Total Variation-Based Reduction of Streak Artifacts, Ring Artifacts and Noise in 3D Reconstruction from Optical Projection Tomography preprint
- 1. Microsc. Microanal., page 1 of 14 1 doi:10.1017/S1431927615015226 2 © MICROSCOPY SOCIETY OF AMERICA 2015 3 Total Variation-Based Reduction of Streak Artifacts, 4 Ring Artifacts and Noise in 3D Reconstruction from 5 Optical Projection Tomography 6 Jan Michálek* 7 Department of Biomathematics, Institute of Physiology of the Czech Academy of Sciences, Videnska 1083, 14220 Prague 4, 8 Czech Republic 9 Abstract: Optical projection tomography (OPT) is a computed tomography technique at optical frequencies for 10 samples of 0.5–15 mm in size, which fills an important “imaging gap” between confocal microscopy (for smaller 11 samples) and large-sample methods such as fluorescence molecular tomography or micro magnetic resonance 12 imaging. OPT operates in either fluorescence or transmission mode. Two-dimensional (2D) projections are taken 13 over 360° with a fixed rotational increment around the vertical axis. Standard 3D reconstruction from 2D OPT 14 uses the filtered backprojection (FBP) algorithm based on the Radon transform. FBP approximates the inverse 15 Radon transform using a ramp filter that spreads reconstructed pixels to neighbor pixels thus producing streak 16 and other types of artifacts, as well as noise. Artifacts increase the variation of grayscale values in the reconstructed 17 images. We present an algorithm that improves the quality of reconstruction even for a low number of projections 18 by simultaneously minimizing the sum of absolute brightness changes in the reconstructed volume (the total 19 variation) and the error between measured and reconstructed data. We demonstrate the efficiency of the method 20 on real biological data acquired on a dedicated OPT device. 21 Key words: optical projection tomography, microscopy, artifacts, total variation, data mismatch 22 23 INTRODUCTION 24 Optical projection tomography (OPT) is a recently (Sharpe 25 et al., 2002) developed implementation of computed tomo- 26 graphy (CT) techniques at optical frequencies. A series of 27 two-dimensional (2D) optical projections through a sample 28 are generated at varying orientations (Fig. 1) from which the 29 3D structure of the sample can be computationally recovered 30 (Bassi et al., 2011). 31 In transmission mode, a collimated light source is used 32 to transmit a parallel beam through the sample to acquire 33 projections at the desired wavelength. Images are recorded 34 on a CCD camera throughout a full 360° rotation. Using 35 computer software the original 3D information is subsequently 36 recalculated. 37 In fluorescence mode, optical tomography projections 38 can be obtained either by recording the autofluorescence 39 emitted by the tissue or by using fluorescent antibody labeled 40 specimens embedded in agarose and made semitransparent 41 in an organic solvent (typically a mixture of benzyl alcohol 42 and benzyl benzoate, BABB). The specimen is exposed to 43 appropriate excitation light and the emitted light is captured 44 from a number of angular positions on the CCD chip of 45 the camera. 46 OPT is suitable for many important model organisms, 47 e.g. insects, animal embryos, or small animal extremities, 48which are too large for techniques such as confocal imaging 49(CLSM), and too small for large-sample methods such as 50fluorescence molecular tomography, X-ray CT or micro 51magnetic resonance imaging (μMRI). OPT covers the range 52of sample sizes from about 0.5 to 15 mm, and thus fills the 53“imaging gap” between CLSM and μMRI. OPT allows 54acquisition of 3D data with proper morphological and spatial 55information without the need to cut the specimen and 56without deformations introduced by cutting. A disadvantage 57of OPT is that its resolution is inferior to that of CLSM 58(CLSM: ~200 nm/pixel; OPT: >2 µm/pixel). 59In 3D reconstruction from tomography projections, a 60slice through the specimen at some height z is reconstructed 61computationally from projections of the rotating specimen 62over a range of angles. The mathematical problem of 63reconstruction from a finite number of projections may be 64underdetermined: if we acquire, e.g., OPT projections from 65400 directions with 512 pixels in each projection, we get 66204,800 measured values. To reconstruct the slice, brightness 67in 512 × 512 = 262,144 pixels needs to be computed, i.e. we 68have 262,144 unknowns, but only 204,800 projected points 69to calculate from. If we represent the Radon transform in 70matrix form R ´ u = b; (1) 71where R is the Radon projection matrix, b the known 204,800 72projection values (sinogram), and u the 262,144 unknown 73pixels, then the equation system is underdetermined and can 74be satisfied by infinitely many solutions u for the sought slice,*Corresponding author. michalek@biomed.cas.cz Received April 30, 2015; accepted September 3, 2015
- 2. 75 many of which do not match the original. Mismatching areas 76 often exhibit conspicuous patterns called artifacts and are 77 visually unacceptable. 78 The standard method for 3D tomographic reconstruc- 79 tion from OPT series is the filtered backprojection (FBP) 80 algorithm, most widely used in other types of parallel beam 81 CT, e.g. X-ray CT. FBP is an approximation of the inverse 82 Radon transform, the theory of which was first published 83 in (Radon, 1917). The FBP tackles the problem of under- 84 determinacy of (1) by applying a ramp filter to acquired data 85 before backprojection. The ramp filter essentially substitutes 86 the missing data by spreading reconstructed pixel values over 87 neighbor points. In FBP-reconstructed images one can 88 notice artifacts, such as: 89 ∙ streak artifacts (Figs. 2e and 3e): fan-shaped streaks in the 90 direction of backprojection, centered at the axis of rotation 91 of the specimen and introduced by the ramp-filter 92 blurring; 93 ∙ ring artifacts centered at the rotation axis (Figs. 2e and 3e) 94 caused by miscalibrated detector elements and not specific 95 only to FBP; 96 ∙ noise (Fig. 3e) obviously does not correspond to any biological 97 structures of the specimen (also not specific to FBP). 98 Though theoretical papers on improving the quality of 99 tomography reconstructions are abundant, examples based 100 on genuine biological data are scarce. Efficacy is usually 101 demonstrated only for artificial, digitized or physical, 102 phantoms, satisfying idealized assumptions that seldom 103 hold for real biological data, such as sparsity (i.e., low 104 number of nonzero values) either of the image (most pixels 105 black) or of its gradient (most image areas are constant such 106 as the well-known Shepp–Logan phantom). In addition, 107 tomography projections are often computer-simulated 108 rather than acquired on a real CT scanner. Since our goal is 109 to reduce artifacts in real-life tomography reconstruction, 110 papers assuming some kind of sparsity or presenting only 111phantom reconstructions were not included in the following 112state-of-the-art review. 