Discrete Tomography in Discrete Deconvolution: Deconvolution of Binary Images Using Ryser s Algorithm

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Discrete Tomography in Discrete

Deconvolution: Deconvolution of Binary

Images Using Ryser’s Algorithm

Behzad Sharif

a,1,2

and Behnam Sharif

b

a _{Coordinated Science Laboratory and ECE Department}

University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

b _{University of Manitoba, Winnipeg, Canada}

Abstract

A new deconvolution algorithm for binary images based on the theory of discrete tomography is proposed. The proposed algorithm is inherently binary as opposed to traditional filtering techniques such as Wiener filtering which require thresholding to produce binary images. Time and space complexity of the proposed algorithm are polynomial in the image size whereas the two-dimensional Viterbi method has an exponential complexity. Application of the proposed method in equalization of two-dimensional inter-symbol interference channels such as page-oriented optical memories is demonstrated. Through numerical simulations, it is shown that the method can outperform the traditional methods such as Wiener filtering especially for low singal-to-noise scenarios.

Keywords: Discrete Tomography, Ryser’s Theorem, Switching Components, Discrete Deconvolution, Two-dimensional Inter-symbol Interference Channels, Page-oriented Optical Memories, Two-dimensional Channel Equalization.

1 _{The ﬁrst author would like to thank professor Richard E. Blahut for introducing and} motivating the problem of binary deconvolution to him.

2 _{Email: [email protected]}

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1 Introduction

In this work, we propose an inherently-binary algorithm for deconvolution of binary images . We apply a fundamental result in discrete tomography, namely Ryser’s theorem and algorithm, to the problem of deconvolution of binary

im-ages. In Section2, using the structure of the point spread function, we reduce

the deconvolution problem to a binary tomography problem with two available

orthogonal projections. In Section3, an initial reconstruction is computed

us-ing Ryser’s algorithm. Next, Ryser’s theorem is utilized to identify variant positions and the blurred image is used to resolve the ambiguities. Perfect reconstruction conditions for an ideal noiseless case are given. Several ideas to battle the noise are presented such as rounding the line sums or a

Metropo-lis algorithm. In Section 4, application of the proposed method in

equaliza-tion of two-dimensional inter-symbol interference (ISI) in page-oriented optical

memories is described. Section5provides extensive numerical experiments to

demonstrate the viability of the method. Finally, Section 6 summarizes the

contributions and concludes the paper.

2 Problem Formulation

2.1 The Discrete Deconvolution Problem

A general two-dimensional deconvolution problem can be written as:

(1) b(x, y) = p(x, y) ∗ ∗c(x, y) + w(x, y)

where c(x, y) is the unknown field, p(x, y) denotes the point spread function (PSF) or convolution kernel and w(x, y) is the additive noise. We refer to b(x, y) as the “dirty image” and to c(x, y) as the “original image” or unknown field. For a nonanalytical solution, c(x, y) must be discretized. In what follows, it is assumed that the unknown field can be sufficiently represented by a weighted sum of shifted versions of a basis function Φ(x, y) as follows:

(2) c(x, y) =

i

j

c_ijΦ(x− i, y − j)

For instance, the basis functions are often chosen to be the set of unit height boxes corresponding to a two-dimensional array of square pixels. In that case,

{cij} are the heights of the square pixels or simply the values of the pixels.

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0 0.2 0.4 0.6 0.8 1 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) −0.5 0 0.5 1 1.5 2 2.5 3 (c)

Fig. 1. (a) Original 64× 64 binary image (original image) (b) The point spread

function (symmetric blurring kernel) (c) The blurred and noisy image (dirty image).

a matrix c of size m× n, and denoting the discretized kernel by P gives the

following matrix equation

b(:) = Pc(:) + w(:)

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where (:) denotes the lexicographic ordering (vector form) of a matrix and w is the measurement noise. Since we are dealing with a binary inverse problem,

the pixel values are restricted to be either one or zero. Figure 1 shows a

sample phantom, a PSF and the resulting dirty image. The measurement noise is additive white Gaussian. The signal-to-noise (SNR) ratio deﬁned as

20 log₁₀1/σ_w is 10dB where σ2_w is the variance of noise samples. Given the

dirty image data b and the kernel P, the task of discrete deconvolution is to

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2.2 Reducing the Deconvolution Problem to a Tomography Problem

