Patent Application: US-201313760790-A

Abstract:
an intensity image is collected at each of a plurality of locations spaced apart in a propagation direction of a light beam . information from the intensity images is combined using a kalman filter which assumes that at least one co - variance matrix has a diagonal form . this leads to considerable reduction in computational complexity . an augmented kalman filter model is used in place of the standard kalman filter model . the augmented kalman filter improves the robustness to noise .

Description:
a flow chart of the embodiment is shown in fig2 . step 11 is to set up experimental conditions , such as those in fig1 , to collect a series of intensity images in different parallel planes . the images are m 1 × m 2 pixels . step 12 is to use the parameters of the experimental set - up to generate a physical model describing it . these steps are the same as used , for example , in [ 1 ], as described above . the remaining steps employ a novel augmented state space model . in step 13 parameters of the model are initialised . in step 14 , the data for a first of the series of images is used to update the model . as described below this makes use of a novel phase reconstruction algorithm , assuming a diagonal covariance matrix . step 14 is performed repeatedly for successive ones of the series of images , until all images have been processed . we aim at estimating the 2d complex - field a ( x , y , z 0 ) at the focal plane z 0 , from a sequence of noisy intensity images i ( x , y , z ) captured at various distances z 0 , z 1 , z 2 , . . . , z n . . . , z n . in the following explanation it is assumed , for simplicity , that the focal plane z 0 is at one end of the series of images , but in fact it is straightforward to generalise this to a situation in which it is in the middle of the set of images ( as shown in fig1 ). we assume a linear medium with homogenous refractive index and coherent ( laser ) illumination , such that the complex - field at z 0 fully determines the complex - field at all other planes . the complex optical field at z is a ( x , y , z )=| a ( x , y , z )| e iφ ( x , y , z ) where | a ( x , y , z )| is the intensity , and φ ( x , y , z ) is the phase . propagation is modeled by the homogeneous paraxial wave equation : ∂ a ⁡ ( x , y , z ) ∂ x = ⅈ ⁢ ⁢ λ 4 ⁢ π ⁢ ∇ ⊥ 2 ⁢ a ⁡ ( x , y , z ) ( 1 ) where λ is the wavelength of the illumination , and ∇ ⊥ is the gradient operator in the lateral ( x , y ) dimensions . the noisy measurements i ( x , y , z ) usually adhere to a ( continuous ) poisson distribution : p ⁡ [ i ⁡ ( x , y , z ) | a ⁡ ( x , y , z ) ] = ⅇ - γ ⁢  a ⁡ ( x , y , z )  2 ⁢ ( γ ⁢  a ⁡ ( x , y , z )  2 ) i ⁡ ( x , y , z ) i ⁡ ( x , y , z ) , ( 2 ) where γ is the photon count detected by the camera . the measurement at each pixel i ( x , y , z ) is assumed statistically independent of any other pixel ( conditioned on the optical field a ( x , y , z )). we can discretize the optical field a ( x , y , z ) as a raster - scanned complex column vector a n , and similarly discretize the intensity measurement i ( x , y , z ) as column vector i n . we denote by b ( u , v , z ) the 2 - d fourier transform of a ( x , y , z ). the column vector b n is again raster - scanned from b ( u , v , z ), and hence can be expressed as b n = ka n , where k where is the discrete fourier transform matrix . since k is unitary , we can write kk h = k h k = u ( with normalization ), where u is the identity matrix and k h denotes the hermitian of k . we can define the propagation matrix at z n as [ 15 ]: h n = diag ⁡ ( exp ⁡ [ - ⅈ ⁢ ⁢ λ ⁢ ⁢ π ⁡ ( u 1 2 l x 2 + v 1 2 l y 2 ) ⁢ δ n ⁢ z ] , … ⁢ , exp ⁡ [ - ⅈλ ⁢ ⁢ π ⁡ ( u m 1 2 l x 2 + v m 2 2 l y 2 ) ⁢ δ n ⁢ z ] ) , ( 3 ) where l x and l y are the width and height of the image , respectively . the relation between two images with distance δ n z in the fourier domain can be written as : we approximate the poisson observation ( 2 ) with a gaussian distribution of same mean and covariance . in particular , we consider the approximate observation