113Reports on reducing artifacts in tomography recon- 114structions from genuine biological data are relatively rare. 115Bruyant et al. (2000) proposed to generate additional 116projections by computational means to reduce the streak 117artifact. They introduced a postacquisition process called 118interpolation of projections by contouring, which creates 119new pseudoprojections by interpolating measured sinogram 120values on a new, denser, grid containing more angles or/and 121detector bins. In a clinical study, they found that increasing 122the number of angles by interpolation can reduce radial 123streaks, while when they interpolated between the bins the 124improvement was not conclusive. 125Yu et al. (2011) presented a model-based iterative 126reconstruction (MBIR) method using spatially non- 127homogeneous iterative coordinate descent (NH-ICD) 128optimization. MBIR algorithms work by first forming an 129objective function which incorporates an accurate system 130model, statistical noise model, and prior model. The image is 131then reconstructed by computing an estimate that minimizes 132the resulting objective function. They consider the image 133and the data as random vectors, and the goal is to reconstruct 134the image by computing the maximum a posteriori 135estimate using a Taylor series expansion to approximate the 136log-likelihood term by a quadratic function. Voxels of 137the image are updated in a spatially NH-ICD to accelerate 138convergence. In order to speed up convergence, the order of 139voxel updates is determined by a voxel selection criterion 140related to the absolute sum of the update magnitudes at the 141last visit. They compared axial slices reconstructed from a 142512 × 512 abdomen scan using FBP and their MBIR NH-ICB 143algorithm. Some of the streak artifacts in the FBP recon- 144struction were no longer visible in the ICB reconstruction. 145When CT projections are acquired at a small number of 146views, the system may become severely underdetermined, 147and analytic methods such as FBP may yield reconstructions 148with considerable aliasing artifacts such as sharp streaks. 149Moreover, data measured in real experiments are con- 150taminated by various physical factors such as noise and 151scatter, and thus they may contain components that are 152inconsistent with the discrete imaging model. Han 153et al. (2011) investigated low-dose micro-CT and its 154application to specimen imaging from substantially reduced 155projection data by using an algorithm referred to 156as the adaptive-steepest-descent-projection-onto-convex-sets 157(ASD-POCS) which reconstructs an image through mini- 158mizing the total variation (TV) of the image and enforcing 159data constraints. The ASD-POCS algorithm minimizes the 160TV of the estimated image subject to data condition and 161nonnegativity constraints: u* = arg min uk kTV s:t: Ru - bk k2 ≤ ε and u ≥ 0; 162where uk kTV, referred to as the image TV, denotes the norm 163of the discrete gradient magnitude of the image, and 164Ru - bk k2 indicates the Euclidean distance between measured 165data and data estimated from the reconstructed image. Figure 1. Photograph of an optical projection tomography system built by Politecnico di Milano (Dipartimento di Fisica) for imaging cleared tissue samples. Telecentric optics is used in order to keep constant magnification throughout the specimen. The white LED is used for transmission optical projection tomography (OPT). Alternatively, the 470 nm LED can be used as excitation light source for the fluorescence OPT. 2 Jan Michálek
- 3. 166 A tolerance parameter ε is introduced to relax the requirement 167 on data distance. The ASD-POCS iterations alternatingly 168 employ the steepest descent (SD) for minimizing the image 169 TV, and POCS methods for minimizing the data distance. 170 Both minimizations are carried out with respect to the sought 171 image u. Both SD and POCS steps are followed by a nonlinear 172 projection operation that enforces the positivity constraint by 173 setting all negative image voxels to zero. Comparison of the 174 ASD-POCS, FDK (Feldkamp, Davis, and Kress algorithm) 175 and POCS (without TV) reconstruction methods applied to CT 176 projections of a porcine heart and kidney specimens suggests 177 that the ASD-POCS algorithm can effectively suppress streak 178 artifacts and noise that were observed in tomography slices 179 reconstructed with FDK and POCS algorithms. 180 Park et al. (2012) proposed a low-dose cone-beam CT 181 using a minimal number of noisy projections. Tomography 182 reconstruction is obtained as the result of constrained 183 minimization with respect to u of the function f ðuÞ = TVðuÞ + λ Ru - bk k2 2 s:t: u ≥ 0: 184 Minimization is done in the gradient direction, but with a 185 special choice of the step size. They use an approximate 186second-order solver, proposed by Barzilai and Borwein. In 187their GP-BB approach, the step size is chosen based on both 188the current and the previous gradient which could result in a 189nonmonotonic, but faster, convergence. For 364 projections 190with 1,024 bins, their algorithm takes about 234 s when 191implemented on a graphic processing unit (GPU). The 192authors report that minimizing the TV-enhanced problem 193results in a visually better quality image than that of the FDK 194(i.e., less noise, streaking artifacts around bones, etc.). 195Leary et al. (2013) have followed a Fourier-based 196approach to electron tomography (ET) reconstruction. 197They found that, for application to ET data sets, where the 198aim is to reconstruct the 3D density of the sample and the 199sample is expected to be approximately piecewise constant, 200simultaneous enforcement of sparsity in both the image and 201gradient domains yields the highest fidelity reconstructions. 202They minimize the weighted sum f ðuÞ = λTVTVðuÞ + λl uk k1 + λ Φu - bk k2 2 Φ::sensing matrix; 203between the data fidelity term and the sparsity term which is 204a blend of TV and nonzero image values evaluated by uk k1. Figure 2. a–d: Four out of 400 optical projection tomography projections from different angles of a muscle specimen of a young pig with nerve fascicles (n. medianus); benzyl alcohol/benzyl benzoate clearing, acquired in transmission mode on our “Milano” scanner with 400 projections 1,004 × 1,002 pixels in size, (e) FBP reconstruction from 400 projections exhibits streak and ring artifacts. Concentric fan-shaped streaks are caused by the way filtered back- projection (FBP) approximates the Radon transform, and by having less measured data than pixels to be reconstructed (400 projections × 1,004 pixels = 401,600 measured values to reconstruct 710 × 710 = 504,100 pixels of the slice). The arrowhead points to the most pronounced ring artifact. Ring artifacts around the center of rotation can be caused by miscalibrated detector elements or by misalignment between the true axis of rotation during data acquisition, and the axis assumed by the reconstruction algorithm, (f) total variation (TV) reconstruction from 400 projections. Reconstruction in Optical Projection Tomography 3