Throughout the paper, we assume that the PSF has the form ⎡ ⎢ ⎢ ⎢ ⎣ h g h g 1 g h g h ⎤ ⎥ ⎥ ⎥ ⎦ for

some g, h∈ R+. The P matrix in Eq. (3) is the discrete convolution kernel

corresponding to this PSF. Assuming that the ﬁrst and last rows and the ﬁrst and last columns of c are zero (that is c has a frame of zeros at the borders), it can be shown that:

i b_ij = (g + 2h)· n i=1 c_i,(j−1)+ (1 + 2g)· n i=1 c_i,j+ (g + 2h)· n i=1 c_i,(j+1)

for 1≤ j ≤ m − 1. For j = 1 the left term is absent and for j = m the right

term is absent. This forms a well-posed system of linear equations that can

be solved to ﬁnd s_j =m_i=1c_ij, the column sums, and similarly r_i=n_j=1c_ij,

the row sums. It can be shown that a general symmetric discrete PSF can be treated by the same token (assuming a thick enough frame of zeros for

c). Having the row and column sums, the problem is reduced to a binary

tomography problem or in other words, reconstruction of a binary matrix c from its row and column sums. We are now in a position to apply the discrete tomography framework to solve the discrete deconvolution problem.

3 The Proposed Method

3.1 Background on Reconstruction from Two Projections

In this subsection, we brieﬂy state theoretical results by Ryser [4,15] which

provide an answer to the uniqueness and existance questions in the binary

tomography problem. This subsection closely follows [12].

Let R = (r₁, . . . r_m) and S = (s₁, . . . s_n) be nonnegative vectors. The

class of all binary matrices c = (c_ij)∈ {0, 1}m×n, with row and column sums

(line sums) equal to R and S respectively, is a member of the tomographic

equivalent classU(R, S).

Deﬁnition 3.1 Consider the matrix ¯c in which i-th row consists of r_i ones

followed by n− r_i zeros. Such a matrix is called maximal and is uniquely

determined by its row sums.

Now, let ¯S be the column sum vector of ¯c. Also, denote the nonincreasing

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theorem answers the existence question, that is whether U(R, S) is empty or not:

Theorem 3.2 (Ryser, 1957) Let S and R be a pair of compatible row and

sum vectors. The class U(R, S) is nonempty if and only if

n j= ´ s_j _≥ n j= ¯ s_j, 2≤ ≤ n (4)

For proof and more details see [12]. Ryser proposed an algorithm for

construction of c when the condition of Theorem 3.2 is satisﬁed which we

refer to as Ryser’s algorithm and is quoted here:

Ryser’s Algorithm:

Input: a compatible pair (R, S) satisfying (4).

Step 1: Construct ´S from S by permutation π. Step 2: Let T = ¯c and k = n.

Step 3: While (k > 1),

While ( ´s_k>m_i=1t_ik),

let j₀= max{j < k|t_ij = 1, t_i,j+1=· · · t_ik= 0}.

let row i₀be where such j₀ was found.

set t_i₀_j₀= 0 and t_i₀_k = 1 (i.e., shift the 1 to right).

k = k − 1.

Step 4: c = π−1_(T).

Step 5: Output c.

The complexity of algorithm is O(mn + n log n).

Deﬁnition 3.3 A “switching component” (SC) of a binary matrix c is a 2×2

submatrix of either of the following forms: A₁=

⎡ ⎣1 0 0 1 ⎤ ⎦ or A₂= ⎡ ⎣0 1 1 0 ⎤ ⎦. A “switching operation” is a transformation of elements of c that changes

a submatrix of type A₁ into type A₂ or vice versa. Collecting the row and

column sums into vectors R and S and denoting the tomographic equivalence

class byU(R, S), the Ryser’s theorem can be stated as follows:

Theorem 3.4 Ryser’s Theorem. If ∃c1, c₂_{∈ U(R, S) such that c}₁_{= c}₂ then

c₁ is transformable to c₂ by a ﬁnite number of switching operations.

The following corollary characterizes the “variant positions” (for deﬁnition

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Corollary 3.5 If position (i, j) is variant, then it is part of a SC. That is

∃i◦_{, j}◦ _{such that either (c}

ij = ci◦_j◦ = 1 and c_i◦_j= c_ij◦ = 0) or

(c_ij = c_i◦_j◦ = 0 and c_i◦_j = c_ij◦ = 1).