model : where v is a gaussian vector with zero mean and covariance r = γdiag ( a * n ) diag ( a n ). state ⁢ : ⁢ [ b n b n * ] = [ h n 0 0 h n * ] ⁡ [ b n - 1 b n - 1 * ] ( 7 ) observation ⁢ : ⁢ ⁢ i n = [ j ⁡ ( b n ) ⁢ ⁢ j * ⁡ ( b n ) ] ⁡ [ b n b n * ] + v , ( 8 ) where v is a gaussian variable with zero mean and covariance r , r = γ ⁢ ⁢ diag ⁡ ( a n * ) ⁢ diag ⁡ ( a n ) ⁢ ⁢ and ⁢ ⁢ j ⁡ ( b n ) = 1 2 ⁢ γ ⁢ ⁢ diag ⁡ ( k t ⁢ b n * ) ⁢ k h . ( 9 ) here s n q or s n p are covariance matrices ( s n p is in fact a pseudo - covariance matrix ). from the update equations of acekf [ 1 , 16 ], we have the following steps : 1 . initialize : b 0 , s q 0 and s p 0 . 2 . predict : { circumflex over ( b )} n = hb n - 1 , ŝ n q = hs n - 1 q h h and ŝ n p = hs n - 1 p h h s n q = ŝ n q −( ŝ n q j h + ŝ n p j t )( jŝ n q j h + jŝ n p j t + j *( ŝ n q )* j t + j *( ŝ n p )* j h + r ) − 1 ( jŝ n q + j ( ŝ n p )*) ( 11 ) s n p = ŝ n p −( ŝ n q j h + ŝ n p j t )( jŝ n q j h + jŝ n p j *( ŝ n q )* j t + j *( ŝ n p )* j h + r ) − 1 ( jŝ n p + j ( ŝ n q )*) ( 12 ) g n =( s n q j h + s n p j t ) r − 1 ( 13 ) b n ={ circumflex over ( b )} n + g n ( i n − γ | a n | 2 ) ( 14 ) the size of s n q or s n p , is n 2 , where n is the number of the pixels in the image . the inversion of the covariance matrix has a computational complexity of o ( n 3 ) in each step . both the storage requirement and computational burden make the above update algorithm impractical for real applications . accordingly , the embodiment makes some constraints and derivations as described below , resulting in a low - complexity algorithm with reduced storage requirement . after some derivation , we can get lemma 1 and theorem 1 and 2 . if h is diagonal and the diagonal entries of h are rotationally symmetric in 2 - d , then ehe = h where e = kk t , and k is the discrete fourier transform matrix . let us consider how to initialize the covariance matrix s 0 . first note that a priori one would expect s n q = e [ b n b n h ]= e [ ka n a n h k h ]= ke [ a n a n h ] k h s n p = e [ b n b n t ]= e [ ka n a n t k t ]= ke [ a n a n t ] k t here e [ . . . ] denotes expectation value . it is assumed that in the complex field every pixel is independently poisson distributed , we can assume that e [ a n a n t ] is equal to a scalar times the identity matrix . thus , the covariance matrix can be initialized as : where q 0 and p 0 are a scalar times the identity matrix . e = kk t can be shown to be a permutation matrix , and symmetric . where q n - 1 and p n - 1 are diagonal . the covariance matrix can be updated as follows q n ={ circumflex over ( q )} n −( { circumflex over ( q )} n +{ circumflex over ( p )} n )( { circumflex over ( q )} n +{ circumflex over ( p )} n +( { circumflex over ( q )} n )*+( { circumflex over ( p )} n )*+ qi ) − 1 ( { circumflex over ( q )} n +( { circumflex over ( p )} n )*) ( 19 ) p n ={ circumflex over ( p )} n −( { circumflex over ( q )} n +{ circumflex over ( p )} n )( { circumflex over ( q )} n +{ circumflex over ( p )} n +( { circumflex over ( q )} n )*+( { circumflex over ( p )} n )*+ qi ) − 1 ( { circumflex over ( p )} n +( { circumflex over ( q )} n )*) ( 20 ) note that q n and p n are diagonal . the covariance matrix s n q and s n p has the same form as the covariance s n - 1 q and s n - 1 p . therefore once the first covariance matrix are initialized as s 0 q = q 0 and s 0 p = p 0 e , the other matrices in the following steps has the same form . the proof of theorem 1 requires the assumption that the value of the phase is small so that , defining d by d = 1 2 ⁢ γ ⁢ ⁢ diag ⁡ ( a n * ) it can be approximated that d * d − 1 equals the identity matrix . g n =( s n q j h + s n p j t ) r − 1 =+( q n + p n )( j ) − 1 q ( 23 ) b n ={ circumflex over ( b )} n + g n ( i n − γ | a n | 2 ) ( 24 ) using these results , the algorithm presented above can be reformulated as a sparse augmented complex extended kalman filter algorithm , used by the embodiment : ( i ) initialization of b 0 , q 0 and p 0 . ( ii ) prediction : q n ={ circumflex over ( q )} n −( { circumflex