- 4. 205 They first Fourier transform the projection images to obtain 206 radial samples of the object in the Fourier domain. This data is 207 then Fourier transformed into the image domain using the 208 nonuniform fast Fourier transform (NUFFT). In conjunction 209 with the NUFFT, for the sparsity-seeking optimization process 210 they have used a conjugate gradient descent algorithm. 211 Niu et al. (2014) start likewise from the TV minimiza- 212 tion framework with a data fidelity constraint u* = arg min uk kTV s:t: 0:5 Ru - bk k2 2 ≤ ε and u ≥ 0; 213 and convert this formulation into minimization of the loga- 214 rithmic barrier function u* = arg min uk kTV - logðε - 0:5 Ru - bk k2 2 ÁÂ Ã s:t:u ≥ 0: 215 Here, the vector b represents the line integral measurements 216 (i.e., after the logarithmic operation on the raw projections), 217 R is the system matrix modeling the forward projection, 218 u the vectorized patient image to be reconstructed, Ru - bk k2 2 219 calculates the L2 norm in the projection space, and uk kTV the 220 TV term defined as the L1 norm of the spatial gradient 221 image. The user-defined parameter ε is an estimate of total 222 measurement errors. The authors propose to minimize the 223 logarithmic barrier function using gradient-based algorithms. 224 The reconstructed images have a size of 512×512 pixels. For 225 tomography reconstruction of one slice from a head-and-neck 226 patient study, Niu et al. (2014) report significant reduction 227of the artifacts from few-view reconstruction using their new 228method in comparison to standard FBP reconstruction. 229The reason why TV minimization is efficient in reducing 230artifacts is that it enforces smoothing of small fluctuations in 231reconstructed slices, yet a large sudden change in image 232brightness is not penalized more than a slow continuous 233brightness change. Thus, small variations like streaks or ring 234artifacts are smoothed, but sharp edges in the specimen remain 235untouched. Figure 4 illustrates this property. 236 MATERIALS AND METHODS 237Tomography Reconstruction through Minimization 238of a Weighted Sum of TV and the Data Mismatch 239Methods of artifact reduction due to Han et al. (2011) and 240Niu et al. (2014) minimize the TV of the reconstructed image u* = arg min uk kTV; 241under the relaxed constraint on the data fidelity 0:5 Au - bk k2 2 ≤ ε; 242and constraints on data positiveness u ≥ 0: 243Such constrained minimization of the TV yields excellent 244tomography reconstruction in cases where the magnitude of the image gradient is nonzero only in a very low number of Figure 3. a–d: Sample optical projection tomography (OPT) projections from four different angles of a 3 × 3 × 3 mm3 block of a rat brain stained in vivo by biotinylated Lycopersicon esculentum (tomato lectin), and cleared using the benzyl alcohol/benzyl benzoate protocol. Data acquisition was done on a commercial OPT scanner Bioptonics 3001 with projections 512 × 512 pixels in size, in fluorescence mode with excitation wavelength 425/40 nm (band pass), and emission wavelength from 475 nm (high pass), (e) noise and ring artifacts in filtered backprojection (FBP) reconstruc- tion from 400 projections, the arrowhead points to the center of the rings, (f) total variation (TV) reconstruction from 400 projections removes the noise-like texture from the reconstruction and brings forward fine details that were hidden by the artifacts in the FBP reconstruction. 4 Jan Michálek
- 5. 245 locations, i.e. is sparse in the terminology of compressed sensing 246 (Candès et al., 2006). Sparsity of the gradient means that the 247 image consists of large regions with constant grayscale values. 248 The well-known Shepp–Logan phantom in Figure 5a is a 249 typical example of a synthetic tomography specimen with 250 sparse gradient. 251 We developed our own algorithm for tomography 252 reconstruction by constrained TV minimization, and tested 253 it using software projections of the 512 × 512 Shepp–Logan 254 phantom. The projections were taken between 0° and 180° 255 with an angle increment of 2°. This yielded 91 projections 256 with 724 bins each, which totals to 724 × 91 = 65,884 known 257 data points. Tomography reconstruction had to recover 258 512 × 512 = 262,144 unknown pixels, i.e. we had only 259 25.133% of the data needed to solve uniquely the underlying 260 equations. Since the Shepp–Logan phantom is piecewise 261 constant, its gradient is sparse, and the constrained TV- 262 based reconstruction (Fig. 5b) matches almost perfectly the 263 original, confirmed also by the profile in Figure 5f. FBP 264reconstruction contains many artifacts (Figs. 5c–5g). 265A justified question arises if the same artifact reduction as 266with TV could be achieved by low-pass filtering of the 267FBP-reconstructed slice, which is much faster than TV 268minimization. The answer is it could not: Figures 5d and 5h 269show the result of FBP filtered with a 10 × 10 pixel Gaussian 270filter with σ = 3. Streaks are visibly reduced, but sharp 271transitions between regions of constant intensity have been 272smeared. This confirms that TV-enhanced tomography 273reconstruction preserves sharp edges, while low-pass filter- 274ing of the reconstructed image does not, even though it also 275reduces the artifacts to some extent. 276The assumption of gradient sparsity rarely holds true for 277biological specimens. We attempted constrained TV-based 278reconstruction from projections of biological tissue with 279slowly varying optical density, i.e. with large areas of nonzero 280gradient. Constrained TV-based reconstruction yielded 281implausible images with watercolor-like large patches of 282constant brightness. Therefore, we decided to replace TV total variation total variation a b Figure 4. Total variation does not penalize discontinuities. Example of a discontinuous and a continuous function whose total variation is the same: (a) discontinuous function and its total variation, (b) continuous function and its total variation. Figure 5. Tomography reconstruction from 91 projections of the 512 × 512 Shepp–Logan phantom. Upper row: (a) the phantom, (b) constrained total variation reconstruction, (c) classical filtered backprojection (FBP) reconstruction, (d) low-pass filtered FBP reconstruction, 10 × 10 pixel Gaussian filter with σ = 3. Lower row: (e–h) brightness profiles along the central horizontal line in the images of the first row. Reconstruction in Optical Projection Tomography 5
