图像反卷积算法研究
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摘要
近半个世纪以来,图像反卷积算法始终是数字图像处理领域研究的热点之一.图像反卷积算法的研究是一个重要但又困难的问题,也是具有理论挑战性的分支以及具有重要的实际应用价值.图像反卷积作为一种改善图像质量的技术,从失真和降质的观测量来估计原来不失真图像,尽量降低和消除图像的失真和噪声.在图像复原的问题中,造成图像模糊的点扩散函数总是一个低通滤波器,这就使输入图像的高频成分受到抑制甚至丧失.反卷积是一个逆过程,目的是要”找回”丢失了的高频成分.不难理解,反卷积过程包含在观测中的噪声会被放大,这意味着反卷积的结果可能偏离真实的解.为了获得尽可能真实的解,在反卷积方法上要加进抑制噪声的考虑,这就需要在复原图像和噪声放大之间做出适当的折衷.
一.正六边形点扩散函数图像反卷积
现有图像复原方法的研究中,点扩散函数有有限的几种,这可以简化‘大类点扩散函数的问题.关于图像退化点扩散函数的三种常用模型：线性运动、圆盘离焦和高斯模型.除了线性运动模型,另外两种模型也都是基于圆形光圈建立起来的.有些图像复原的研究工作虽然超出了这三种模型,但仍然要求点扩散函数具有圆对称性质.
我们注意到,随着科技的不断进步,具有可变光圈(一般是正多边形光圈)的光学系统越来越广泛地应用于各行各业.为了适应客观实际需求,有必要在图像反卷积方法的研究中,根据这一类降质图像的成像机理,研究相应的反卷积方法,以提高图像复原的精度和效率.而在现有文献中,关于可变(正多边形)光圈降质图像的反卷积方法的研究至今未见相关报导.本论文的主要研究之一是对正六边形光圈离焦图像反卷积方法的研究.
图像的降质模型可以写成如下形式：y(l,s)=(h*x)(t,s)+γ(t,s)(1)其中,y,h,x和r分别表示观测图像,真实图像,点扩散函数和噪声.图像复原的目的就是通过观测图像y(t,s)来得到真实图像x(t,s)的一个精确估计.根据Fourier变换的卷积定理,上述模型在频域的表达为：Y(υ,υ)=H(υ,υ)X(υ,υ)+Γ(υ,υ)(2)其中Y,H,X,Γ分别是y,h,x,r的Fourier变换.
对于处理正六边形可变光圈造成的图像离焦,本文的贡献有如下几个方面：
1)基于正六边形光圈光学系统的成像机理,提出了正六边形离焦模型(3.2节)：其中a为正六边形的边长,Rθ=(?).
该点扩散函数模型刻画了正六边形光圈离焦的本质,对于精确估计正六边形光圈离焦点扩散函数是必要的.
2)本文通过分析和推导,我们从理论上提出并证明了正六边形离焦点扩散函数Fourier变换H(u,υ)的零点分布定理(3.3节).
第一圈零点分布定理(定理3.7)：当0 第二圈零点分布定理(定理3.14)：f(x)的第二个零点在区间(?)≤x≤(?)上.在这个区间上f(x)有且仅有一个零点.
其中f(x)是H(u,υ)的一种数学变形.
这两个定理从理论上证明了正六边形光圈离焦点扩散函数在频域存在零点,并可以对零点具体位置的搜索提供指导.根据这两个定理,用数值方法先一次性计算出H(u,υ)在频域中心附近的第一圈零点集合L1和第二圈零点集合L2.