over ( q )} n +{ circumflex over ( p )} n )( { circumflex over ( q )} n +{ circumflex over ( p )} n +( { circumflex over ( q )} n )*+( { circumflex over ( p )} n )*+ qi ) − 1 ( { circumflex over ( q )} n +( { circumflex over ( p )} n )*) ( 29 ) p n ={ circumflex over ( p )} n −( { circumflex over ( q )} n +{ circumflex over ( p )} n )( { circumflex over ( q )} n +{ circumflex over ( p )} n +( { circumflex over ( q )} n )*+( { circumflex over ( p )} n )*+ qi ) − 1 ( { circumflex over ( p )} n +( { circumflex over ( q )} n )*) ( 30 ) b n ={ circumflex over ( b )} n +( q n + p n )( j ) − 1 q ( i n − γ | a n | 2 ) matrices q , and p n are diagonal , hence they can be stored as two vectors . the storage burden of equations ( 11 ), ( 12 ) in the update step is reduced from n 2 to n . the inverse of j in equation ( 31 ) can be computed by a fast fourier transform ( fft ). since q n and p n are diagonal , the matrix multiplications and inversions in equations ( 29 ) and ( 30 ) have a computational complexity of o ( n ). the overall computational complexity of the sparse acekf is at the scale of o ( n z n log ( n )) due to the fft . three sets of data have been considered to assess the performance the augmented kalman filter . data set 1 consists of 100 images of size 100 × 100 pixels artificially generated to simulate a complex field propagating from focus in 0 . 5 μm steps over a distance of 50 μm with illumination wavelength of 532 nm . pixels are corrupted by poisson noise so that , on average , each pixel detects γ = 0 . 998 photons . data set 2 comprises 50 images of size 150 × 150 pixels acquired by a microscope . the wavelength was again 532 nm , and the defocused intensity images were captured by moving the camera axially with a step size of 2 μm over a distance of 100 μm . data set 3 has 101 images of size 492 × 656 pixels acquired by a microscope . the wavelength was 633 nm , and the images were captured by moving the camera axially with a step size of 2 μm . fig3 shows the images of simulated data data set 1 ( fig3 ( a )) and experimental data data set 2 ( fig3 ( b )) and data set 3 ( fig3 ( c )). table 2 summarizes the results of data set 1 using three methods : acekf ( augmented complex extended kalman filter ) [ 1 ], diagonalized cekf ( diagonalized complex extended kalman filter ) [ 8 ], and the method sparse acekf ( sparse augmented complex extended kalman filter ) used in the embodiment . the acekf method has a high computational complexity of o ( n z n 3 ) and storage requirement of o ( n 2 ). in order to alleviate the computational burden of acekf , the images are divided into independent blocks of size 50 × 50 , but it still takes 13562 . 8 seconds by a general personal computer . on the other hand , the computational complexity of the sparse acekf is o ( n z n log n ), and it takes 0 . 40 seconds to process the 100 ( full ) images . fig4 shows recovered intensity and phase images obtained by processing the dataset of fig3 ( a ) by acekf , diagonalized cekf , and the embodiment . the results shown are those at the focal plane . as illustrated in table 2 , the computational complexity of the diagonalized cekf is lower than that of acekf . however , the latter yields better results in terms of phase error . in order to reduce the error of the diagonalized cekf , forward and backward sweeps ( iterations ) are applied in [ 8 ]. however , the iteration increases the computational complexity linearly , and makes the method no longer recursive . the sparse acekf method has an intensity error of 0 . 0071 , and a phase error of 0 . 