- 6. 283 minimization under the constraint of a perfect data match 284 with the minimization of a weighted sum of TV and the data 285 mismatch. In the minimization of a weighted sum of TV and 286 the data mismatch, our aim is to find a reconstructed image 287 ur which minimizes the function FðuÞ = TVðuÞ + μ 2 Ru - bk k2 2; 288 i.e. ur = arg min u TVðuÞ + μ 2 Ru - bk k2 2 |fflfflfflfflffl{zfflfflfflfflffl} L2 2 6 4 3 7 5: (3) 289 This problem statement differs from that of Park et al. 290 (2012) only in the absence of the constraint u ≥ 0. However, 291 since Niu et al. (2014) report stability problems of the 292 Barzilai–Borwein gradient algorithm used by Park et al. 293 (2012), we developed our own algorithm based on variable 294 splitting and alternating direction method of multipliers 295 (ADMM) for the minimization in (3). In the following, 296 we will use the shorthand TV-L2 for this algorithm. The 297 algorithm is different from all algorithms reviewed in this 298 subsection. 299 The Algorithm 300 The algorithm is derived in the Appendix. We present here 301 only the resulting steps. The meaning of the math symbols is 302 explained in Table 1. 303 Using the procedure of variable splitting described in the 304 Appendix, the minimization of the total-variation regular- 305 ized mismatch between the measured projections and 306 reprojected tomography slices in (3) can be replaced by 307 the minimization of the augmented Lagrangian, in which 308 the function LAðw; uÞ = wk k1 + β 2 Du - w - ck k2 2 + μ 2 Ru - bk k2 2; 309 is minimized iteratively by alternating minimizations with 310 respect to u and w (ADMM). The algorithm operates in three 311 phases: 312 Step 1. Keeping w, c constant, do one optimal descent step in 313 the gradient direction of LA(w, u) 314 ∙ gradient: gk = βDT (Duk − wk − ck ) + μRT (Ruk − b) 315 ∙ optimum step size: αk = (gkT gk )/(gkT Agk ), where 316 A = βDT D + μRT R 317 ∙ SD step: uk + 1 = uk − αk gk 318 Step 2. Keeping u, c constant minimize LA(w, u) with respect 319 to w: 320 ∙ min w LAðw; uk + 1 Þ )wk + 1 p = max Dpuk + 1 - ck p - 1 β ;0 n o 321 ´ sgn Dpuk + 1 - ck p for all pixels p, where all the operations 322 are done componentwise 323 Step 3. Update the vector of Lagrangian multipliers c: 324 ∙ ck + 1 = ck − (Duk + 1 − wk + 1 ) 325 until a termination criterion is satisfied. 326Relationship between our Method and the 327Compressed Sensing Approach 328In its pure form, image reconstruction based on compressive 329sensing assumes that there exists a representation of 330the image in form of a linear combination of some basis 331functions such that the number of the basis functions needed 332to represent the image is much lower than the number of 333pixels of the image (Candès and Wakin, 2008; Romberg, 3342008). The image is reconstructed so as to minimize the 335number of needed basis elements under the constraint of 336data fidelity, which in case of computer tomography means 337that Radon reprojection of the reconstructed image matches 338the measured Radon projections. This is the approach Table 1. List of Math Symbols. F(u) Weighted sum of total variation and the data mismatch TV Total variation u The unknown image (a slice in the three-dimensional volume) to be reconstructed (N-vector) ur Minimizer of F(u) (the reconstructed slice) R Radon projection matrix μ/2 Weight of the data mismatch term b Measured values of parallel Radon tomography projections ν 2N-vector of Lagrange multipliers c 2N-vector of scaled Lagrange multipliers w Auxiliary variable (2N-vector) Dp 2 × N matrix for calculation of 1st order horizontal/ vertical differences at the pixel p Dpu Discrete gradient (a 2 × 1 vector) at the pixel p cp = c1 p c2 p # Scaled Lagrange multipliers (horizontal/vertical) at the pixel p corresponding to discrete gradient at p, Dpu D 2N × N matrix formed of first and second components of all N rows of all matrices Dp Du Discrete gradient ordered in a 2N-vector β/2 Weight of the mismatch between the discrete gradient Du and the auxiliary variable w ΛA Augmented Lagrangian LA Scaled augmented Lagrangian m Number of image rows n Number of image columns M Matrix in the unconstrained optimization problem (7) and Theorem 1 N Number of image pixels k Iteration counter Qk (u) Augmented Lagrangian with wk kept constant at the k-th step gk = ∇Qk (uk ) Gradient of Qk (u) at the current image iterate uk (new search direction) αk Optimum step size at the current iterate uk A Positive definite matrix computed from D and R f, h Functions in the unconstrained optimization problem (7) ηk, λk Summable sequences in Theorem 1 Rn n-dimensional Euclidian space u* Limit of the sequence of iterates uk 6 Jan Michálek
- 7. 339 adopted by Han et al. (2011) and Niu et al. (2014). They 340 minimize the TV of the reconstructed image (which results 341 in minimum number of nonzero elements of the image 342 gradient) while requiring relaxed data fidelity allowing an 343 L2 error less than some ε. 344 In contrast to that, Park et al. (2012) and Leary et al. 345 (2013) minimize a weighted sum between the sparsity term 346 and the data fidelity term, i.e. they trade some of the sparsity 347 for data fidelity. Although they claim to promote sparsity 348 (Park in the sense of TV minimization, Leary in the sense of a 349 mixture of TV and nonzero image values), this differs from 350 the notion of compressed sensing by Romberg (2008), and 351 the sparse solution will, in general, not be found. 352 Although our method formally solves a problem similar 353 to that of Park et al. (2012), we do not assume any form 354 of sparsity of the reconstructed image. In this sense, our 355 method can rather be considered an edge preserving 356 tomography reconstruction with artifact removal. This has 357 a practical consequence: while the weighting of the sparsity 358 term can be high in cases where the sparsity assumption 359 is justified, in our case it may result in watercolor-like recon- 360 structed slices with unrealistic patches of constant gray values. 