3)在上面工作的基础上,本文提出了正六边形光圈离焦图像快速复原方法(3.4节),方法的主要步骤如下：
为了得到点扩散函数,利用观测图像y(t,s)的Fourier变换和H(u,υ)的第一圈、第二圈零点集合L,可以辨识出正六边形点扩散函数发生离焦现象时正六边形的边长a和偏转角度θ.接下来的问题就是一个已知点扩散函数的非盲反卷积问题了.为了算法的执行效率,加之图像的信噪比(SNR)比较高,可以采用Wiener滤波的方法.具体步骤如下：
a).计算观测图像y(t,s)的Fourier变换Y(u,υ).
b).求解问题得到点扩散函数的两个参数的估计：正六边形的边长a和偏转角度θ.并由此得到点扩散函数的估计Ha,θ(υ,υ)=H(n(υ,υ)Rθ).
c).利用Wiener滤波(或者其他有效的非盲图像反卷积算法),通过Ha,θ来估计真实图像.
应用此方法对于有较高信噪比(SNR)的图像进行了图像复原实验,得到比较满意的效果.
二.保持纹理结构的图像反卷积算法
在自然图像中,纹理特征广泛的存在.例如指纹图像,地震剖面图像等等,这些图像中包含着丰富的弯曲的振荡纹理和定向纹理信息.由于纹理特征是一种高频信息,而图像模糊过程是一个低通滤波的过程,因此在图像反卷积时,就需要将这种重要的图像特征恢复出来.现今的绝大多数的反卷积算法,采用的先验约束主要是小波、曲波、全变差等,由于它们对纹理信息表达的局限性,因此这些算法难以将图像的纹理信息很好的复原.这些图像反卷积算法往往会将图像中的纹理丢失,造成图像失真和视觉上的不自然.为了很好的解决纹理图像复原这一类特殊而且重要的问题,就需要采用适合纹理信息表达的数学工具.波原子变换是新进提出的一种半尺度半方向变换,它的重要特性就是能够较其他多尺度工具更能稀疏的表达振荡纹理和定向纹理特征.由于这一重要特点,很自然的就会想到将波原子这一工具应用到纹理保持的图像处理中.
在一些基于多尺度变换的图像反卷积算法中,我们发现得到的复原图像包含一些振铃,产生这一现象的原因是阈值收缩时导致的振铃效应.同样,振铃效应也出现在了基于波原子的反卷积算法中.因此为了即能够保证图像的纹理信息在反卷积过程中得到很好的保持,又能够减少处理后的振铃效应,我们提出：
1).提出了一种基于波原子变换的图像反卷积算法.这个算法可以有效的处理纹理信息丰富的图像,而且能够很好的保持图像的纹理结构.
对于图像反卷积而言,Fourier域是一个基本的选择,因为根据卷积定理有：Y(κ1, κ2)=H(κ1,κ2)X(κ1,κ2)+Γ(κ1,κ2)(6)Y,H,X,Γ分别是y,h,x,r的离散Fourier变换.
a). Fourier正则化反卷积(FoRD).图像FoRD估计x。的Fourier形式为其中,其中H表示H的复共轭,(?)(k1,k2)≥0是正则项,用来控制收缩的幅度.
b).波原子域Wiener滤波.
对于含有有色噪声的图像x。,波原子域的Wiener收缩估计为：其中cα,μ是xα在下标为μ的波原子变换系数,cα,μh是波原子域硬阈值估计xαh的波原子变换系数,σα,μ2是波原子域下的噪声方差,β是正则化参数.计算cα,μw的逆波原子变换,这样我们就得到了基于波原子域Wiener滤波估计xαw.
2).提出联合非局部均值滤波来解决振铃效应,并且依然能够很好的保持图像的纹理信息.使用联合非局部均值滤波去处理由波原子反卷积得到的图像,这种方法有效的解决了由波原子收缩导致的振铃现象.
a).使用估计图像xαw作为联合非局部均值滤波器中的参考图像.估计图像xαw保留了图像大部分的纹理特征和细节,所以联合非局部均值滤波能够有效的改进图像质量,得到的估计图像xJ按如下公式计算：
b).加权估计.如果直接使用联合非局部均值滤波,那么在输出的反卷积图像中会出现一些噪声圆斑,尤其是当噪声水平很高时(噪声方差很大时).为了抑制这些圆斑和保持处理上的简单有效,我们在联合非局部均值滤波估计的基础上加上基于波原子的反卷积估计.也就是说,最终的反卷积结果x*由下式得到：χ*=βχαω+(1-β)χJ,β∈[0,1)(11)这个加权方法同时也可以平衡波原子估计和联合非局部均值估计,改进图像质量.
