Blockiterative RichardsonLucy methods for image deblurring
 NamYong Lee^{1}Email author
https://doi.org/10.1186/s1364001500692
© Lee. 2015
Received: 30 August 2013
Accepted: 9 May 2015
Published: 4 June 2015
Abstract
In this paper, we extend the RichardsonLucy (RL) method to blockiterative versions, separated BIRL, and interlaced BIRL, for image deblurring applications. We propose combining algorithms for separated BIRL to form block artifactfree output images from separately deblurred block images. For interlaced BIRL to accelerate the iteration, we propose an interlaced blockiteration algorithm on downsampled blocks of the observed image. Simulation studies show that separated BIRL and interlaced BIRL achieve desired goals in Gaussian and diagonal deblurrings.
Keywords
Parallel computation Block artifacts Ordered subsets Accelerated iteration1 Introduction
The image deblurring problem has many applications in science and engineering fields, and many methods have been proposed for it [1]. Among them, the RichardsonLucy (RL) method, which was proposed independently by Richardson [2] and Lucy [3] in 1970s, has been one of the most widely used iterative deblurring methods. Applications of RL in microscopy, astronomy, or motion deblurring can be found in [4, 5] and references therein. Variants of RL with specific purposes such as adaptivity, parallel implementation, acceleration, suppression of ringing/boundary artifacts, or edgepreserving can be found in [6–12].
In the paper by Shepp and Vardi [13], the same algorithm was rederived from the maximization likelihood principle in the application to emission tomography and the expectation maximization (EM) method in [14]. In the paper by Hudson and Larkin [6], the ordered subsets expectation maximization (OSEM) method was proposed as an acceleration technique for EM. Since its introduction, OSEM has been successfully used for many applications in emission tomography [15, 16].
The use of RL in image deblurring is, however, computer intensive and often suffers from slow convergence. To deal with these obstacles, blockiterative RL (BIRL) methods have been proposed [17, 18]. BIRL decomposes the main problem into several subproblems by grouping pixels of the observed image into several subsets and applies RL to each subproblem. In this paper, image parts defined on those grouped pixel subsets will be called blocks. In [17], the observed image was decomposed to four rectangular blocks, RL was applied to each block separately, and the final image was obtained by combining four deblurred block images. In [18], an OSEMlike method was proposed to accelerate the iteration by using multiviews of the image. The term ‘multiviews of the image’ means that the image to be recovered is observed several times under different imaging environments. The observed data at each imaging environment is often called a view of the image. The problem of deblurring images from multiviews is also called multiple image deconvolution [4].
In this paper, we extend methods in [17, 18] to separated BIRL and interlaced BIRL, for image deblurring applications.
The goal of separated BIRL is to combine separately deblurred subimage blocks into the resulting output image without block artifacts. Obviously, this is not a new approach. What is new in the proposed separated BIRL is that the proposed method can suppress block artifacts efficiently for arbitrarily shaped blocks. This flexibility is important for the success of separated BIRL; simulation results of this paper will show that diagonally shaped blocks produce better results than rectangularly shaped blocks for separated BIRL in the deblurring problem modeled by a diagonal point spread function (PSF).
The goal of interlaced BIRL is the accelerated iteration by using an OSEMlike method as in [18]. Image deblurring problems of this paper, however, are assumed to have only one view of the image, unlike in [18]. To overcome this ‘one view’ limitation, the proposed interlaced BIRL decomposes the observed image into several downsampled blocks and treats those downsampled blocks as multiviews of the image. Interlaced BIRL accelerates the iteration by using the onestep RL block iterates as the starting image in the next block iteration, as OSEM does in emission tomography.
The performance of proposed methods depends on how blocks are formed. There are many possibilities for the block partition. It is, however, not clear how blocks should be formed for a specific PSF.