3.2 Resolving the Switching Component Ambiguities

Having the row and column sums, we can get a reconstruction, say ˆc, by

running the Ryser’s algorithm stated above. Next, using Corollary 3.5, we

know that all of the ambiguities (variant positions) in ˆc are SCs. For each SC

in ˆc with corners at (i, j) and (i◦, j◦), we can “correct” it using the following rule:

Discrete Deconvolution (DD) rule: if b_i,j+ b_i◦_,j◦ ≥ b_i,j◦+ b_i◦_,j then

decide ˆc_i,j= ˆc_i◦_,j◦ = 1 and ˆc_i,j◦ = ˆc_i◦_,j = 0 and vice versa.

Note that here we are using information from the dirty image b. This is the key diﬀerence between the deconvolution and tomography problems. Using the extra information available in the dirty image, we aim to resolve the SC

ambiguities. A “good” SC in ˆc is also a SC in the original image c, i.e., one

of the cases in Corollary 3.5. If it is not a SC in c we refer to it as a “bad”

SC. The following Lemma asserts that assuming the PSF given in Section2.2

with h = 0, the DD rule we make a correct decision for every good SC in the noiseless case provided that g is small enough.

Lemma 3.6 If g < 1

4 then the DD rule will make the correct decision for

every SC in the noiseless case.

Proof. Consider a good SC with corners at (i, j) and (i◦_{, j}◦_{). Without loss}

of generality, assume c_i,j = c_i◦_,j◦ = 1 and c_i,j◦ = c_i◦_,j = 0. The worst-case

situation for the DD rule happens when the (i, j◦) and (i◦, j) positions are

surrounded by ones in c (original image) and because of the non-ideal PSF,

each of the surrounding ones contribute a nonzero value to the (i, j◦) and

(i◦, j) positions in b (dirty image). Considering the form of the PSF given

above and assuming h = 0, the maximum possible value of b_i◦_,j or b_i,j◦ is

hence 4g. Therefore, the DD rule makes a correct decision if 4g + 4g < 1 + 1

which is equivalent to g < 1₄. 2

The Noiseless-DD algorithm is the following:

Step 1. Solve for row and column sums using the method in Section2.2.

Step 2. Apply the Ryser’s algorithm to obtain an initial reconstruction ˆc.

Step 3. For each undecided SC at (i, j) and (i◦_{, j}◦_{) do the following:}

If b_i,j+ b_i◦_,j◦ ≥ 2 or b_i,j◦+ b_i◦_,j ≥ 2 then

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(a) (b)

(c) (d)

Fig. 2. (a) Original binary image (b) Original Ryser’s algorithm reconstruction. Panels (c) and (d) show the evolution of the proposed method. Notice that the diﬀerence between (b) and (c) or (c) and (d) or (d) and (a) is only in one switching operation. The ﬁnal reconstruction result is exactly the one is (a) and we have perfect reconstruction.

else do not change the SC and continue. Step 4. Return ˆc as the reconstruction result.

In the ideal conditions mentioned above, one can detect the good SCs and only correct those. That is, although we do not have access to c but we can

say whether a SC in ˆc is a SC in c. It is straightforward that this can be done

simply by summing up the corresponding diagonal elements in b as mentioned

in the algorithm (note that we have assumed g < 1₄ and h = 0). Since the only

diﬀerence between c and ˆc is in good SCs (according to Ryser’s Theorem) and

because the DD rule corrects any good SC (Lemma 3.6), it follows that the

Noiseless-DD algorithm gives a perfect reconstruction of c.

Figure2 shows the simulation result for the noise-free scenario where the

blurring kernel is 3× 3 symmetric with g = 0.24 and h = 0. Panel (a)

shows the original binary image. Panel (b) is the original Ryser’s algorithm reconstruction. The image in (c) is derived from (b) by one switching operation based on DD rule. Similarly, the next iteration step gives the image in (d) and ﬁnally, with one more switching operation, we get the original image of (a).

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3.3 Rounding and Modiﬁed Ryser’s Algorithm

In practice, the dirty image contains noise and the perfect reconstruction results of the last part do not apply. The simplest idea to battle the noise is rounding the row and column sums which will give the perfect line sums only if the noise power is small enough. Otherwise, the reconstruction will be based on perturbed projections and will result in imperfect reconstruction. A more severe problem occurs when after rounding, the resulting (R, S) pair fail

to satisfy the conditions of Theorem3.2. In that case, the classU(R, S) will

be empty and Step 3 of the Ryser’s algorithm may fail since it is possible that

no (j₀,i₀) exist that satisﬁes the required condition. One may think of a way

to optimally alter the faulty line sums to get a nonempty class. For instance,

the optimality measure may be minimum ₁ perturbation in the measured

line sums and will lead to a combinatorial optimization problem. Adopting a diﬀerent approach, we propose to modify the Ryser’s algorithm in order to get a reconstruction and then hope that application of DD rule will reduce the number of errors. The modiﬁcation is to add a line in the inner loop of Step 3 as follows:

Step 3: While (k > 1),

While ( ´s_k>m_i=1b_ik), ...