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 PSF
There 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, and
This 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 Algorithm
There 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 χαω
2). 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.
一.正六边形点扩散函数图像反卷积
现有图像复原方法的研究中,点扩散函数有有限的几种,这可以简化‘大类点扩散函数的问题.关于图像退化点扩散函数的三种常用模型：线性运动、圆盘离焦和高斯模型.除了线性运动模型,另外两种模型也都是基于圆形光圈建立起来的.有些图像复原的研究工作虽然超出了这三种模型,但仍然要求点扩散函数具有圆对称性质.
我们注意到,随着科技的不断进步,具有可变光圈(一般是正多边形光圈)的光学系统越来越广泛地应用于各行各业.为了适应客观实际需求,有必要在图像反卷积方法的研究中,根据这一类降质图像的成像机理,研究相应的反卷积方法,以提高图像复原的精度和效率.而在现有文献中,关于可变(正多边形)光圈降质图像的反卷积方法的研究至今未见相关报导.本论文的主要研究之一是对正六边形光圈离焦图像反卷积方法的研究.
图像的降质模型可以写成如下形式：y(l,s)=(h*x)(t,s)+γ(t,s)(1)其中,y,h,x和r分别表示观测图像,真实图像,点扩散函数和噪声.图像复原的目的就是通过观测图像y(t,s)来得到真实图像x(t,s)的一个精确估计.根据Fourier变换的卷积定理,上述模型在频域的表达为：Y(υ,υ)=H(υ,υ)X(υ,υ)+Γ(υ,υ)(2)其中Y,H,X,Γ分别是y,h,x,r的Fourier变换.
对于处理正六边形可变光圈造成的图像离焦,本文的贡献有如下几个方面：
1)基于正六边形光圈光学系统的成像机理,提出了正六边形离焦模型(3.2节)：其中a为正六边形的边长,Rθ=(?).
该点扩散函数模型刻画了正六边形光圈离焦的本质,对于精确估计正六边形光圈离焦点扩散函数是必要的.
2)本文通过分析和推导,我们从理论上提出并证明了正六边形离焦点扩散函数Fourier变换H(u,υ)的零点分布定理(3.3节).
第一圈零点分布定理(定理3.7)：当0
其中f(x)是H(u,υ)的一种数学变形.
这两个定理从理论上证明了正六边形光圈离焦点扩散函数在频域存在零点,并可以对零点具体位置的搜索提供指导.根据这两个定理,用数值方法先一次性计算出H(u,υ)在频域中心附近的第一圈零点集合L1和第二圈零点集合L2.
3)在上面工作的基础上,本文提出了正六边形光圈离焦图像快速复原方法(3.4节),方法的主要步骤如下：
为了得到点扩散函数,利用观测图像y(t,s)的Fourier变换和H(u,υ)的第一圈、第二圈零点集合L,可以辨识出正六边形点扩散函数发生离焦现象时正六边形的边长a和偏转角度θ.接下来的问题就是一个已知点扩散函数的非盲反卷积问题了.为了算法的执行效率,加之图像的信噪比(SNR)比较高,可以采用Wiener滤波的方法.具体步骤如下：
a).计算观测图像y(t,s)的Fourier变换Y(u,υ).
b).求解问题得到点扩散函数的两个参数的估计：正六边形的边长a和偏转角度θ.并由此得到点扩散函数的估计Ha,θ(υ,υ)=H(n(υ,υ)Rθ).
c).利用Wiener滤波(或者其他有效的非盲图像反卷积算法),通过Ha,θ来估计真实图像.
