporal data with 15 % white noise added . a clear image of a single absorber on the correct position was obtained . these results show that our approach is noise - resistant . this inversion method can include a change of diffusion coefficients , δd ( r ) or δμ s ( r ), as described before for detecting scattering defects in a highly scattering host medium . it should be noted that a reconstruction algorithm which is suitable for clinical applications must be reasonably rapid in running time while maintaining mathematical integrity . an inverse algorithm mainly based on fourier transform runs in the fastest way . in order to make a fourier transform deconvolution available , the condition of &# 34 ; translation invariance &# 34 ; should be satisfied . in other words , the propagator k ( y , x ) must be the function of y - x . this condition is violated in experimental settings in the following three ways : ( 1 ) the media is not uniform ; ( 2 ) there are boundaries on which the light sources and detectors are setting ; ( 3 ) the propagator here is a function of ( r s , r d , r ); therefore , it is impossible to make a fourier transform for both r s - r and r d - r . among above listed violations , ( 1 ) is relatively soft , since we can assume a reference with uniform background , and put non - uniform as an image . on the other hand , reconstruction based on the inverse matrix method does not require the condition of translation invariance and can be used for more general cases . for purposes of clinical applications , a three - dimensional ( 3d ) reconstruction algorithm is necessary . because light diffuses in 3d space , the measurements on a specific layer are affected by the hidden objects on other layers . in extending our approach to the 3d case , we face a new difficulty . the computational complexity to inverse a matrix is about n 3 , where n is the size of the matrix . if one supposes that the size on the third dimension is divided into 30 grids , the computing complexity to inverse a matrix in the 3d case is about 27 , 000 times that in the 2d case , which makes the precomputation of an inverse matrix for 3d case unfeasible . to overcome this difficulty , we propose adding a 1d fourier transform to the 2d matrix inversion . therefore , a novel inverse algorithm for obtaining images of 3d objects hidden in scattering media is taught in this invention , the method combining a 2d matrix inversion with a 1d fourier transform inversion . in our approach the conditions for fourier transform available are properly treated . this method greatly reduces the computational burden , compared to standard 3d matrix inversion methods . this approach can be used for both time - resolved data and frequency domain data . using the above mentioned pre - computation and database storage strategy , we tested the present technique . it takes about 30 minutes on a silicon graphic indy 4400 computer to obtain a 3d image of 28 × 28 × 32 voxels of mm objects in cm &# 39 ; s scattering media . the source - detector pairs are arranged on the boundary of scattering media , such as breast and brain , in a 2d plane ( the x - y plane ), which is scanned along the z direction . accordingly , z coordinates of sources and detectors are the same , thereby eliminating the violation ( 3 ) mentioned above . the reference system is assumed to be uniform and infinite along the z direction . this assumption leads to some deviation in the reconstructed image due to finite z - size in experimental setting . since there is no source and detector setting on the z - end plane , this end effect becomes less serious . under the above experimental setting and assumption , w satisfies z - translation invariance . in other words , it is a function of z - z 0 , where z 0 is z - position of source - detector plane and z is z - coordinate of a voxel . after making a one - dimensional fourier transform over z - z 0 we obtain k independent 2d matrices , w 2d ( k ), parametrized by k , with k the number of grid - points in the fourier k - space . we separately calculate the k inverse matrices w 2d ( k ) t w 2d ( k )+ λ ( k )! - 1 , k = 1 , 2 , . . . , k , where λ ( k ) is a matrix for regularization . these inverse matrices are stored as database for later reconstruction of image in different hidden object cases . the computational complexity now is k times that of inverse of a w 2d matrix , which is much less than that of inverse of a w 3d matrix . the experimental data in time - resolved ( or frequency domain ) are y ( r d , r s , t ( ω ), z 0 ), where r d and r s , are ( x , y ) coordinates of detector and source , respectively , and t is the slice ( or ω is frequency ). we make a fourier transform over z 0 to obtain y ( r d , r s , t ( ω ), k ), k = 1 , 2 , 3 , . . . , k . we then obtain the image in the k space using the following matrix multiplication : then , the inverse 1d fourier transform of x ( k ) over k generates the 3d distribution of the change of absorption coefficients or scattering coefficients . a simulated experimental setting for a 3d image obtained using a plurality of source - detector ( s - d ) pairs is shown in fig7 ( a ) and 7 ( b ). the s - d pairs are arranged around a square of size 60 × 60 mm 2 . this arrangement of s - d pairs is then rotated 90 degrees and 180 degrees and the plane is scanned 32 steps along z in a range of 120 mm . for each simulated temporal profile , intensities at 40 time slices uniformly distributed from 602 ps to 2045 ps are taken . the number of sampling data are 7 × 3 × 40 × 32 . data both with and without hidden objects are computed to produce y =- 100ln ( i / i 0 ), adding uniform distributed noise in ± 10 % range , where i and i 0 are intensity with and without hidden objects , respectively . the volume of sample is divided into 28 × 28 × 32 voxels . two arrangements of hidden absorption objects are separately shown in fig8 ( a ) and 8 ( b ). the cross - sections of the hidden objects are 4 × 4 mm 2 and 6 × 6 mm 2 . in case ( a ), both objects are located from the 9th z - layer to the 13th z - layer ( about 20 mm in length ). in case ( b ), one is located from the 9th layer to the 13th layer ( about 20 mm in length ) and another is located from 17th layer to the 20th layer ( about 16 mm in length ). the w is calculated using green &# 39 ; s function method based on the diffusion equation . for absorbing coefficients , it is given by the following equation : ## equ7 ## where g 2d 0 is 2d green &# 39 ; s function for reference , d is the diffusion constant and c is the speed of light in the medium . scattering objects with different l t can be also calculated as mentioned above for 2d case . the background transport mean free path is chosen as l t 0 = 2 . 