The work in this paper will be restricted to the test of rectangular or diagonal subimage blocks for separated BIRL and rectangularly or diagonallydownsampled blocks for interlaced BIRL in Gaussian and diagonal deblurrings. While explaining test results, possible extensions of proposed methods to general deblurring problems will be presented briefly.
2 Definition and background
2.1 Image deblurring problem
Here, \(g_{i_{1},i_{2}}\) represents the intensity of the image g at the pixel (i _{1},i _{2}). We will use the same convention, the use of boldface alphabet for the image and the normal alphabet with the pixel subscript for the intensity, throughout this paper. Notations ℓ ^{2}(Ω) and ℓ ^{2}(Λ) are used to denote image spaces defined on Ω and Λ, respectively. Here, inner products of ℓ ^{2}(Ω) and ℓ ^{2}(Λ) are the usual dot product of two images (sums of pixelbypixel multiplications).
Here \({\mathcal S}_{\mathbf {k}}\), the support of k, is \(\phantom {\dot {i}\!}\{(j_{1},j_{2})\mid k_{j_{1},j_{2}}>0\}\). We also assume that the PSF k is nonnegative, its components have sum 1, and the point \((0,0)\in {\mathcal S}_{\mathbf {k}}\); \({\mathcal T}\mathbf {p}\) is defined on Λ, where (i _{1},i _{2})∈Λ if and only if \((i_{1},i_{2}){\mathcal S}_{\mathbf {k}}\subset \Omega \). Thus, Λ⊂Ω.
2.2 Computation of \({\mathcal T}\) and \({\mathcal T}^{\ast }\)
for q∈ℓ ^{2}(Λ).
2.3 Richardson–Lucy iteration
Here, the notation I _{ A } means the allone image on the pixel subset A and.∗ is the pixelbypixel multiplication, and the division between two images, \({\frac {\mathbf {g}}{{\mathcal T}\mathbf {f}^{n}}}\), is the image resulted from the pixelbypixel division of g and \({\mathcal T}\mathbf {f}^{n}\).
Performing RL as described in (8) often results in a very slow convergence. For the acceleration of RL, several methods have been suggested [7, 19–21]. Among them, the technique in [7] has been noted for its success. For example, the function ‘deconvlucy,’ the RL implementation of the software MATLAB®; (The MathWorks, Natick, MA, USA), uses the technique described in Section 2.3 and 2.4 in [7]. Interlaced BIRL, the proposed method for the acceleration of RL in this paper, will be compared with the technique in [7] in Section 4.
2.4 Boundary artifacts
Boundary artifacts are one of key obstacles in the development of BIRL; block boundaries introduced by the block partition may cause artifacts in combining deblurred block images. It is also true that boundary artifacts are one of key obstacles in many image deblurring problems [22].
To reduce boundary artifacts in image deblurring, many methods have been proposed. One group of methods imposes certain conditions on pixels in Ω−Λ. Examples include periodic, reflective, and antireflective boundary conditions [23–27].
Other group of methods [17, 28, 29] does not impose any conditions on pixels in Ω−Λ, and let the iteration itself determine results in Ω−Λ. In [29], this approach is called the free boundary condition method.
Before we close this section, it is worth to mention some research works related to fast direct deblurring methods. It is well known that if the imposed boundary condition is one of periodic, reflective, or antireflective boundary conditions, then the image deblurring with symmetric PSFs (for a periodic boundary condition, the symmetry of PSF can be omitted) can be directly computable by using FFT for periodic boundary condition, discrete cosine transform (DCT) for reflective boundary condition, and discrete sine transform (DST) for antireflective boundary condition [1, 23, 25].
These fast transformbased direct deblurring methods, however, often present severe boundary artifacts. To reduce boundary artifacts, one can smooth the boundaries of the observed image to decay to 0 (to make the imposed boundary condition to be more feasible) before those direct deblurring methods are applied to. This approach can reduce boundary artifacts in some degree, but, at the same time, makes it more difficult to recover near boundary image pixels.
The performance of direct deblurring methods depends heavily on the feasibility of imposed boundary conditions. The difficulty of imposing correct boundary conditions is the main reason why iterative deblurring approaches with free boundary conditions have been considered, despite the fact that fast direct deblurring methods are available [17, 28, 29].