361 Specimen Preparation 362 Biological tissue specimens have poor optical transmission 363 characteristics due to light scattering at the refractive index 364 interface between the cell membrane (lipid) and intracellular 365 and extracellular tissue fluids (aqueous). Therefore, a process 366 called optical clearing is essential for OPT. In the process of 367 optical clearing, tissue specimens are made transparent to 368 light by sequential perfusion with fixing, dehydrating, 369 and clearing agents. Perfusion of dehydrated tissue with a 370 solution that has a refractive index similar to that of proteins 371 makes it transparent and the light does not scatter. 372 A standard clearing protocol uses BABB to make tissues 373 nearly transparent. The BABB clearing protocol is detailed, 374 e.g., in Dodt et al. (2007). After fixation, the samples are 375 dehydrated with methanol. Optical clearing of the samples is 376 done using a 1:2 mixture of BABB. However, BABB has the 377 drawback of breaking down fluorescent dyes such as green 378 fluorescent protein (GFP) in the tissues and thus depleting 379 the GFP signal in fluorescent OPT tomography. 380 For that reason, a number of novel clearing protocols 381 have emerged lately. One that we tested was ScaleA2, in which 382 the clearing agent consists of 240 g of urea, 10 mL of Triton 383 X-100, 100 g of glycerol, and water added to 1,000 mL. The 384 resulting transparency of ScaleA2-cleared samples is lower 385 than that of BABB, yet the GFP signal is preserved which 386 allows OPT imaging of small parts of tissues. 387 A new promising method of clearing tissues called 388 CLARITY is described in the study by Tomer et al. (2014). 389 Clarity is a method for chemical transformation of intact 390 biological tissues into a hydrogel-tissue hybrid, which 391 becomes amenable to interrogation with light and macro- 392 molecular labels while retaining fine structure and native 393 biological molecules. CLARITY involves the removal of 394lipids in a stable hydrophilic chemical environment to 395achieve transparency of intact tissue, preservation of 396ultrastructure and fluorescence, and accessibility of native 397biomolecular content to antibody and nucleic acid probes. 398The CLARITY protocol has been successfully applied to the 399study of adult mouse, adult zebrafish, and adult human 400brains. Since the CLARITY protocol was published more 401than a year after we started working on the tomography 402reconstruction method presented here, we have no OPT data 403from CLARITY-prepared samples to present in this 404manuscript. 405Four biological tissue specimens were used for OPT 406reconstructions in this study: 4071. Figure 2 shows 4 out of 400 total tomography projections 408of a muscle specimen of a young pig with nerve fascicles 409(n. medianus). The specimen was optically cleared using 410the BABB clearing protocol. No staining was applied. 4112. Figure 3 shows four sample OPT projections of a 4123 × 3 × 3-mm block of mouse brain. To avoid washing 413out the fluorescence dye during sample preparation, 414BABB clearing was applied only after the specimen was 415perfused in vivo with tomato lectin (Lycopersicon 416esculentum). Thanks to the in vivo perfusion, the staining 417was firmly fixed in the tissue, and we were able to discern 418nicely the inner structures of the brain, especially the 419blood vessels. 4203. The third specimen in Figures 6a to 6d, a cut through the 421heart of a 1-day-old mouse, differed from specimens 1 422and 2 in that the ScaleA2 optical clearing (Hama et al., 4232011), rather than BABB, was used. The specimen was 424stained by GFP. 4254. The fourth specimen in Figures 7a to 7d shows the middle 426part of an earthworm. The specimen was dehydrated in 427methanol and cleared by immersion in BABB. No 428staining was applied. 429 RESULTS 430Tomography Reconstruction from Radon 431Projections Acquired with an OPT Device 432In this subsection, we compare FBP and TV-L2 tomography 433reconstructions from the full length OPT series for the spe- 434cimens depicted in the upper rows of Figures 2, 3, 6, and 7. 435Reconstructed slices are shown in panels (e) for FBP and 436(f) for TV-L2 of the respective figures. 437Figures 2a to 2d show an example of the muscle speci- 438men of a young pig with nerve fascicles (n. medianus) cleared 439using the BABB protocol, without staining. The OPT series 440was acquired in transmission mode illuminated by white 441laser. The OPT device had the resolution of 1,004 × 1,002 442pixels and was built in our lab in cooperation with Poli- 443tecnico di Milano (hereinafter referred to as “Milano” scan- 444ner). The device design is similar to that in Figure 1. TV-L2 445reconstruction from 400 projections in Figure 2f provides 446remarkably better reconstruction than FBP in Figure 2e. Reconstruction in Optical Projection Tomography 7
- 8. Figure 6. a–d: Four out of 1,000 optical projection tomography projections of a cut through the heart of a 1-day-old mouse, ScaleA2 optical clearing, with green fluorescent protein staining, “Milano” scanner, 1,024 × 1,024 pixels, fluorescence mode: excitation 425/40 nm (band pass), emission from 475 nm (high pass), (e) filtered backprojection (FBP) reconstruction with artifacts in form of concentric rays (f) the new TV-L2 reconstruction algorithm removes the fan-like texture from the reconstruction while preserving sharp boundary between the specimen and the background. Figure 7. a–d: Four optical projection tomography (OPT) projections of the middle part of an earthworm, benzyl alcohol/benzyl benzoate clearing, no staining, acquired with the Bioptonics 3001 scanner in fluorescence mode: excitation 425/40 nm (band pass), emission from 475 nm (high pass), (e) filtered backprojection (FBP) reconstruction of the middle section of the sample reconstructed from 400 OPT projections exhibits streak artifacts, (f) TV-L2 reconstruction from 400 projections with streak artifacts largely suppressed. 8 Jan Michálek