If no such j₀found decrease k by 1 and go to start of Step 3.

...

Furthermore, it is no longer possible to detect good SCs and the algorithm applies the DD rule to every SC. This may result in complications in case of bad SCs. The theoretical analysis of this issue is still open. However, through

numerical experiments, we learned that instead of just scanning ˆc once for SCs

and deciding for each of them, it is better to do multiple passes. In all of the

experiments done in Section5, the number of bit errors, i.e.,|c − ˆc|, converges

in at most three passes. A summary of the proposed algorithm, referred to as

the DD algorithm, is shown in Figure 3.

3.4 Further Discussion on Battling the Noise

It is known that for an ill-posed inverse problem such as limited-angle tomog-raphy, forcing the solution to satisfy all the noisy measurements is not a good

strategy [3]. Reconstruction of binary images from the their line sums in a

ﬁnite number of directions is also ill-posed. Even some small perturbation in the measurements can lead to dramatically diﬀerent yet still unique solutions.

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tomog-Fig. 3. Flowchart of the proposed algorithm (DD algorithm).

raphy problem the perfect match to the noisy line sums maybe far from the

true image while a nonperfect match can be quite close. In [1], it is asserted

that the instability result of [2] is not valid for reconstruction from two

pro-jections, which is our case. The results are proved for perturbations of size 2

(in the ₁ norm) and a general result is sill to come. It should be noted that

the studies on stability of the reconstruction algorithms which incorporate a

priori knowledge also exist in the literature [7,13]. In [5], the authors focus

on stability results for special lattice sets especially the convex ones. In [8],

experimental studies on reconstruction of convex binary bodies from noisy projections are presented. It is worth mentioing that, the applications that we have focused on do not allow for any prior assumption on the structure of the binary ﬁeld, e.g., convexity, connectivity, etc. Here we introduce some practical ideas than can be combined with the DD algorithm:

• Metropolis Algorithm on Ryser’s Graph: In the Ryser graph of a

tomo-graphic equivalence class U, two vertices are adjacent if and only if the

images corresponding to the two vertices diﬀer only by one switching

op-eration [11]. Given a real positive function deﬁned on the vertices of the

graph, the Metropolis algorithm seeks to maximize the function using an

iterative stochastic method (for details see [11]). In the problem at hand,

one can postprocess the reconstructed binary image ˆc using the Metropolis

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E( ˆci)≈ |

1

m · nb − P ˆci22− ˆσw2|1

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where ˆc_iis the image associated with the i-th vertex and ˆσ_w2 is the estimated

noise variance (|.|₁ and .₂ denote the ₁ and ₂ norms respectively). The

idea behind the proposed objective function is that if ˆc_i≈ c, then E(_m·n1 b−

P ˆc_i2

2) ≈ E(m·n1 w22) ≈ ˆσ2w. This is similar to the discrepancy principle

used in parameter estimation. Since the computational complexity of the Metropolis algorithm is much higher than Ryser’s algorithm, i.e., it takes much longer for it to reach a close-to-optimal solution, it makes sense to apply it as a postprocessor to the output of the DD algorithm.

• Two-dimensional Wiener Denoising: Assuming Gaussian noise statistics,

one can denoise the dirty image b prior to computing the line sums and

then apply the DD algorithm. The problem with this idea is that the

Wiener ﬁlter tends to smoothen the edges and if the original binary image is not smooth, it may worsen the situation by introducing extra errors in the dirty image. Therefore, for applications where the underlying images are not typically smooth, one should avoid ﬁltering of any kind.

• Divide and Conquer: For data storage applications, it is in our power to

predesign the system so that instead of having to reconstruct the whole unknown ﬁeld c(x, y), the problem would decompose into smaller subprob-lems. This can be accomplished by dividing the c(x, y) matrix into, say,

N2_{submatrices and enforcing frames of zeros around them (which wastes a}

negligible number of storage cells). In this way, we can deal with each of the

N2 _{subproblems individually which will reduce the number of terms in the}

line sums by a factor of N . Assuming independent identically distributed noise, this will reduce the noise power in a row sum by a factor of N .