0143 ( radian ). compared with the diagonalized cekf , the sparse acekf has the same computational complexity and storage requirement , but returns lower error images . the error is here calculated by root mean square error ( mse ). however , mse might not be optimal to evaluate the error . the proposed sparse acekf has an error near to acekf , while the recovered phase and intensity images of the sparse acekf in fig4 might look better . the images recovered by acekf exhibit a block effect as straight lines crossing the images , whereas the result of sparse acekf is free of this block effect . it is because the sparse acekf has a much lower complexity that the embodiment avoids the need to divide the images into independent blocks . the images recovered by acekf and the diagonalized cekf contain traces of phase in the intensity images . however , the trace of phase is almost removed on the estimated intensity image of the sparse acekf . fig5 compares the estimated intensity image and phase image of data set 2 using acekf , the diagonalized cekf , and the sparse acekf . stripes in the phase image recovered by the diagonalized cekf look darker , while the strips in the recovered phase image of the sparse acekf method have stronger contrast . all images are as at the focal plane . fig6 ( a ) shows the recovered phase [ nm ] of the data set 3 by acekf , the diagonalized cekf , and the sparse acekf . the real depth of the sample in data set 3 is around 75 nm ± 5 nm . the proposed embodiment takes 20 . 24 seconds to process 101 images of size 492 × 656 . however , the acekf method takes 54 . 15 hours and each image is separated into 117 pieces of 50 × 50 blocks . fig6 ( b ) compares the depth along the black line in fig6 ( a ). the sparse acekf method shows a result much closer to the true value , compared to acekf and the diagonalized cekf . there are other state space models for which the concept of a diagonal covariance matrix can be applied . for example , based on ( 4 )-( 6 ) we have a state space model : we can define the follow three steps using a standard kalman filter [ 16 ]: ( 1 ) initialization : b 0 and error covariance matrix , m 0 . ( 2 ) prediction : { circumflex over ( b )} n = hb n - 1 ; { circumflex over ( m )} n = hm n - 1 h h ( 3 ) update : g n ={ circumflex over ( m )} n j h ( j { circumflex over ( m )} n j h + r ) − 1 b n ={ circumflex over ( b )} n + g n ( i n − j { circumflex over ( b )} n ) it can be shown that provided m 0 is initialized with a diagonal covariance matrix ( specifically m 0 is a scalar times u ), the state covariance matrix for all n is diagonal . in this case the update procedure becomes simply : the inverse of j can be computed efficiently by means of a fast fourier transform ( fft ) algorithm . both the embodiment described in the previous sections , and this variation , are low - complexity algorithms . as compared the embodiment , this variation takes more iterations to converge , but it has the advantage of being more stable . the method could efficiently recover phase and amplitude from a series of noisy defocused images . it is recursive , and feasible for the real time application . the phase from intensity techniques could find applications in areas such as biology and surface profiling . due to the scalability of the wave equations and the simplicity of the measurement technique , this method could find use in phase imaging beyond optical wavelengths ( for example , x - ray or neutron imaging ), where high - quality images are difficult to obtain and noise is significant and unavoidable . digital holographic microscopy ( dhm ) has been successfully applied in a range of application areas [ 5 ]. however , due to dhm &# 39 ; s capability of non - invasively visualizing and quantifying biological tissue , biomedical applications have received most attention . wave propagation based methods , and the proposed method in particular , may be applied to the same range of applications . examples of biomedical applications are [ 5 ]: label - free cell counting in adherent cell cultures . phase imaging makes it possible to perform cell counting and to measure cell viability directly in the cell culture chamber . today , the most commonly used cell counting methods , hemocytometer or coulter counter , only work with cells that are in suspension . label - free viability analysis of adherent cell cultures . phase imaging has been used to study the apoptotic process in different cell types . the refractive index changes taking place during the apoptotic process are easily measured