- 9. 447 Fan-like streak artifacts have been reduced to a large 448 extent, and the tissue resolution inside the specimen has 449 significantly improved. 450 In Figures 3a to 3d, OPT projections of a 3 × 3 × 3 mm3 451 of a rat brain stained in vivo with tomato lectin and cleared 452 with BABB are shown. Tomography reconstructions from 453 400 projections by the FBP and TV are compared in 454 Figure 3e and f. TV-based reconstruction brings forward fine 455 details almost hidden by FBP artifacts, and removes the 456 noise-like texture from the reconstruction. 457 Figures 6a to 6d show 4 out of 1,000 projections of a cut 458 through the heart of a 1-day-old mouse made transparent by 459 ScaleA2 optical clearing with GFP staining. The OPT series 460 was acquired on our “Milano” scanner resolving 461 1,004 × 1,002 pixels in fluorescence mode. The excitation 462 LED operated at 425/40 nm (band pass), the emission 463 was recorded from 475 nm (high pass). The new TV-L2 464 reconstruction algorithm removes the fan-like artifact 465 texture from the reconstruction, shown in Figure 6f, while 466 preserving fine structures of the specimen and drawing sharp 467 boundary between the specimen and the background. 468 Figure 7 displays OPT projections of the middle part of 469 an earthworm after BABB clearing without staining, 470 acquired with the Bioptonics 3001 scanner in fluorescence 471 mode (excitation 425/40, emission from 475 nm). FBP 472 reconstruction from 400 projections (Fig. 7e) exhibits streak 473 artifacts, which are largely suppressed in TV reconstruction 474 (Fig. 7f). 475 TV-Enhanced Tomography in Reconstruction from 476 a Limited Number of Projections 477 We use the specimen in Figures 3a to 3d (a 3 × 3 × 3 mm3 478 block of a rat brain stained by tomato lectin) to demonstrate 479 the power of TV reconstruction from a limited number of 480 views, 100 projections in this case (Figs. 8a, 8b). Figure 8a 481shows that when the number of tomography projections is 482reduced from 400 to 100, artifacts in FBP reconstruction 483make it almost useless, while TV reconstruction (Fig. 8b) still 484yields acceptable images. 485Comparison between TV-based Tomography 486Reconstruction and Low-Pass Filtered FBP 487Reconstruction 488Image processing practitioners often ask if the objective of 489reducing tomography artifacts could be achieved using faster 490and simpler algorithms than constrained or unconstrained 491TV minimization. For the artificial Shepp–Logan phantom 492with only a finite number of grayscale values we showed in 493Figure 5 that constrained TV reconstruction preserved sharp 494edges in the reconstructed images in Figures 5b and 5f, while 495low-pass filtered FBP reconstruction (Figs. 5d, 5h) blunted 496the signal jumps. 497Figure 9 shows that this is also true for TV-L2 498tomography reconstruction of a real-life tomography series. 499Gauss-filtered reconstruction (Fig. 9a) was generated from a 500conventional FBP reconstruction by Gaussian low-pass 501filtering with σ = 2. Contrary to Gauss filtering, TV 502reconstruction (Fig. 9b) preserves sharp edges and empha- 503sizes fine structures in the slice. Hence, low-pass filtering of 504the FBP is no alternative to TV-based reconstruction. 505Reconstruction Accuracy 506To compare accuracy of our new TV-L2 method and the 507standard FBP reconstruction, Table 2 lists the respective 508values of the TV and the normalized root mean square 509error (NRMSE). For all four tested specimens, the TV-L2 510algorithm achieves much smoother reconstruction as well as 511a better match between the measured Radon projections and 512reprojected final iterates of reconstructed slices. Figure 8. a, b: Tomography reconstruction from a limited number of views (100 projections in this case) of a 3 × 3 × 3 mm3 block of a rat brain stained by tomato lectin, and cleared using the benzyl alcohol/benzyl benzoate protocol: (a) filtered backprojection (FBP) reconstruction from a limited number of views exhibits a disturbing level of artifacts (streak artifacts and noise in the reconstructed slice), (b) TV-L2 reconstruction significantly improves both streak artifacts and noise in the reconstruction. Reconstruction in Optical Projection Tomography 9
- 10. 513 Implementation and Computation Speed 514 We implemented the algorithm for unconstrained 515 TV-enhanced tomography reconstruction in MATLAB. 516 To achieve maximum speed, time-critical procedures 517 (notably Radon projection, Radon adjoint, gradient compu- 518 tation, and gradient adjoint) were implemented in C. 519 To enable parallel processing on multicore CPUs, OpenMP 520(www.openmp.org) was used for parallelization of the code 521wherever possible. 522Table 3 summarizes execution times for unconstrained 523TV-enhanced tomography reconstructions of Figures 2, 3, 6, 524and 7. There were a fixed number of 200 iterations in each 525reconstruction. The execution times were measured on a PC 526with 48 GB RAM and an Intel Core i7 950 processor with Figure 9. Comparison of (a) Gauss-filtered (σ = 2) filtered backprojection (FBP) reconstruction and (b) total variation (TV)-enhanced tomography reconstruction of real-life tomography sections. It is obvious that—while both procedures reduce noise to some extent— TV-L2 reconstruction produces crisp edges whereas Gauss filtering blurs them. Table 2. Comparison of the Values of Total Variation (TV) and the Normalized Root Mean Square Error (NRMSE) between Radon Projections and the Reprojection of Final Iterates of the Reconstructed Slices. Specimen 1 (Fig. 2) Specimen 2 (Fig. 3) Specimen 3 (Fig. 6) Specimen 4 (Fig. 7) FBP: value of TV 37206858.9 6163021.9 5288472.4 2536355.9 TV-L2 : value of TV 1536827.9 951828.3 954099.6 548130.9 FBP: value of NRMSE 59.8% 17.5% 13.9% 11.19% TV-L2 : value of NRMSE 38.9% 9.7% 12.9% 9% NRMSE is defined as: NRMSE = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ru - bk k2 2 bk k2 2 r . It is evident that the TV-L 2 algorithm consistently outperforms FBP both in the smoothness of the reconstructed image (measured by its TV) and in accuracy of the data match (measured by NMRSE). FBP, filtered backprojection. Table 3. Execution Times of Unconstrained Total Variation-Enhanced Tomography Reconstruction. Number of OPT Projections Size of OPT Projection Size of Reconstructed Slice Number of Iterations Execution Time (s) Specimen 1 (Fig. 2) 400 1,004 × 1,002 710 × 710 200 372.3554 (FBP: 0.62) Specimen 2 (Fig. 3) 400 512 × 512 362 × 362 200 98.855 (FBP: 0.18) Specimen 3 (Fig. 6) 1,000 1,024 × 1,024 762 × 762 200 923.4592 (FBP: 1.66) Specimen 4 (Fig. 7) 400 512 × 512 362 × 362 200 103.2811 (FBP: 0.18) It is seen that execution times are proportional to the size and total number of OPT projections. For comparison, FBP reconstruction times are quoted in the last column. OPT, optical projection tomography; FBP, filtered backprojection. 10 Jan Michálek