应用此方法对于有较高信噪比(SNR)的图像进行了图像复原实验,得到比较满意的效果.
二.保持纹理结构的图像反卷积算法
在自然图像中,纹理特征广泛的存在.例如指纹图像,地震剖面图像等等,这些图像中包含着丰富的弯曲的振荡纹理和定向纹理信息.由于纹理特征是一种高频信息,而图像模糊过程是一个低通滤波的过程,因此在图像反卷积时,就需要将这种重要的图像特征恢复出来.现今的绝大多数的反卷积算法,采用的先验约束主要是小波、曲波、全变差等,由于它们对纹理信息表达的局限性,因此这些算法难以将图像的纹理信息很好的复原.这些图像反卷积算法往往会将图像中的纹理丢失,造成图像失真和视觉上的不自然.为了很好的解决纹理图像复原这一类特殊而且重要的问题,就需要采用适合纹理信息表达的数学工具.波原子变换是新进提出的一种半尺度半方向变换,它的重要特性就是能够较其他多尺度工具更能稀疏的表达振荡纹理和定向纹理特征.由于这一重要特点,很自然的就会想到将波原子这一工具应用到纹理保持的图像处理中.
在一些基于多尺度变换的图像反卷积算法中,我们发现得到的复原图像包含一些振铃,产生这一现象的原因是阈值收缩时导致的振铃效应.同样,振铃效应也出现在了基于波原子的反卷积算法中.因此为了即能够保证图像的纹理信息在反卷积过程中得到很好的保持,又能够减少处理后的振铃效应,我们提出：
1).提出了一种基于波原子变换的图像反卷积算法.这个算法可以有效的处理纹理信息丰富的图像,而且能够很好的保持图像的纹理结构.
对于图像反卷积而言,Fourier域是一个基本的选择,因为根据卷积定理有：Y(κ1, κ2)=H(κ1,κ2)X(κ1,κ2)+Γ(κ1,κ2)(6)Y,H,X,Γ分别是y,h,x,r的离散Fourier变换.
a). Fourier正则化反卷积(FoRD).图像FoRD估计x。的Fourier形式为其中,其中H表示H的复共轭,(?)(k1,k2)≥0是正则项,用来控制收缩的幅度.
b).波原子域Wiener滤波.
对于含有有色噪声的图像x。,波原子域的Wiener收缩估计为：其中cα,μ是xα在下标为μ的波原子变换系数,cα,μh是波原子域硬阈值估计xαh的波原子变换系数,σα,μ2是波原子域下的噪声方差,β是正则化参数.计算cα,μw的逆波原子变换,这样我们就得到了基于波原子域Wiener滤波估计xαw.
2).提出联合非局部均值滤波来解决振铃效应,并且依然能够很好的保持图像的纹理信息.使用联合非局部均值滤波去处理由波原子反卷积得到的图像,这种方法有效的解决了由波原子收缩导致的振铃现象.
a).使用估计图像xαw作为联合非局部均值滤波器中的参考图像.估计图像xαw保留了图像大部分的纹理特征和细节,所以联合非局部均值滤波能够有效的改进图像质量,得到的估计图像xJ按如下公式计算：
b).加权估计.如果直接使用联合非局部均值滤波,那么在输出的反卷积图像中会出现一些噪声圆斑,尤其是当噪声水平很高时(噪声方差很大时).为了抑制这些圆斑和保持处理上的简单有效,我们在联合非局部均值滤波估计的基础上加上基于波原子的反卷积估计.也就是说,最终的反卷积结果x*由下式得到：χ*=βχαω+(1-β)χJ,β∈[0,1)(11)这个加权方法同时也可以平衡波原子估计和联合非局部均值估计,改进图像质量.
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 PSF
There 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, and
This 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 Algorithm
There 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 χαω
2). 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.
引文
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