Variational Regularized Bayesian Estimation forJoint Blur Identification and Edge-Driven Image

Restoration

Hongwei Zheng and Olaf Hellwich

Computer Vision & Remote Sensing, Berlin University of TechnologyFranklinstrasse 28/29, Office FR 3-1, D-10587 Berlin

hzheng,hellwich@cs.tu-berlin.de

Abstract. The paper presents a novel method for joint blur identifica-tion and edge-driven image restoration in variational double regularizedBayesian estimation. The motivation is that the degradation of imagesincludes not only additive, random noises but also multiplicative, spa-tial degradations, i.e., blur. Traditional nonlinear filtering techniques areobserved in underutilization of blur identification techniques, and viceversa. To improve blur identification and image restoration, a designednonlinear diffusion operator and a point spread function (PSF) learningterm are needed to integrate the proposed approach. A newly introducedprior solution space provides prior information to the PSF learning. Thecost functions are optimized iteratively in an alternate minimization withrespect to the estimation of images and PSFs. Numerical experimentsshow that the proposed algorithm is efficient and robust, handling im-ages that are formed in different environments with different types andamounts of blur and noise.

1 Introduction

The challenge of blind image deconvolution (BID) is to uniquely define the op-timized signals from the degraded images with unknown blur information. BIDincluding blur identification and image restoration is an important and widelystudied problem in computer vision and image processing. An ideal image f inthe object plane is normally degraded by two unknown processes. These twoprocesses are linear space-invariant point spread function (PSF) h and additivewhite Gaussian noise n which are formulated using the lexicographic notationg = h∗f +n. This equation provides a good working model for image formation.

In two decades, there has been considerable progress in image restorationand edge-preserving image diffusion. Nonlinear filtering techniques are typicallybased on either partial differential equations (PDEs) [1] incorporated in varia-tional methods, or nonlinear estimation incorporated in statistical methods [2].A great number of nonlinear PDE models have been successfully introduced to awide variety of applications in computer vision [3], including the total variationmodel [4], the shock filter [5], the cartoon-texture model [6], the coherence en-hancing flow [7], the Mumford and Shah model [8] and its PDE version [9]. The

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(a) (b) (c) (d)

Fig. 1. (a) Ground truth. (b) Spatially degraded blurring image, Gaussian noise 15dB.(c) Restored image b using shock filter. (d) Restored image b using the normal TV.

nonlinear PDE models have proven to be very effective for image restoration andimage enhancement, but only for the well adapted input data with some under-lying geometric assumptions [10]. Furthermore, most nonlinear image diffusiontechniques address the removal of additive, random noise without consideringmultiplicative, spatial degradations such as blur and its noise, e.g., Fig. 1.

On the other hand, there is somehow a gap between blur identification, imagediffusion and image restoration. Most of the blur identification approaches arebased on the regularization theory [11, 12] which replaces an ill-posed problem bya well-posed problem with an acceptable approximation to the solution. Gradu-ally, a set theoretic approach based iterative regularization [13], projection-basedconjugate-gradient methods [14, 15] and parametric model-based image restora-tion methods [16] have been developed. However, the restoration results cannotbe improved progressively due to isotropic diffusion operators, even if the blurkernel is accurately identified. The restored results are also influenced by un-derutilization of prior information. Even if a unique solution exists, a properinitialization value is still intractable, e.g., when the cost function is non-convex,convergence to local minima often occurs without proper initialization. Molina etal. [17] have also reported that the estimates for the PSF could vary significantly,depending on the initialization.

Hence, general strategies are still needed that are effective for solving theblur problem as well as achieving maximum fidelity of image restoration. TheBayesian estimation framework provides a structured way to include prior knowl-edge concerning the quantities to be estimated [18]. This process supports moreeffective prior information or constraints to yield a unique solution. In this pa-per, we present a PSF learning term and an edge-driven nonlinear diffusion termwhich is used to replace an isotropic diffusion operator in a variational regular-ized Bayesian estimation. The introduced prior solution space of PSFs supportsaccurate parametric PSF models to reduce the search space in Bayesian PSFlearning. The chosen diffusion operator automatically adjusts to the strengths ofedge gradient. It has several important effects: firstly, it provides a good initialvalue in regularization via a statistic method for blur identification. Secondly, itshows a theoretically and experimentally sound way of how local diffusion oper-ators are changed automatically. Finally, the double cost functions are optimized

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iteratively in an alternating minimization procedure with respect to the imageand the PSF. The experimental results show that the method yields encouragingresults under different kinds and amounts of noise and blur.