- 11. 527 four parallel cores running at 3,207 MHz. Faster recon- 528 struction could be achieved with parallelization using a GPU. 529 SUMMARY 530 In an attempt to remove artifacts originating from the FBP 531 algorithm for 3D reconstruction in OPT, we presented a new 532 algorithm denoted TV-L2 which simultaneously minimizes 533 the mean square error between the measured Radon 534 projections of the sample and the reconstructed projections, 535 and the TV of the reconstructed tomography slice. In all 536 tested cases, the new TV-L2 algorithm outperforms the FBP 537 algorithm (and other tomography reconstruction algorithms 538 like conjugate gradients) regarding both the smoothness of 539 the reconstructed slice in terms of its TV and the data fidelity 540 measured by the NRMSE. The algorithm was developed 541 without assumptions about special properties of the samples 542 such as sparsity of the reconstructed images or sparsity 543 of their gradient, since such assumptions in general do not 544 hold for OPT. 545 We show the feasibility of TV-based tomography 546 reconstructions for real-life, not simulated, data sets as big as 547 1,024 detector pixels in 1,000 projections. The authors 548 reviewed in the Introduction presented true biological data 549 reconstructions from at most an 1,188 bin detector in 550 361 views (Han et al., 2011). Contrary to some algorithms we 551 reviewed, our algorithm behaved stably in all tested cases. 552 Typically, it reduced both TV and the data mismatch 553 monotonically, and in the fixed number of 200 iterations 554 limits were reached. Finally, our TV-L2 tomography recon- 555 struction algorithm is not restricted to OPT and is applicable 556 to any other parallel beam tomography geometry. 557 ACKNOWLEDGMENTS 558 The author wishes to express his gratitude to prof. David 559 Sedmera (Institute of Physiology) who provided some 560 of the cleared specimens, and to Dr. Martin Capek (formerly 561 from the Institute of Physiology) for acquiring the optical 562 projection tomography (OPT) data used in this study on the 563Bioptonics and “Milano” scanners. The author also feels 564obliged to Andrea Bassi of Politecnico di Milano for kindly 565consenting to reprint the photograph of their OPT design in 566Figure 1. The author acknowledges funding from the Czech 567Republic’s public funds provided by Czech Academy of 568Sciences (RVO: 67985823), Ministry of Education, Youth 569and Sports (KONTAKT LH13028), and Science foundation 570of the Czech Republic (13-12412S). 571 REFERENCES 572BASSI, A., FIERAMONTI, L., D’ANDREA, C., MIONE, M. VALENTINI, G. 573(2011). In vivo label-free three-dimensional imaging of zebrafish 574vasculature with optical projection tomography. J Biomed Opt 57516, 100502. 576BOYD, S., PARIKH, N., CHU, E., PELEATO, B. ECKSTEIN, J. (2010). 577Distributed optimization and statistical learning via the 578alternating direction method of multipliers. Foundations 579Trends Mach Lear 3, 1–122. 580BRUYANT, P.P., SAU, J. MALLET, J.J. (2000). Streak artifact reduction 581in filtered backprojection using a level line-based 582interpolation method. J Nucl Med 41, 1913–1919. 583CANDÈS, E., ROMBERG, J. TAO, T. (2006). 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Nat Neurosci 14, 6001481–1488. 601HAN, X., BIAN, J., EAKER, D.R., KLINE, T.L., SIDKY, E.Y., RITMAN, E.L. 602PAN, X. (2011). Algorithm-enabled low-dose micro-CT imaging. 603IEEE Trans Med Imaging 30, 606–620. Table 4. Comparison of the Exact and the Inexact Alternating Direction Method of Multipliers (ADMM). Exact ADMM Inexact ADMM Initialize u0, w0, c0 u0, w0, c0 Repeat Step 1 Keeping w, c constant minimize LA(w, u) exactly with respect to u Keeping w, c constant, do one optimal steepest descent step in the gradient direction of LA(w, u) Step 2 Keeping u, c constant minimize LA(w, u) with respect to w Keeping u, c constant minimize LA(w, u) with respect to w Step 3 Update the vector of Lagrangian multipliers c Update the vector of Lagrangian multipliers c Until a termination criteria achieved Inexact ADMM is used for tomography reconstruction in this manuscript. It is seen that the inexact method differs from the exact one only in Step 1 where the steepest descent step is computationally much cheaper than the exact minimization. Reconstruction in Optical Projection Tomography 11
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A New Alternating 639 Minimization Algorithm for Total Variation Image Reconstruction. 640 SIAM J. Imaging Sci. 1(3), 248–272. 641 APPENDIX 642 Algorithm for Tomography Reconstruction through 643 Minimization of a Weighted Sum of Total 644 Variation and the Data Mismatch 645 Derivation of the Function to be Minimized: the 646 Augmented Lagrangian 647 The unconstrained minimization in (3) ur = arg min u TVðuÞ + μ 2 Ru - bk k2 2 h i (3) 648 cannot be solved using a pure SD approach due to 649 nondifferentiability of the TV term at u = 0. We therefore 650 suggest the approach of variable splitting akin to that of 651 Li (2009). 652 We are using vector notation for images where the 653 pixels of an m × n image are ordered into an N-vector u 654 with N = (m × n). For discrete functions, it is convenient 655 to use the anisotropic form of the TV term in (3), in 656 which total variation is defined as the sum of absolute 657differences between pixels of u. Total variation (TV) of u is 658then TVðuÞ = XN p = 1 ∇huðpÞj j + ∇vuðpÞj j = X p2u Dpu 1 p :: pixel of the image u ∇huðpÞ; ∇vuðpÞ ::: forward differences between neighbor pixels in horizontal and vertical direction; respectively; 659where Dp are 2× N matrices such that the first and the second 660component of the vector Dpu contain the horizontal and vertical 661forward difference at the pixel p, respectively. The reconstructed 662image in equation (3) is obtained by minimization of ur = arg min u X p Dpu 1 + μ 2 Ru - bk k2 2 # : (4) 663First rows of the N matrices Dp, p = 1, … , N can be 664collected into an N × N horizontal difference matrix D(1) , and 665analogically, from the second rows an N × N vertical difference 666 matrix D (2) is created. The 2N × N matrix D = D 1ð Þ D 2ð Þ ! will 667 allow us to simplify notation to ur = arg min u Duk k1 + μ 2 Ru - bk k2 2 h i : 668The l1 norm :k k1 in (4) is nondifferentiable. To bring Du 669out of the nondifferentiable :k k1 term, we introduce an 670auxiliary variable w and use an update scheme that makes w 671eventually approach the components of Du, the vector of 672horizontal and vertical differences between all neighboring 673pixels of the image u: w Du: 674The unconstrained optimization (4) is the equivalent to 675constrained optimization wk k1 + μ 2 Ru - bk k2 2 s:t: w = Du: (5) 676The constraint drives the auxiliary variable w towards the 677discrete gradient components Du. 678For constrained optimization such as in (5), an impor- 679tant class of methods seeks the minimizer or maximizer by 680approaching the original constrained problem by a sequence 681of unconstrained subproblems. One of these is the aug- 682mented Lagrangian method. The constrained minimization 683(5) can be solved by iteratively minimizing the augmented 684Lagrangian function ΛAðw; uÞ = wk k1 - νT ðDu - wÞ + β 2 Du - wk k2 2 + μ 2 Ru - bk k2 2; 685in which the 2N-vector ν contains Lagrange multipliers, and 686β is the weight of the mismatch between the TV and the 687auxiliary variable w. 688Minimization of ΛA(w, u) is equivalent to the mini- 689mization of the scaled augmented Lagrangian LAðw; uÞ = wk k1 + β 2 Du - w - ck k2 2 + μ 2 Ru - bk k2 2; 690since, by carrying out the square of the :k k2 norm, 12 Jan Michálek