Considering these facts, we suggest the free boundary conditionbased RL approach for the image deblurring problem (1). To reduce computational burden and accelerate the slow convergence of RL, we will propose blockiterative methods in the next section.
3 Proposed method
3.1 Block partition
for i=1,2,…,t. Notice that pixels only in Ω _{ i } can contribute the observation g ^{[i]} (defined on Λ _{ i }) and Λ _{ i }⊂Ω _{ i } for i=1,2,…,t.
Throughout this paper, Λ _{ i } are assumed to be mutually disjoint, unless stated otherwise.
3.2 Separated BIRL
The weights r ^{[i]} in (11) are motivated by the following interpretation. Recall that \({\mathcal T}\) represents the truncated convolution by the PSF k that is nonnegative and has 1 as its sum of all components. Thus, with the assumption that mutually disjoint blocks Λ _{ i }, i=1,2,…,t are selected to form \(\Lambda = \cup _{i=1}^{t}\Lambda _{i}\), \(({\mathcal T}^{\ast }\textbf {I}_{\Lambda _{i}})_{j_{1},j_{2}}\) can be interpreted as the probability with which the pixel (j _{1},j _{2})∈Ω contributes the observation on Λ _{ i }. This argument shows that weights r ^{[i]} make the intensity at the pixel (j _{1},j _{2}) to depend on f ^{[i],N } (the deblurred image from g ^{[i]} defined on Λ _{ i }) proportionally to the probability of the contribution of the pixel (j _{1},j _{2}) to g ^{[i]} (the observation on Λ _{ i }).
For certain deblurring problems, it is desirable to use overlapped blocks Λ _{ i }. In such case, separated BIRL separately deblurs g ^{[i]} on overlapped blocks Λ _{ i } to produce a deblurred block image f ^{[i],N } for each i, and, after cropping out some of overlapped pixels of f ^{[i],N }, i=1,…,t, combines them to the final output image.
3.3 Interlaced BIRL
where Ω _{ i } is computed by (10) from Λ _{ i }. In this case, one iteration of interlaced BIRL (from steps I3 to I5) updates most pixel values on Ω, including all pixel values on Λ, t times, while RL updates pixel values on Ω just once. Notice that, in the algorithmic point of view, the described benefit of interlaced BIRL is identical to that of OSEM [6] in emission tomography.
RL often uses the allone image I _{ Ω } as the initial guess f ^{0}. Interlaced BIRL, however, uses f ^{0}=RL(g,I _{ Ω },k,Λ) as the initial guess (see the step I1). This suggestion is made to update pixel values of f ^{0} on Ω _{ i }, without causing many discontinuities.
In the next subiteration (m=0 and i=2), pixel values on Ω _{2} are updated by \(\textbf {RL}\left (\mathbf {g}^{[2]}, \mathbf {f}^{1/4} \mid _{\Omega _{2}}, \mathbf {k}, \Lambda _{2}\right)\). Once pixel values are updated on remaining blocks, Ω _{3} and Ω _{4}, the resulting image is f ^{1} as shown in Fig. 1.
3.4 Algorithmic limitations of proposed methods
In the algorithmic point of view, separated BIRL can be used for any PSFs. Roughly speaking, the size of the PSF determines the minimum size of blocks that can be used for separated BIRL, and the number of blocks determines the maximum gain in separated BIRL by parallel computations.
where the denominator t in the lefthand side is used by considering that interlaced BIRL with well chosen t downsampled blocks accelerates iterations t times. The condition (13) does not hold for PSFs with large numbers of nonzero elements.
In our simulation, we used PSFs k with \({\mathcal S}_{\mathbf {k}} \le 441\) and 480×480 sized images. For such PSFs and images, the FFTbased RL implementation optimized for 512×512 sized images was slower than the pixelwise computationbased RL implementation. The condition \({\mathcal S}_{\mathbf {k}} \le 441\) includes PSFs that are used in many important image deblurring applications. For instance, any PSFs that have not more than 441 nonzero elements (e.g., a PSF of the form of the 441×441 diagonal matrix) satisfy this condition. Thus, interlaced BIRL is computationally more efficient than FFTbased RL in those image deblurring applications.
3.5 Free boundary condition
Proposed methods use the free boundary condition to suppress block artifacts. This suggestion is based on the observation that the free boundary condition successfully suppresses boundary artifacts for arbitrarily shaped images; note that periodic, reflective, and antireflective boundary conditions can be applied to rectangularshaped images only. For details, see [23–26].
The use of the free boundary condition appears in the step S4 and the step I4 by imposing no restriction on pixels on Ω _{ i }−Λ _{ i }. The step I1 of interlaced BIRL also uses the free boundary condition, again, by imposing no restriction on pixels on Ω−Λ.
3.6 Examples of blocks