The paper is organized as follows. Variational regularized Bayesian estimationis described in Sect. (2). Blur identification and edge-driven image restoration arediscussed in Sect. (3) and Sect. (4). Experimental results are shown in Sect. (5).Conclusions are summarized in Sect. (6).

2 Double Regularized Bayesian Estimation

We propose to solve the joint PSF identification and edge-driven image restora-tion in a double regularization functional according to:

E(f , h) = 12

∫Ω

(g − h ∗ f)2dA︸ ︷︷ ︸fidelityTerm

+λ

∫Ω

φε(x,∇f)dA︸ ︷︷ ︸imageSmoothing

+β

∫Ω

|∇h|dA︸ ︷︷ ︸psfSmoothing

+γ

∫Ω

|h− hf |dA︸ ︷︷ ︸psfLearning

where dA = dxdy. The estimates of the ideal image f and the PSF h are denotedby f and h respectively, g is an observed image. The image smoothing term is avariable exponent, nonlinear diffusion term. The PSF smoothing term representsthe regularization for the blur kernel. The flexibility of the last term γ

∫Ω|h −

hf |dA is the proposed PSF learning term which denotes the PSF learning errorbetween the current estimated PSF and the final selected PSF model which canadjust and incorporate the parametric model of the PSF throughout the processof blur identification and image restoration.

Following the Bayesian paradigm, the estimated true image f , the estimatedPSF h and the observed image g fulfill,

p(f , h|g) =p(g|f , h)p(f , h)

p(g)∝ p(g|f , h)p(f , h) (1)

This Bayesian MAP approach can be seen as a regularization approach whichcombines optimization methods for minimizing two proposed cost functions inthe image domain and the PSF domain. The cost function of the restored imagef and the estimated PSF h from Eq. (1) are deducted respectively according tothe following,

E(f(g,h)) ∝ p(g|f , h)p(f),E(h(g,f)) ∝ p(g|f , h)p(h)

(2)

Some constraints are assumed for the application of these equations, e.g., theimage pixel correlations are independent identically distributed (i.i.d.). However,manipulation of probability density functions in Bayesian estimation is difficult[19]. Several forms of the prior distribution, e.g., Gibbs distribution [20], smooth-ness prior or maximum entropy have been suggested by researchers from differentscientific disciplines. However, they are based on general knowledge about im-ages. For most real blurred images, power spectral densities vary considerably

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from low frequency domain in the uniform smoothing region to medium and highfrequency domain in the discontinuity and texture regions. Moreover, most PSFsexist in the form of low-pass filters. Blur identification can be based on thesecharacteristic properties. The proposed prior solution space supports parametricPSFs in a Bayesian MAP estimation, as PSFs of numerous real blurred imagessatisfy parametric models up to a certain degree.

3 Regularized Bayesian Estimation in the PSF Domain

3.1 Solution Space of Blur Kernel Priors

The proposed prior supports the parametric structure of PSFs in Bayesian esti-mation and reduces the effective search space.

Fig. 2. PSFs in the prior solution space. (a) Original synthetic image. (b) Pill-boxPSF. (c) Gaussian PSF. (d) Linear motion PSF.

We define a set Θ as a solution space of Bayesian estimation which consistsof primary parametric blur models as Θ = hi(θ), i = 1, 2, 3, ..., N presentedin Fig. 2. hi(θ) represents the ith parametric model of PSF with its definingparameters θ, and N is the number of blur kernel types.

hi(θ) =

h1(θ) ∝ h(x, y;Li, Lj) = 1/K, if |i| ≤ Li and |j| ≤ Lj

h2(θ) ∝ h(x, y) = K exp(−x2+y2

2σ2 )h3(θ) ∝ h (x, y, d, φ) = 1/d, if

√x2 + y2 ≤ D/2, tanφ = y/x

(3)

h1(θ) is a pill-box blur kernel with radius K. h2(θ) is a Gaussian PSF character-ized by its variance σ2 and a normalization constant K. h3(θ) is a simple linearmotion blur PSF with a camera motion d and a motion angle φ. The other blurstructures like out-of-focus and uniform 2D blur [21], [16] are also built in thesolution space as a priori information. A set of parametric PSFs construct apredefined prior solution space for Bayesian MAP Estimation.