- 13. 691 we get LAðw; uÞ = wk k1 - βcðDu - wÞ + β 2 Du - wk k2 2 + β 2 ck k2 + μ 2 Ru - bk k2 2; 692 and for the special choice of ν = βc, LAðw; uÞ = ΛAðw; uÞ + β ck k2 2 ; 693 β ck k2 2 is independent of (w, u), hence LA(w, u) and ΛA(w, u) 694 have the same minimizer (w, u): arg min w; u LAðw; uÞ = arg min w; u ΛAðw; uÞ: 695 The function LAðw; uÞ = wk k1 + β 2 Du - w - ck k2 2 + μ 2 Ru - bk k2 2; (6) 696 is called the scaled augmented Lagrangian. 697 A Method to Minimize the Augmented Lagrangian: 698 the Exact Alternating Direction Method of 699 Multipliers (ADMM) 700 The augmented Lagrangian (6) can be efficiently minimized 701 using the ADMM (e.g., Boyd et al., 2010). Using ADMM, the 702 augmented Lagrangian (6) can be minimized by iterating the 703 following steps until convergence: 704 1. keeping w, c constant minimize LA(w, u) with respect to u; 705 2. keeping u, c constant minimize LA(w, u) with respect to w; 706 3. update the vector of Lagrangian multipliers c. 707 For a given triple wk , uk , ck the iterations are as follows: 708 1. uk + 1 = arg min |fflfflfflffl{zfflfflfflffl} u wk 1 + β 2 Du - wk - ck 2 2 + μ 2 Ru - bk k2 2 ' ; 709 710 711 2. wk + 1 = arg min |fflfflffl{zfflfflffl} w wk k1 + β 2 Duk + 1 - w - ck 2 2 + μ 2 Ruk + 1 - b 2 2 ' ; 712 713 714 3. ck + 1 = ck − (Duk + 1 − wk + 1 ). 715 Since the minimizations in Steps 1 and 2 are exact, we 716 call this method the exact ADMM. Convergence of the exact 717 ADMM is proven in the study by Boyd et al. (2010). 718 The Method used in this Work: Replacing the 719 Costly Exact Minimization in Step 1 by a SD Step 720 with Line Search: Inexact ADMM 721 Minimization with respect to w in Step 2 is easy, since the 722 objective, being a sum of nonnegative terms corresponding 723 to individual pixels, is separable. The result of pixelwise 724 minimization is given by the soft thresholding (or shrinkage) 725operation (Wang et al., 2008): wk + 1 p = max Dpuk + 1 - ck p - 1 β ; 0 ' ´ sgn Dpuk + 1 - ck p for all pixels p; where all the operations are done componentwise 726where 727cp = c1 p c2 p # and wp = w1 p w2 p # ; 728 729are the respective horizontal/vertical values of scaled 730Lagrange multipliers c and of the auxiliary variable w at the 731pixel p corresponding to discrete gradient at p, Dpu. 732Minimization with respect to u in Step 1 does not 733depend on wk 1 , and is thus equivalent to the minimization 734of a sum of two quadratic functions of the image being 735reconstructed: arg min u LAðwk ; uÞ = arg min u Qk ðuÞ with Qk ðuÞ = β 2 Du - wk - ck 2 2 + μ 2 Ru - bk k2 2; 736Qk (u) is thus a positive definite function of u, and its 737minimum could actually be computed in closed form by 738setting the gradient = 0 and resolving for u. This approach 739is, however, intractable for images of realistic size, since 740the matrices involved are huge: for an image of 741N = 1,000 × 1,000 = 106 pixels the resulting matrices would 742have 1012 elements. For this reason, we chose to replace the 743exact mimization in Step 1 by an approximate minimization 744using only a single gradient step. The gradient of Qk at the 745current image iterate uk gk = ∇Qk ðuk Þ = βDT ðDuk - wk - ck Þ + μRT ðRuk - bÞ; 746is used as the new search direction. Since Qk (u) is a quadratic 747function, the optimum step size can be calculated explicitly as αk = gkT gk gkTAgk ; 748where 749A = βDT D + μRT R. 750Based on these considerations, approximate minimiza- 751tion in Step 1 is carried out as follows: uk + 1 = uk - αk gk : 752Replacing exact minimization in Step 1 by an approx- 753imate one (one gradient step) is justified by a Theorem due 754to Eckstein Bertsekas (1992) below. We first need to 755introduce the concept of variable splitting. Consider an 756unconstrained optimization problem min u2Rn f ðuÞ + hðMuÞ f : Rn ! - 1; 1ð Š; h : Rd ! - 1; 1ð Š M 2 Rd ´ n : ð7Þ 757Variable splitting is a procedure that transforms the 758unconstrained optimization problem (7) of a single variable, 759u, into a constrained optimization problem by creating an Reconstruction in Optical Projection Tomography 13
- 14. 760 auxiliary variable, w = Mu. The constrained optimization 761 problem equivalent to (7) is then min u2Rn f uð Þ + h wð Þ subject to w = Mu: (8) 762 Variable splitting makes sense in cases where con- 763 strained optimization (8) is easier to solve than its uncon- 764 strained counterpart (7). The following Theorem establishes 765 convergence of the Inexact ADMM. 766 Theorem 1 The generalized alternating direction method 767 of multipliers. 768 Consider the minimization problem (7), where M has 769 full column rank and f, g are closed, proper, convex functions. 770 Consider arbitrary β 0 and w0, c0 ∈ Rd . Let {ηk ≥ 0, 771 k =0,1, …}and {λk ≥0, k = 0,1,…} be two sequences such that X1 k = 0 ηk 1; X1 k = 0 λk 1: 772 Suppose {uk ∈Rn }k = 1 ∞ , {wk ∈Rd }k = 0 ∞ , and {ck ∈Rd }k = 0 ∞ 773 conform, for all k, to uk + 1 - arg min u f ðuÞ + β 2 Mu - wk - ck 2 ' ≤ ηk wk + 1 - arg min w hðwÞ + β 2 Muk + 1 - w - ck 2 ' ≤ λk ck + 1 = ck - ðMuk + 1 - wk + 1 Þ: Then if (7) has a solution, the sequence {uk } converges 774to a solution of (7) and {ck } converges to a solution of the 775dual problem. Furthermore, {wk ∈Rd }k = 0 ∞ converges to Mu*, 776where u* is the limit of {uk }k = 1 ∞ . If the dual problem has no 777optimal solution, then at least one of the sequences {wk }k = 0 ∞ or 778{ck }k = 0 ∞ is unbounded. 779The above Theorem states that it is not necessary to 780exactly solve the minimizations in Steps 1 and 2 in 781the ADMM algorithm; as long as sequences of errors are 782absolutely summable, convergence is still guaranteed. 783Obviously, with the choice f ðuÞ = μ 2 Ru - bk k2 2; M = D; hðMuÞ = Muk k1 784 785 786our Algorithm 1 fits into the framework of Theorem 1. We 787exploit the Theorem in Step 1 of the Algorithm, where exact 788minimization is replaced by a single gradient step, since exact 789minimization would be intractable for images of 790realistic size. We were not able to prove summability of the 791error in Step 1 theoretically, but empirical results confirm 792that for tomography reconstructions of practical interest the 793algorithm converges to an acceptable solution. The exact 794ADMM method (not used here) and the inexact algorithm 795(used here) are compared in Table 4. 796 14 Jan Michálek