For separated BIRL, select blocks Λ _{ i }, i=1,2,…,t, that make Ω _{ j }∩Ω _{ n } as small as possible for all pairs (j,n), 1≤j<n≤t.

For interlaced BIRL, select the block Λ _{ i } that makes Ω _{ i }≈Ω and \({\mathcal T}^{\ast }\textbf {I}_{\Lambda _{i}}\) to be uniform as much as possible for each i.
Simulation results in Section 4 will show why these two suggestions are important.
3.6.1 3.6.1 Rectangular blocks
Figure 2a shows 4×4rectangular blocks of the observed image (Fig. 5a). Separated BIRL with 2×2, 4×4, 8×8, and 16×16 rectangular blocks will be tested in the Gaussian deblurring.
Overlapped rectangular blocks can be formed by adding several pixel rows and columns to boundaries of disjoint rectangular blocks. This type of overlapped blocks will be also tested in separated BIRL.
3.6.2 3.6.2 Diagonal blocks
Figure 2b shows 8 diagonal blocks of the observed image (Fig. 5b). Separated BIRL with 2, 4, 8, and 16 diagonal blocks will be tested in the diagonal deblurring.
3.6.3 3.6.3 Rectangularlydownsampled blocks
3.6.4 3.6.4 Diagonallydownsampled blocks
Figure 2d shows 8 diagonallydownsampled blocks of the observed image (Fig. 5b).
Interlaced BIRL with 2, 4, 6, and 8 diagonallydownsampled blocks will be tested in the diagonal deblurring.
4 Simulation results and discussion
As mentioned in Section 3, interlaced BIRL is useful only when pixelwise computations (2) and (5), instead of FFTbased ones, are used for \({\mathcal T}\) and \({\mathcal T}^{*}\). In simulation studies, we used a personal computer equipped with 2.0 GHz Intel Core 2 Duo CPU and 8 GB RAM. In this computing environment, one RL iteration with FFTbased computations took 5.04 s both for Gaussian and diagonal deblurrings. On the other hand, one round of interlaced BIRL (the forloop in steps I3, I4, and I5) with pixelwise computations took 4.84 s for 4×4 rectangularlydownsampled blocks for Gaussian deblurring (\({\mathcal S}_{\mathbf {k}_{G}} = 441\)) and 0.88 s for 8 diagonallydownsampled blocks for the diagonal deblurring (\({\mathcal S}_{\mathbf {k}_{D}} = 61\)). This result implies that interlaced BIRL is, at least computational point of view, useful for both deblurring problems in our simulation. Considering these facts, we used pixelwise computations (2) and (5) in our simulation.
4.1 Standard RL
4.2 Separated BIRL
4.2.1 4.2.1 Rectangular blocks for Gaussian deblurring
Computation times for separated BIRL with rectangular blocks for the Gaussian deblurring and their smallest RSE results
NB  BP  SB  CB  RSE(%)  IN  PC 