3.2 From Learnt PSF Statistics to PSF Estimation

The cost function for estimating PSF according to Eq. (2) is described in a totalvariational regularization with a PSF learning term,

E(h(g,f)) = arg maxh

p

(g

∣∣∣h, f)

pΘ

(h)

(4)

5

= α22

∫Ω

(g − h ∗ f)2dA + β

∫Ω

|∇h|dA + γ

∫Ω

|h− hf |dA

where p(g|f , h) ∝ exp−α2

2

∫Ω

(g − h ∗ f)2dA

, pΘ(h) is the prior knowledgeand needs to be computed. Θ is a set of primary parametric blur priors. Sincean image represents intensity distributions that cannot take negative values, thePSF coefficients are always nonnegative, h(x) ≥ 0. Furthermore, since imageformation system normally do not absorb or generate energy, the PSF shouldfulfill

∑x∈Ω h (x) = 1.0 , x ∈ Ω and Ω ⊂ R2.

During the estimation process, each estimated PSF h for a sampling imageregion normally has some noise error hj = h + noise. To achieve a better esti-mated PSF with less noise, the estimated PSF is convolved with several low-passfilters to remove noise and get several PSF neighbors hj , j ∈ (1, ...,K). In or-der to study the interaction between statistical blur kernel knowledge and blurdegraded image information, we define the likelihood P (hj) of the estimatedPSF h of an observed image in resembling the ith parametric model hi(θ) in amultivariate Gaussian distribution, P (hj) ∝

arg maxθ

log

1

(2π)LB2 |

∑dd|

12· exp

[−1

2

(hi (θ)− hj

)T ∑−1dd

(hi (θ)− hj

)]

The first subscript i denotes the index of blur kernel. The modeling error d =hi(θ)− h is assumed to be a zero-mean homogeneous Gaussian distributed whitenoise process with covariance matrix

∑dd = σ2

dI independent of image. LB isan assumed support size of blur. Then the Gaussian probability corresponds toa PSF learning likelihood:

lij(hj) =12exp[(hi(θ)− hj)t

∑−1

dd(hi(θ)− hj)] (5)

In reality, most of blurs satisfy up to a certain degree of parametric structures. Abest fit model hi(θ) for h is selected according to the Gaussian distribution anda weighted mean filter. The mean value of PSF learning likelihood li(h) is thatlij(hj) is weight divided by d(h, hj). d(h, hj) is the Euclidean distance betweenh and its neighbor hj , li(h) =

∑Kj=1[E(hj)d2(h, hj)]/[d2(h, hj)].

The weighted mean likelihood li(h) depends on two conditions. The firstcondition is the likelihood value of the blur manifold lij(hj), and the second isthe distance between h and its neighbor hj . The estimated output blur modelhf is obtained from the parametric blur models using

hf = [l0(h)h +∑C

i=1li(h)P (hj)]/[

∑C

i=1li(h)] (6)

where l0(h) = 1 −max(li(h)), i = 1, ..., C. The main objective is to assess therelevance of current estimated blur h with respect to parametric PSF models,and to integrate such knowledge progressively into the computation scheme.

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If the current blur h is close to the estimated PSF model hf , that means h

belongs to a predefined parametric blur structure. Otherwise, if h differs fromhf significantly, this means that current blur h may not belong to the predefinedPSF priors.

4 Edge-Driven Regularization in the Image Domain

The fundamental goal is to find f given an observed image g and an estimatedPSF h by minimizing the image cost function. In the image domain, the costfunction can be minimized according to the following formulation,

E(f(g,h)) =α1

2

∫Ω

(g − h ∗ f)2dA+ λ min

f∈BV ∩L2(Ω)

∫Ω

φ(x,∇f)dA (7)

where p(g|f , h) ∝ exp−α1

2

∫Ω

(g − h ∗ f)2dA

and p(f) is extended to a non-linear diffusion functional with variable exponent [22]. Edge-preserving piecewisesmoothing can be considered as a priori knowledge for the estimation of image,then p(f) ∝ exp

−

∫Ω

φ(x,∇f)dA

, where

∫Ω

φ(x,∇f)dA =

1

q(x) |∇f |q(x), |∇f | < β

|∇f | − βq(x)−βq(x)

q(x) , |∇f | ≥ β(8)

where β > 0 is fixed, and 1 ≤ q(x) ≤ 2. The term q(x) is chosen as q(x) =1 + 1

1+k|∇Gσ∗I(x)|2 based on the edge gradients shown in Fig. 3, I(x) is the

observed image g(x), Gσ(x) = 1σ exp(−|x|

2

2σ2 ) is a Gaussian filter. k > 0, σ > 0 arefixed parameters. We solve the minimizaion problem for the image cost functionEq. (7) numerically using the flow of its associated Euler-Lagrange equation,

∂f

∂t− λdiv(φ(x,∇f)) + α1(f ∗ h− g) = 0, inΩ × [0, T ] (9)

where we indicate with div the divergence operator, and with ∇ and ∆ respec-tively the gradient and Laplacian operators, with respect to the space variables.The Neumann boundary condition ∂f

∂n (x, t) = 0 on ∂Ω × [0, T ] and the initialconditionf(x, 0) = f0(x) = g in Ω are used, where n is the direction perpendic-ular to the boundary, g is the observed image.

