The Study On Image Deconvolution Algorithm

Deconvolution is an inverse problem existing in a wide variety of signal and image processing fields including physical, optical, medical, and astronomical applications. For example, practical satellite images are often blurred due to limitations such as aperture effects of the camera,camera motion, or atmospheric turbulence. Deconvolution becomes necessary when we wish a crisp deblurred image for viewing or further processing. Due to the point spread function is the low-pass filter, the high frequency of image is suppressed and even lost. The aim of deconvolution is to find back the high frequency part. It is not difficult to understand that the observation noise will be amplified, which means that the deconvolution results may deviate from the true solution. So the deconvolution method should compromise between the restored image and noise amplification.I. Image Deconvolution based on Regular Hexagonal PSFThere are three common types PSF of image degradation:motion blur,out of focus and Gaussian model. The last two supposes using circular aperture. For most of other researches on image restoration beyond these three models, to satisfy the characteristics of circle symmetry is required.With the development of science and technology, optical systems having the regular polygon apertures generally, are increasingly broadly applied in most fields. For the aim to increase the accuracy and efficiency of image restoration, it is necessary to study the corresponding deblurring methods in the research of image restoration according to the imaging mechanism of degraded images. In the existing literatures, there are few researches on restoration of degraded images from iris diaphragms.The degradation procedure is often modeled as the result of a convolution with a low-pass filter y(t,s)=(h*χ)(t,s)+γ(t,s)(1) where x and y are the original image and the observed image, respectively.γ is the noise introduced in the procedure of image acquisition, and it is generally assumed to be independent and identically distributed (i.i.d.) zero-mean additive white Gaussian noise (AWGN) with variance a2."*" denotes convolution, and h denotes the point spread function (PSF) of a linear space invariant system H. The process of deblurring is known to be an ill-posed problem. Thus, to obtain a reasonable image estimate, a method of reducing controlling noise needs to be utilized.The Fourier domain is the traditional choice for deblurring because convolution sim-plifies to scalar Fourier operations: Y(υ,ν)=H(υ,ν)X(υ,ν)+Γ(υ,ν)(2) where Y, H, X, F are the2-D Fourier transforms (DFTs) of y, χ, h andγ, respectively. For the out-of-focus image degradation of regular-hexagon aperture, the contributions of this paper are as follows:1). Present the Point spread function model of regular hexagon: where the a is the side length of regular hexagon, andThis point spread function model depicts the out-of-focus degradation of regular-hexagon aperture in nature, and is important to accurately compute the out-of-focus PSF of regular-hexagon aperture. 2). Present and prove the zero-point distribution theorem of Fourier trans-form H(υ,υ).The first zero point ring theorem:There is one and only one zero point of f(χ) in the interval (0,3π/κ+√3).The second zero point ring theorem:There is one and only one zero point of f(χ) in the interval [4π/√3,9π/2√3].where f(χ) is the mathematical transform of H(υ,υ).These two theorems prove that there exists zero points for regular hexagon aperture defocused PSF in some intervals of frequency domain and guide the searching of zero points. Define zero point set L1contain all the points of the first zero point ring. And likely, L2contain all the points of the second zero point ring (It is not like a "ring" but some kind of twisted regular hexagon). According to the two theorems, we can use numerical methods to get L1and L2.3). With all the works above, we propose a fast regular hexagon aperture defocused image restoration method, and the main steps are as follows:We use the Fourier transform of observed image y(t, s) and the first zeros point ring and the second point ring of H(υ,υ) to estimate the side length a and the angle θ of the regular hexagon. The next problem is conversion into the non-blind deconvolutin problem. For improve the computational efficiency, while the SNR of image is very high, we can use Wiener method to solve the deconvolution problem:a). Compute the Fourier transform of y(t,s) to obtain Y(υ,υ).b). Solve the problem: to compute the two parameter of PSF:the side length a and the deflection angle θ. So we obtain the Haα.θ(υ,υ)=H(α(υ,υ)Rθ).c). Use Wiener filter (or other non-blind deconvolution methods) and the Ha,0to estimate the real image.The results of experiments on high SNR classic images show our algorithm works well.II.Texture-Preserving Image Deconvolution AlgorithmThere are many the warped oscillatory functions or oriented textures (e.g., figer-print, seismic profile, engineering surfaces) in the nature image. The texture charac-teristic is the high frequency information, while image degradation is the low-pass filter process. Due to limitation of the priori estimate, most deconvolution methods using wavelets,curvelet,total variance model could not preserve the image texture informa-tion. Because image textures are important visual information to the human eye, the results with textures lost may show unnatural looks. Wave atom is a new transform which is half multi-scale and half multi-directional. An important advantage in the use of this redundant wave atom transform implementation for deconvolution is that it has the ability to adapt to arbitrary local directions of a pattern, and the ability to sparsely represent anisotropic patterns aligned with the axes. The texture-shape elements of wave atoms also own very high directional sensitivity and anisotropy. Obviously it is natural to apply the wave atoms for texture-preserving image deconvolution.Although the wave atom-based method is efficient in texture-preserving image de-convolution, it is prone to producing edge ringing which relates to the structure of the underlying wave atom. In order to reduce the ringing, we develop an efficient joint non-local means filter by using the wave atom deconvolution result.1). Present the image deconvolutin algorithm based on Wave Atom trans-form. This method could suppress the leaked colored noise while preserving image texture.The Fourier domain is the traditional choice for deblurring because convolution sim-plifies to scalar Fourier operations: y(k1,k2)=H(k1,k2)X(k1,k2)+Γ(k1,k2)(6) where Y, X, H and F are the2-D discrete Fourier transforms (DFTs) of y, χ, h andγ, respectively.a). Fourier-Tikhonov Shrinkage (FoRD). The image estimate xa in the Fourier domain is given by Xα(k1,k2)=Y(k1,k2)Ha(k1,k2)(7) and where H is the complex conjugate of H, and A(k1,k2)>0commonly referred to as regularization terms, control the amount of shrinkage.b). Wave Atom-based Wiener Shrinkage Filter.The wave atom-based Wiener Shrinkage estimate of colored noisy image xa is: where cα,μ and cαh,μ denote the wave atom coefficients of the still noisy image χα and the hard-thresholding estimate χαh using wave atom decomposition μ,σα2,μ are the noise's variance at wave atom subscriptμ, and βis regularization parameters. Computing the inverse wave atom transform with cαω,μ, we can get the wave atom-based Wiener estimate χαω2). Present the joint non-local means filer to reduce the ringing, and this filter could suppress the leaked colored noise while preserving image details. The joint non-local means filter is proposed to surmount the problems of boundary effects, and to be effective in regularizing the approximate deconvolution process.a). Use the estimate image χαω as a reference image in the spatial non-local mean filter for the noisy image. Considering the wave atom deblurring image χαω preserves most of the important image texture features, so the joint non-local means filter could improve the image quality, the estimate image χJ can be computed as the following formula: where b). Weighted Estimate. If we use the joint non-local means filter directly, there are some noise spots in the output debluring image, especially when the noise level is high. In our work, for simplicity, we add the wave atom-based estimate in the joint non-local means filter to suppress the spots. χ*=βχαω+(1-β)χJ,β∈[0,1)(12) This method also can balance the wave atom-based estimate and joint non-local means-based estimate, and improve the image quality.