4 Applications

As stated in the introduction, the main application of this method is in equal-ization (deconvolution) of two-dimensional communication or storage systems. One instance of such problems is removal of ISI in page-oriented (page-access) optical memories where the ISI pattern is two-dimensional and the unknown

ﬁeld is inherently binary [6,9].

During the past decade, volume holographic data storage has emerged as a promising digital storage technology because of its ideal properties such as high data storage density, high data rate, and short access times. A typical

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Fig. 4. Typical setup of a digital holographic data storage system

are input to the system via a spatial light modulator (SLM) and in the simplest case, each bit is mapped to a single SLM pixel. Several data pages are recorded holographically by a reference beam. The ability to multiplex several pages into a ﬁxed volume allows for high data densities. As storage density increases, the performance of the storage channel is degraded due to ISI and increase in noise level. Any such deviation from ideal imaging results in inter-pixel cross talk and generally in a higher bit-error rate (BER). Recently, researchers have proposed use of signal processing and estimation techniques to decrease the BER of the retrieved data which results in easing the design tolerances

of the optical components and permits the use of large data pages. The

techniques range from Wiener ﬁltering or linear minimum mean square error

estimation (LMMSE) [6,10] to combination of Viterbi algorithm with decision

feedback equalizer [9]. The ﬁltering-type methods are unable to incorporate

the knowledge that the original image is binary. The problem with Viterbi-type algorithms is their exponential computational complexity.

The developed deconvolution algorithm (Figure 3) seems like an ideal

match to this application. It is inherently binary and also has polynomial time complexity and linear space complexity. In all cases it is assumed that the input date page is much larger than the optical system blur, so that edge

eﬀects are neglected. This validates the assumption made in Section2.2about

the frame of zeros around the image. The blurring occurs in page-access opti-cal memories because of optiopti-cal defocusing, aberrations of the optiopti-cal system, or blooming at the CCD array. The blurring kernel can be reasonably

mod-eled by a 3× 3 symmetric kernel and can be reliably estimated a priori [9].

Therefore, the method of Section 2.2is applicable. Another interesting point

is that the DD algorithm does not assume any speciﬁc noise statistics and this makes it applicable for both the Gaussian and non-Gaussian noise situations.

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0 0.2 0.4 0.6 0.8 1 (a) 0 0.2 0.4 0.6 0.8 1 (b) 0 0.2 0.4 0.6 0.8 1 (c) 0 0.2 0.4 0.6 0.8 1 (d)

Fig. 5. Reconstruction results for the 1st experiment. The original image is shown

in Figure1. (a) The thresholded Wiener ﬁltering (LMMSE) result (# bit errors =

597) (b) Original Ryser’s algorithm reconstruction (# bit errors = 1227)(c) Final DD algorithm result (# bit errors = 434) (d) Output of the Metropolis algorithm performed on the image in (c) (# bit errors = 394).

In fact, the Gaussian noise assumption is only valid for the case of electronic-noise-dominated storage channels and if the channel is dominated by optical

noise the output will obey a Rician statistics [6].

Restoration of astronomical images [14] in situations where the PSF can

be accurately estimated can be another application. A typical astronomi-cal image consists of sparse point sources on a smooth background which can be estimated and subtracted from the measurements, leaving an image with sparse nonzero pixels. In some cases the intensities are almost uniform which, with appropriate preprocessing, will result in a discrete deconvolution problem. However, the application of the developed framework is limited to situations where the underlying image can be modeled as a binary image.

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0 0.2 0.4 0.6 0.8 1 (a) −1 0 1 2 3 4 (b) 0 0.2 0.4 0.6 0.8 1 (c) 0 0.2 0.4 0.6 0.8 1 (d)

Fig. 6. (a) Original binary image of the 2nd experiment (b) The dirty image (PSF same as 1st experiment) with SNR = 10dB (c) The thresholded Wiener ﬁltering (LMMSE) result (# bit errors = 154) (d) Final DD algorithm result (# bit errors = 64).