The numerical implementation of the nonlinear diffusion operator div(φ(x,∇f))is based on central differences for coefficient and the isotropic term, minmodscheme for the curvature term, and upwind finite difference scheme developedby Osher and Sethian for curve evolution [4] for the hyperbolic term. The diffu-

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(a) (b)

Fig. 3. Strength of q(x) in the Lena image. (a) Strength of q(x) between [1,2] in theLena image. (b) Strength of q(x) is shown in a cropped image with size [50, 50].

sion operator becomes,

div(φ(x,∇f)) =

|∇f |p(x)−2︸ ︷︷ ︸Coefficient

[(p(x)− 1)∆f︸ ︷︷ ︸IsotropicTerm

+(2− p(x))|∇f |div(∇f

|∇f |)︸ ︷︷ ︸

CurvatureTerm

+∇p · ∇f log |∇f |︸ ︷︷ ︸HyperbolicTerm

]

with

p(x) =

q(x) ≡ 1 + 1

1+k|∇Gσ∗I(x)|2 , |∇f | < β

1, |∇f | ≥ β

(10)

The image is restored by denoising in the process of edge-driven image diffu-sion as well as deblurring in the process of image deconvolution. Firstly, thechosen of variable exponent of p(x) is based on the computation of gradientedges in the image. In homogeneous flat regions, the differences of intensity be-tween the neighboring pixels are small; then the gradient ∇Gσ become smaller(p(x) → 2). The isotropic diffusion operator (Laplace) is used in such regions. Innon-homogeneous regions (near a edge or discontinuity), the anisotropic diffu-sion filter is chosen continuously based on the gradient values (1 < p(x) < 2) ofedges. The reason is that the discrete chosen of anisotropic operators will ham-per the recovery of edges. Secondly, the nonlinear diffusion operator for piecewiseimage smoothing is processed during the image deconvolution based on a previ-ously estimated PSF . Finally, coupling estimation of PSF (deconvolution) andestimation of image (edge-driven piecewise smoothing) are alternately optimizedapplying a stopping criteria. Hence, over-regularization or under-regularizationis avoided by pixels at the boundary of the restored image.

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(a) (b) (c)

Fig. 4. Comparison of the normal TV method and the suggested edge-driven diffusionmethod. (a) Un-blurred image with 8dB Gaussian noise. (b) Denoising using the normalTV method. (c) Denoising using the suggested edge-driven diffusion method.

5 Experiments and Results

5.1 Alternate Minimization of PSF and Image Energy

To achieve the joint results, a scale problem arises between the minimizationof the PSF and the image via steepest descent. The reason is that the ∂E/∂h

is∑

x∈Ω f(x) times larger than ∂E/∂f . Also, the dynamic range of the image[0, 255] is larger than the dynamic range of the PSF [0, 1]. The scale factorchanges dynamically with space coordinates (x, y). To avoid the scale problem,an alternate minimization method following the idea of coordinate descent [15]is applied. The alternating minimization decreases complexity. The equationsnumerically derived from Eq. (1) are using finite differences which approximatethe flow of the Euler-Lagrange equation associated with it,

∂E/∂f = α1h(−x,−y) ∗ (h ∗ f − g)− λdiv(φ(x,∇f)) (11)

∂E/∂h = α2f(−x,−y) ∗ (f ∗ h− g)− β∇h · div(∇h

|∇h|) + γ|1− hf/h| (12)

Neumann boundary conditions are assumed [23]. In the alternate minimization,blur identification including deconvolution, and image restoration including de-noising are processed alternatingly for the estimation of the image and the PSF.The partially recovered PSF is the prior for the next iterative image restorationand vice versa. The algorithm is described in the following:

Initialization: f0(x) = g(x), h0(x) is random numberswhile (nmse > threshold

(1). nth it. fn(x) = arg min(fn|hn−1, g), fix hn−1(x)(2). (n + 1)th it. hn+1 = arg min(hn+1|fn, g), fix fn(x), h(x) > 0

end

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The global convergence of the algorithm to the local minima of cost func-tions can be established by noting the two steps in the algorithm. Since theconvergence with respect to the PSF and the image are separate and optimizedalternatively, the flexibility of this proposed algorithm allows us to use conjugategradient algorithm for computing the convergence. Conjugate gradient methodsutilize the conjugate direction instead of local gradient to search for the minima.Therefore, it is faster and also requires less memory when compared with theother methods. If an image has M ×N pixels, the above conjugate method willconverge to the minimum of Ef (f |g, h ) after m MN steps based on par-tial conjugate gradient method. A meaningful measure called normalized meansquare-error (NMSE) is used to evaluate the performance of the identified blur,NMSE = (

∑x

∑y (h(x, y)− h(x, y))2)1/2/(

∑x

∑y h(x, y)). The selected PSFs

of NMSE normally has a range [0, 0.1] depending on the different PSFs.