5 mm , and the absorption length l a 0 = 500 mm . the absorbing objects have l a = 2 . 5 mm . in making fourier transform and computing inverse matrix w 2d ( k ) t w 2d ( k )+ λ ( k )! - 1 , the regularization matrix is set as λ ( k )=( λ + βk 2 ) i , with λ and β are parameters for regularization . in this calculation λ = 10 2 and β = 0 are taken . the images of the absorption objects are separately shown in fig9 ( a ) and fig9 ( b ), which correspond to fig8 ( a ) and fig8 ( b ), respectively . in fig9 ( a ) and 9 ( b ), index k is the index of layer in z coordinate . the images of hidden objects appear in the right layers where the objects are arranged , with a resolution of about 4 mm . there are some aliased images appearing near the z - end because of discrete sampling . the following comments , observations , objects , features , uses , applications and / or advantages may be made about the present invention : ( 1 ) temporal intensities extracted at multiple time slices of scattered light profile measured using multiple source detector pairs around turbid media are used as input data for the inversion to obtain the image maps of the media . ( 2 ) an image reconstruction algorithm for imaging highly scattering turbid media is developed . the inversion procedure consists of input scattered light intensity data at different time slices , physical modeling of light propagation in random media , inversion computation algorithm , reconstructed image of spatial distributions of key optical parameters of random media , and image displays . ( 3 ) using different theoretical models for describing photon migration in random media for imaging . the theory includes transport theory for photon migration , and its approximations , such as the diffusion theory and the telegraph equations . ( 4 ) key optical parameters , such as the absorption coefficient , the index of refraction , the transport scattering length , the scattering length , and the diffusion constant , are mapped for imaging the internal structure of the turbid media at different light wavelengths . a difference in both absorption and scattering can be imaged to form a map of the internal structure of the turbid media in 3 - d . ( 5 ) a fast inversion algorithm is developed based on shaw &# 39 ; s principle , eq . ( 4 ), in which the weight function matrix , w ( eq . ( 3 )), which relates a change in the properties of turbid media to the changes in measured light intensities , can be pre - computed and stored for fast image inversion for each patient . ( 6 ) a cumulant formula is developed to relate the experimentally measured intensities to the prediction from theoretical models based on properties of the turbid media . ( 7 ) 3 - d images of spatial distribution of key optical parameters of turbid media can be obtained using this algorithm . 2 - d slices of tomographic images can be obtained by taking a section of a 3 - d image and displaying it to highlight the location of tumors . ( 8 ) use the algorithm to image non - invasively tumors and abnormality in human body , such as tumor growths in breasts , tumors in brain , cerebral hemorrhage in brain , hemorrhage in internal organs , prostate . ( 9 ) using the algorithm with temporal data at various nir wavelengths from 700 to 1500 nm for image reconstruction for monitoring metabolic states of a given part of human body in vivo . ( 10 ) this novel inverse reconstruction method can be combined with shadowgram type of image techniques to further improve spatial resolution and enhance the presence of the abnormality . ( 11 ) use single source optical fiber and multiple detector fibers for collecting temporal profiles of the scattered pulses . ( 12 ) use multiple input source fibers and multiple detector fibers for collecting temporal profiles of the scattered pulses . a time sharing beam deflector will be incorporated to switch the input from one source fiber to another . ( 13 ) multiple s - d optical fibers are mounted on bra - like or hat - like holders surrounding a breast or a head ( brain ) to detect temporal intensity profiles of scattered pulses from breast or head ( brain ) for imaging the internal structure of optical properties . ( 14 ) the lasers used for imaging are cr 4 + forsterite , cr 4 + yag , semiconductor lasers , nd : yag , and / or ti : sapphire lasers . the laser pulses at different wavelengths will be coupled to optical fiber by a time sharing scheme to deliver to the scattering media for spectral imaging . ( 15 ) temporal intensity profiles of scattered pulses around scattering media are measured using time resolved detection methods , including streak camera , ultrafast pin diodes , fast photomultipliers . ( 16 ) reconstructed images are to be displayed in 3 - d or in 2 - d tomography slices . ( 17 ) manipulating images obtained from claim 9 at different wavelengths to construct difference , sum , ratio and / or sum of ratio image maps at two wavelengths or several wavelengths to highlight tumors and to differentiate benign from malignant tumors . the embodiments of the present invention described above are intended to be merely exemplary and those skilled in the art shall be able to make numerous variations and modifications to it without departing from the spirit of the present invention . all such variations and modifications are intended to be within the scope of the present invention as defined in the appended claims .
a method for imaging objects in highly scattering turbid media . according to one embodiment of the invention , the method involves using a plurality of intersecting source / detectors sets and time - resolving equipment to generate a plurality of time - resolved intensity curves for the diffusive component of light emergent from the medium . for each of the curves , the intensities at a plurality of times are then inputted into the following inverse reconstruction algorithm to form an image of the medium : ## equ1 ## wherein w is a matrix relating output at source and detector positions r s and r d , at time t , to position r , λ is a regularization matrix , chosen for convenience to be diagonal , but selected in a way related to the ratio of the noise , & lt ; nn & gt ; to fluctuations in the absorption x j that we are trying to determine : λ . sub . ij = λ . sub . j δ . sub . ij with λ . sub . j =& lt ; nn & gt ;/& lt ; δxjδxj & gt ; y is the data collected at the detectors , and x k is the kth iterate toward the desired absoption information . an algorithm , which combines a two dimensional matrix inversion with a one - dimensional fourier transform inversion is used to obtain images of three dimensional hidden objects in turbid scattering media .