5 Simulation Results

In this section we provide simulation results corresponding to realistic

sit-uations in the optical storage application described in Section 4. The ﬁrst

experiment is the one introduced in Figure1. The PSF is of the form in

Sec-tion2.2 with g = 0.44 and h = 0.11 and the SNR (deﬁned in Section 2.1) is

10dB. Figure5 shows the simulation results comparing the standard

thresh-olded Wiener ﬁltering to the DD algorithm. Note the dramatic improvement in panel (c) compared to panel (b) that is accomplished by applying the DD rule. The ﬁgure also depicts the output of the Metropolis algorithm described

in Section 3.4. As can be seen from the images and the number of bit

er-rors, the DD algorithm outperforms the Wiener filter in this case and the Metropolis algorithm performed on the DD algorithm’s output gives a slight improvement. The explanation is that the DD algorithm is able to formu-late the prior knowledge that the underlying field is binary whereas Wiener filtering uses knowledge of noise statistics.

The second experiment is shown in Figure6. The SNR and blurring kernel

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−3 10−2 10−1 100 g BER Wiener 28dB DD 28dB Wiener 21dB DD 21dB Wiener 14dB DD 14dB (a) 40 24 19 16 14 12 10 9 10−3 10−2 10−1 100 SNR BER Wiener g=0.3 DD g=0.3 Wiener g=0.4 DD g=0.4 Wiener g=0.7 DD g=0.7 (b)

Fig. 7. (a) Bit-error rate (BER) for the 1st experiment as a function of blurring kernel intensity g comparing DD and Wiener methods for 3 diﬀerent SNRs (b) BER as a function of SNR comparing DD and Wiener methods for 3 diﬀerent kernels.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−4 10−3 10−2 10−1 100 g BER Wiener 28dB DD 28dB Wiener 21dB DD 21dB Wiener 14dB DD 14dB (a) 40 24 19 16 14 12 10 10−3 10−2 10−1 100 SNR BER Wiener g=0.3 DD g=0.3 Wiener g=0.4 DD g=0.4 Wiener g=0.7 DD g=0.7 (b)

Fig. 8. (a) Bit-error rate (BER) for the 2nd experiment as a function of blurring kernel intensity g comparing DD and Wiener methods for 3 different SNRs (b) BER as a function of SNR comparing DD and Wiener methods for 3 different kernels. as random as before and is much smoother. As can be seen from the results, this time the improvement factor in using DD instead of Wiener filtering is almost twice as the first experiment (2.4 compared to 1.3). This implies that not only the SNR or the kernel parameters are a factor in the performance of the DD algorithm, but also the properties of the original image itself, e.g., smoothness or connectivity, can have a significant effect on the reconstruction performance.

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In order to compare the performance of Wiener ﬁltering and the DD algo-rithm in more detail, we have computed the BER as a function of g and SN R

for both methods (h is taken to be 1₄g). The results for the ﬁrst experiment are

shown in Figure7. Each point in the plots is the average BER of 20

Monte-Carlo runs. As can be seen from the ﬁgure, for low SNRs (less than 14dB) and moderate values of g (between 0.3 to 0.5), the DD algorithm outperforms the

Wiener ﬁlter. And in other cases its performance is worse. Figure 8

demon-strates the same results for the second experiment. As expected, here the DD algorithm outperforms Wiener ﬁltering even for moderate SNRs (20dB and lower) and almost all of the g range (except for g < 0.3). It should be noted that the ﬁrst experiment is much closer to reality for a data storage application whereas the second experiment resembles an imaging application. It is important to note that the DD algorithm implemented here does not use any knowledge of the statistics of the noise (except for the Metropolis

result in Figure5). Overall, the results of this section have demonstrated that

the proposed algorithm is a viable candidate for computationally eﬃcient de-convolution of binary images when the underlying inverse problem is severely ill-posed.

6 Summary and Conclusions

We have proposed a new deconvolution algorithm for binary images that is built upon the theory of discrete tomography especially Ryser’s theorem. Ex-tensive numerical results have shown that the method can outperform the traditional methods such as Wiener ﬁltering, especially for low SNR scenar-ios. Overall, the attributes of the proposed DD algorithm are as follows:

(i) The proposed algorithm utilizes the fact that the unknown field is a binary. Other standard inverse filtering methods such as Wiener filtering are incapable of doing so.

(ii) The time complexity of the proposed algorithm is polynomial as opposed to the two-dimensional Viterbi method which has exponential complexity. (iii) The proposed algorithm does not use any assumed knowledge of statistics of the noise or signal. In contrast, all standard methods for ISI equal-ization methods assume a prior statistics for noise and/or the underlying ﬁeld.

(iv) To our knowledge, the proposed method is the ﬁrst combinatorial ap-proach to the binary deconvolution problem.

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conditions but theoretical results of the stability and robustness of the algorithm are still open for investigation.

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