5.2 Edge-Driven Blind Deconvolution of Degraded Images

Experimental results are presented to illustrate the efficiency and the effective-ness of the proposed method. BID techniques can remove multiplicative blur,however, it is heavily influenced and restricted by some noise. Nonlinear PDEdiffusion techniques have advantages regarding the removal of additive, randomnoise but can not deblur due to the underlying principle.

In the first experiment, we have studied some nonlinear PDE diffusion fil-tering techniques. Although nonlinear PDE diffusion methods have advantagesfor image denoising and enhancement, they still need to incorporate image de-convolution to recover structural and textural parts for degraded images suchas those shown in Fig. 1 and Fig. 5. The experiment in Fig. 4 also shows thatthe suggested edge-driven diffusion term performs better than the normal TVmethod for unblurred noisy image. Compare to the blur identification methodin Fig. 6, the blurred noisy (30 dB) image is blur identified and restored usingthe suggested method.

We compare the traditional Lucy-Richardson (L-R) deconvolution methodwith known PSF and the suggested method with unknown PSF, the ”Lena” testimage is heavily blurred with two different signal-to-noise ratio (SNR) 20 dB and12 dB shown in Fig. 7(a)(d) respectively. However, the noise is heavily amplifiedduring the L-R deconvolution with known PSF in Fig. 7(b)(e). In the suggestedmethod, the PSF is computed in the Bayesian estimation so that the alternateminimization (AM) algorithm gets an accurate PSF initial value for the imagedeconvolution. The coefficient of diffusion operator p(x) is then varying in theinterval [1, 2]. The estimated PSF supports the varying of the image smoothingcoefficient λ progressively till the best recovered image is reached, e.g., shownin Fig. 7 (c) and (f). From the restored images, we can find Lena’s face is moresmooth while the fine details of eyes, mouth, nose, and feather are preserved viaanisotropic operators.

This experiment demonstrates the flexibility of this proposed algorithm whichcan further utilize the advantages of nonlinear PDE diffusion in an adaptive andcontinuous way. The results of the proposed approach also show that the image

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(a) (b) (c)

Fig. 5. (a) Blurred image with white Gaussian noise 12dB, size:[256,256]. (b) Diffusionusing the Shock filter. (c) Diffusion using the normal TV method.

(a) (b) (c) (d) (e)

Fig. 6. (a) Ground truth image. (b) Ground truth PSF. (c) Blurred image with whiteGaussian noise 30dB. (d) Blind deconvolution for image c. (e) Estimated PSF

(a) (b) (c)

(d) (e) (f)

Fig. 7. (a)(d) Blurred image with white Gaussian noise SNR=20dB and 12dB respec-tively, image size:[256,256]. (b)(e) Restoration using L-R method with known PSF.(c)(f) Blind deconvolution using the suggested method.

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and the PSF can be recovered even under the presence of stronger noise andblur.

(a) (b) (c)

(d) (e) (f)

Fig. 8. (a)(d) Real video frames. (b)(e) Blurred parts in video. (c)(f) Results of blinddeconvolution using the suggest method

We also illustrate the capability of the proposed algorithm to handle blurdegraded objects in real-life video data by non-standard blur in Fig. 8. Thevideo frames are captured from films or video test data. The degraded videoobjects are separated into RGB colour channels and each channel is performedrespectively. Based on the estimated PSFs and parameters, piecewise smoothand accurate PSF model helps to suppress the ringing effects and preserve theedges or discontinuities during the BID.

6 Conclusions

This paper validates the hypothesis that the challenging task of nonlinear PDEdiffusion and BID are tightly coupled in a variational regularized Bayesian es-timation. Mutual support of image deconvolution and adaptive nonlinear im-age diffusion is demonstrated. Accurate prior information directly supports thecomputation for solving an ill-posed inverse problem. The suggested method canspeed up the minimization of related cost functions progressively based on theaccurate initialization. The partially recovered PSF is the prior for the next it-erative image deconvolution and diffusion, and vice versa. The nonlinear PDEdiffusion improves the results significantly even with stronger noise. It is clear

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that the proposed method is instrumental in discontinuity-preserving BID andcan easily be extended in practical environments.

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