1×1  n.a.  4.400  n.a.  0.52  420  1,848 
2×2  10.1  1.098  0.0080  0.53  441  494 
4×4  10.3  0.275  0.0093  0.53  432  129 
8×8  10.6  0.068  0.0126  0.54  499  44 
16×16  11.6  0.017  0.0208  0.55  513  20 
As ‘the number of blocks’ (hereafter abbreviated by NB) increased, computation times for the block partition (BP) and the combining of block images (CB) increased. Increments in computation times were, however, very small. The time for the single block iteration, however, linearly decreased as NB increased. There were also some slight increments in RSE and IN as NB increased.
Results in Table 1 and Fig. 7 show that separated BIRL with, at least up to 4×4, rectangular blocks achieves the desired goal (combining deblurred block images to final images without block artifacts, while maintaining deblurring quality and approximation rate) in the Gaussian deblurring.
4.2.2 4.2.2 Diagonal blocks for diagonal deblurring
Computation times for separated BIRL with diagonal blocks for the diagonal deblurring and their smallest RSE results
NB  BP  SB  CB  RSE(%)  IN  PC 

1  n.a.  0.658  n.a.  0.65  102  67 
2  2.2  0.327  0.0074  0.65  104  36 
4  2.2  0.163  0.0076  0.65  104  19 
8  2.2  0.082  0.0079  0.65  103  10 
16  2.2  0.040  0.0086  0.65  104  6 
Results in Table 2 and Fig. 8 show that separated BIRL with, at least up to 16, diagonal blocks achieves the desired goal (combining deblurred block images to final images without block artifacts, while maintaining deblurring quality and approximation rate) in the diagonal deblurring.
4.3 Interlaced BIRL
4.3.1 4.3.1 Rectangularlydownsampled blocks for Gaussian deblurring
Computation times for interlaced BIRL with rectangularlydownsampled blocks for the Gaussian deblurring and their smallest RSE results
NB  BP  BI  RSE(%)  IN  TC 

1×1  n.a.  4.40  0.52  420  1,848 
2×2  10.3  4.51  0.52  107  492 
4×4  10.8  4.84  0.52  26  136 
6×6  11.3  5.40  0.60  11  70 
8×8  12.1  6.22  0.91  2  24 
RSE results in Table 3 show that as NB increases, interlaced BIRL reaches its smallest RSE at an earlier iteration in the Gaussian deblurring (the iteration is accelerated by about NB times).
Results in Table 3 and Fig. 9 show that interlaced BIRL with up to 4×4 rectangularlydownsampled blocks achieves the desired goal (the accelerated iteration, with deblurring quality maintained) in the Gaussian deblurring.
4.3.2 4.3.2 Diagonallydownsampled blocks for diagonal deblurring
Computation times for interlaced BIRL with diagonallydownsampled blocks for the diagonal deblurring and their smallest RSE results
NB  BP  BI  RSE(%)  IN  TC 

1  n.a.  0.65  0.65  102  66 
2  2.2  0.69  0.65  52  38 
4  2.3  0.75  0.65  26  21 
6  2.3  0.81  0.65  17  16 
8  2.4  0.88  0.67  12  12 
Results in Table 4 and Fig. 11 show that interlaced BIRL with up to 6 diagonallydownsampled blocks achieves its desired goal (the accelerated iteration, with deblurring quality maintained) in the diagonal deblurring.
4.4 Miscellaneous results

In Gaussian deblurrings, as the size of Gaussian PSF increased, separated BIRL maintained the convergence rate of RL to smaller ranges of rectangular blocks.

In diagonal deblurrings, separated BIRL maintained the convergence rate of RL at least up to 16 diagonal blocks for all diagonal PSFs.

In Gaussian deblurrings, as the size of Gaussian PSF increased, interlaced BIRL accelerated iterations to wider ranges of rectangularlydownsampled blocks.

In diagonal deblurrings, as the size of diagonal PSF increased, interlaced BIRL accelerated iterations to wider ranges of diagonallydownsampled blocks.

Smoother images had more chance of exhibiting block artifacts in separated BIRL with rectangular blocks for Gaussian deblurrings. In separated BIRL with diagonal blocks for diagonal deblurrings, the test image did not affect block artifacts.

The use of Gaussian noise models itself did not affect the performance of proposed methods.

Deblurring with noisier data had less chance of exhibiting block artifacts in separated BIRL with rectangular blocks for Gaussian deblurrings. In separated BIRL with diagonal blocks for diagonal deblurrings, the noise level did not affect block artifacts.