through phase imaging . label - free cell cycle analysis . the phase shift induced by cells has been shown to be correlated to the cell dry mass . the cell dry mass can be combined with other parameters obtainable by phase imaging , such as cell volume and refractive index , to provide a better understanding of the cell cycle . label - free morphology analysis of cells . phase imaging has been used in different contexts to study cell morphology using neither staining nor labeling . this can be used to follow processes such as the differentiation process where cell characteristics change . phase imaging has also been used for automated plant stem cell monitoring , and made it possible to distinguish between two types of stem cells by measuring morphological parameters . label free nerve cell studies . phase imaging makes it possible to study undisturbed processes in nerve cells as no labeling is required . the swelling and shape changing of nerve cells caused by cellular imbalance was easily studied . label - free high content analysis . fluorescent high content analysis / screening has several drawbacks . label - free alternatives based on phase shift images have therefore been proposed . the capability of phase imaging to obtain phase shift images rapidly over large areas opens up new possibilities of very rapid quantitative characterization of the cell cycle and the effects of specific pharmacological agents . red blood cell analysis . phase shift images have been used to study red blood cell dynamics . red blood cell volume and hemoglobin concentration has been measured by combining information from absorption and phase shift images to facilitate complete blood cell count by phase imaging . it has furthermore been shown that phase shift information discriminates immature red blood cells from mature , facilitating unstained reticulocyte count . flow cytometry and particle tracking and characterization . phase images are calculated from the recorded intensity images at any time after the actual recording and at any given focal plane . by combining several images calculated from the same intensity images , but at different focal planes , an increased depth of field may be obtained , which is vastly superior to what can be achieved with traditional light microscopy . the increased depth of field makes it possible to image and characterize the morphology of cells and particles while in suspension . observations may be done directly in a microfluidic channel or statically in an observation chamber . time - lapse microscopy of cell division and migration . the autofocus and phase shift imaging capabilities of dhm and the proposed method makes it possible to effortlessly create label - free and quantifiable time - lapse video clips of unstained cells for cell migration studies . tomography studies . phase imaging allows for label - free and quantifiable analysis of subcellular motion deep in living tissue . l . waller , m . tsang , s . ponda , s . yang , and g . barbastathis , “ phase and amplitude imaging from noisy images by kalman filtering ,” optics express 19 , 2805 - 2814 ( 2011 ). r . paxman , t . schulz , and j . fienup . “ joint estimation of object and aberrations by using phase diversity ”, j . opt . soc . am . a , 9 ( 7 ): 1072 - 1085 , 1992 . j . m . huntley , “ phase unwrapping : problems and approaches ”, proc . fasig , fringe analysis 94 . york university , 391 - 393 , 1994a . m . takeda , “ recent progress in phase - unwrapping techniques ”, proc . spie , 2782 : 334 - 343 , 1996 . zhong jingshan , justin dauwels , manuel a . vazquez , laura waller . “ efficient gaussian inference algorithms for phase imaging ”, proc . ieee icassp , 617 - 620 , 2012 . r . gerchberg and w . saxton , “ a practical algorithm for the determination of phase from image and diffraction plane picture ”, optik , 35 : 273 - 246 , 1972 . j . fienup , “ phase retrieval algorithms : a comparison ”, appl . opt ., 21 , 1982 . m . teague , “ deterministic phase retrieval : a green &# 39 ; 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