Studying of Algorithms of Ridgelet, Curvelet and Partial Differential Equations for Image Processing 

Author  BaiJian 
Tutor  FengXiangChu 
School  Xi'an University of Electronic Science and Technology 
Course  Applied Mathematics 
Keywords  wavelet ridgelet ridgelet transform curvelet digital ridgelet Radon transform anisotropic diffusion fractionalorder partial differential equations fractional order difference image denoising image smoothing image decomposition Besov space Modulus Of Smoothness 
CLC  TP391.41 
Type  PhD thesis 
Year  2007 
Downloads  899 
Quotes  3 
Many image processing tasks take advantages of sparse representations of image data where most information is packed into a small number of samples.Typically, these representations are achieved via invertible and nonredundant transforms. Currently,the most popular choices for this purpose are the wavelet transform.The success of wavelets is mainly due to the good performance for piecewise smooth functions in one dimension.Unfortunately,such is not the case in two dimensions.For twodimensional linear,surface singularities functions,the wavelet can’t approximate them very well.In fact,the decay of the error of the M term approximationε_{n}（M）=‖f_{M}f‖^{2} only reach O(M^{1}).The lack of the wavelet makes reseacher try to seek the other harmonic analysis tools.Emmanuel J.Candes introduced the concept of the ridgelet in 1998.Such system is very well at representing the smooth functions with line singularities.Candes also proposed the concept of the curvelet later.The curvelet has the best ability at representing the smooth functions with curvilinear singularities.Abilities of ridgelet and curvelet make them superior to the wavelet for image processing.They are valued by more and more researchers.Relative to the computational harmonic analysis method in the image processing,another method is partial differential equations.Since the work of Perona and Malik,which replaced the isotropic diffusion by an anisotropic difuusion,reserchers have proposed many anisotropic diffusion models which preserve important structures in images,while removing noise.These models make good effects in image denoising,image enhancement,edge detetion et al.Until now,the partial differential equation（PDE） method is still an important method and a hotspot in image processing.In this paper we will study some aspects of algorithms of ridgelet,curvelet and PDE and obtain some new instructive algorithms.（1）The finite ridgelet transform is one of the digital methods of ridgelet transform. The key is the implementation of the finite Radon transform.The sequencing problem of each slope projection in finite Radon transform is studied on this paper.We use the theory of modulus of smoothness of the Besov space and construct a smooth discriminant function in time domain.The optimal sequence has the smallest functional value,so a new adaptive sequencing algorithm is proposed.When applying it to the image compression and denoising,the good results are obtained.（2）The other is the digital ridgelet transform based on true ridge fuction.Compare with the finite ridgelet transform,it eliminates the "wrap around" and consists a frame. Thus it can’t provide the most sparer representations of the image.A global dual frame （GDF）representation for the digital ridgelet reconstruction algorithm is discussed,then a novel concept of the local dual frame（LDF）is presented.Based on the properties of LDF,we propose a new digital ridgelet reconstruction algorithm.The method reduces the redundancy in the digital ridgelet reconstruction while keeping the characteristics of low compute cost.When applying it to the image compression and denoising,the good results are obtained. （3）For an anisotropic image,wavelets lose their effects on singularity detection because discontinuities across edges are spatially distributed.Based on the idea of the curvelet,a new digital curvelet reconstruction algorithm is proposed.Our algorithm provides sparser representations and keeps low computational complexity.When applying it to the image denoising,much better results than the original algorithm are obtained.（4）This paper introduces a new class of fractionalorder anisotropic diffusion equations for noise removal.These equations are EulerLagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function,so the proposed equations can be seen as generalizations of secondorder and fourthorder anisotropic diffusion equations.We use the discrete Fourier transform（DFT）to implement the numerical algorithm and give an iterative scheme in the frequency domain.It is one important aspect of the algorithm that it considers the input image as a periodic image.To overcome this problem,we use a folded algorithm by extending the image symmetrically about its borders.And finally,we list various numerical results on denoising real images. Experiments show that the proposed fractionalorder anisotropic diffusion equations yield good visual effects and better signaltonoise ratio.（5）Recent years,decomposing an image into cartoon component（bounded variation component）and oscillating component（texture component）is an important problem in the field of image processing.The cartoon component of an image is modeled by a bounded variation（BV）function;the corresponding incorporation of BV penalty terms in the variational functional leads to solve PDE equations.Daubechies replaced the BV penalty term by a Besov term and wrote the problem in a wavelet framework. Following this idea,we propose a new image decomposition algorithm based on the digital curvelet transform.By designing a digital curvelet transform algorithm and a scaledependent thresholding rule,elegant and numerically efficient schemes are obtained.We can see that this approach is very robust to additive noise and can keep the image edges stable.