Separated BIRL worked better for larger sized images, while the performance of interlaced BIRL did not depend on the image size.
Figure 13b, c shows 100th iterates of RL and separated BIRL with 4 horizontal blocks, respectively. No noticeable difference between Fig. 13b, c indicates that separated BIRL achieved its desired goal (the block artifactfree block deblurring) in this simulation. Figure 13d shows the 25th iterates of interlaced BIRL with 4 horizontallydownsampled blocks. Again, no noticeable difference between Fig. 13b, d indicates that interlaced BIRL achieved its desired goal (the accelerated iteration) in this simulation.
We tested proposed methods with an image taken from a real camera. Figure 13a shows a blurred and noisy image (of size 316×480) taken by the famous photographer Robert Capa [30]. Figure 13b, c, d shows deblurred images by RL, separated BIRL with 4 horizontal blocks, and interlaced BIRL with 4 horizontallydownsampled blocks, respectively. Here, horizontal blocks for separated BIRL and horizontallydownsampled blocks for separated BIRL were selected by following two suggestions explained in Section 3.6 (the estimated PSF for Fig. 13a is known to be spread horizontally [30]; see Fig. 15a).
Figure 13b, c shows 100th iterates of RL and separated BIRL with 4 horizontal blocks, respectively. No noticeable difference between Fig. 13b, c indicates that separated BIRL achieved its desired goal (the block artifactfree block deblurring) in this simulation. Figure 13d shows the 25th iterates of interlaced BIRL with 4 horizontallydownsampled blocks. Again, no noticeable difference between Fig. 13b and 13d indicates that interlaced BIRL achieved its desired goal (the accelerated iteration) in this simulation.
Block artifacts in Fig. 16b were caused by boundary artifacts generated in diagonal deblurrings of rectangular blocks. In fact, the Gaussian deblurring also generated boundary artifacts for both rectangular and diagonal blocks. Those boundary artifacts were, however, confined only in outermost part of Ω _{ i }−Λ _{ i } and did not appear in the final image (Fig. 16a), since r ^{[i]} in (11) were very small for image pixels (j _{1},j _{2})∈Ω _{ i }−Λ _{ i } where boundary artifacts were strong. In diagonal deblurrings of rectangular blocks, however, boundary artifacts appeared in the final image (Fig. 16b) because of the exact opposite reason. The asymmetrical and slow decay of the diagonal PSF k _{ D } makes such difference.
On the other hand, diagonal deblurrings of diagonal blocks do not produce block artifacts, as shown in Fig. 8; most of boundary artifacts from diagonal deblurrings of diagonal blocks appear only in Ω−Λ and hence can be easily removed by cutting out pixels in Ω−Λ. These results show that, in case when the formula (11) is used for the combining of deblurred block images, the shape of blocks is important for the diagonal deblurring but not for the Gaussian deblurring.
Selecting blocks depending on the PSF is not an easy task. To deal with this problem, we suggest to use overlapped rectangular blocks Λ _{ i } for separated BIRL, in the case when it is not certain which blocks should be chosen.
Separated BIRL with these overlapped rectangular blocks produces 256 deblurred block images and combines 256 deblurred block images of size 30×30, after cutting out required amount of rows and columns from 256 deblurred block images, to the final image (Fig. 17a). Figure 17b can be obtained by a similar procedure.
5 Conclusions
In this paper, we extended RL to blockiterative versions, separated BIRL, and interlaced BIRL, for image deblurring applications. We conducted simulation studies to test proposed methods in Gaussian and diagonal deblurrings. Simulation results showed that separated BIRL can have a benefit of parallel computations for Gaussian and diagonal deblurrings, with a wide range of rectangular and diagonal blocks, respectively. Simulation results also showed that interlaced BIRL can accelerate the iteration for Gaussian and diagonal deblurrings but only with a limited range of rectangularly and diagonallydownsampled blocks, respectively.
In this work, proposed methods were tested only for Gaussian and diagonal PSFs, k _{ G } and k _{ D } in Fig. 3. It is clear that proposed methods can be extended to other PSFs as long as admissible blocks can be selected. But, as mentioned earlier, it is not clear how blocks should be formed for a specific PSF. Simulation results in Section 4 indicates that overlapped rectangular blocks can be used as universal admissible blocks for separated BIRL, at the cost of additional computations caused by overlapped pixels. For interlaced BIRL, however, such universal admissible blocks are not known yet.
Declarations
Acknowledgements
This work was supported by the Inje Research and Scholarship Foundation in 2012 and conducted while the author was a visiting scholar at the Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University. The author would like to express his sincere appreciation to Prof. Lucier at Purdue University for his helpful suggestions.
Authors’ Affiliations
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