Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power ( 1 55), which, while slower than the optimal rate ( 2), is much better than ( 1) achieved with wavelets, but at similar complexity. Index Terms—Directional vanishing moments, directionlets, filter banks, geometry, multidirection, multiresolution, separable filtering, sparse image representation, wavelets. I.

It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a new discrete multiscale directional representation called the Discrete Shearlet Transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the Discrete Shearlet Transform is very competitive in denoising applications both in terms of performance and computational efficiency.

Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a pre-specified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a proper dictionary can be done using one of two ways: (i) building a sparsifying dictionary based on a mathematical model of the data, or (ii) learning a dictionary to perform best on a training set. In this paper we describe the evolution of these two paradigms. As manifestations of the first approach, we cover topics such as wavelets, wavelet packets, contourlets, and curvelets, all aiming to exploit 1-D and 2-D mathematical models for constructing effective dictionaries for signals and images. Dictionary learning takes a different route, attaching the dictionary to a set of examples it is supposed to serve. From the seminal work of Field and Olshausen, through the MOD, the K-SVD, the Generalized PCA and others, this paper surveys the various options such training has to offer, up to the most recent contributions and structures.

Abstract—We propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks. We convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks, where we form a nonredundant transform family, which exhibits both directional and nondirectional basis functions. We demonstrate the potential of the proposed transforms using nonlinear approximation. In addition, we employ the proposed family in two key image processing applications, image coding and denoising, and show its efficiency for these applications. Index Terms—Directional filter banks (DFBs), geometrical image transforms, image coding, image denoising, nonlinear approximation (NLA), wavelet transform (WT). I.

Abstract—Image priors based on products have been recognized to offer many advantages because they allow simultaneous enforcement of multiple constraints. However, they are inconvenient for Bayesian inference because it is hard to find their normalization constant in closed form. In this paper, a new Bayesian algorithm is proposed for the image restoration problem that bypasses this difficulty. An image prior is defined by imposing Student-t densities on the outputs of local convolutional filters. A variational methodology, with a constrained expectation step, is used to infer the restored image. Numerical experiments are shown that compare this methodology to previous ones and demonstrate its advantages. Index Terms—Constrained variational inference, image restoration, product prior, Student’s-t prior, Variational Bayesian Inference.

Abstract—In this paper a new image prior is introduced and used in image restoration. This prior is based on products of spatially weighted Total Variations (TV). These spatial weights provide this prior with the flexibility to better capture local image features than previous TV based priors. Bayesian inference is used for image restoration with this prior via the variational approximation. The proposed restoration algorithm is fully automatic in the sense that all necessary parameters are estimated from the data and is faster than previous similar algorithms. Numerical experiments are shown which demonstrate that image restoration based on this prior compares favorably with previous state-of-the-art restoration algorithms. I.

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”.

Multidimensional multirate systems have been used widely in signal processing, communications, and computer vision. Traditional multidimensional multirate systems are tensor products of one-dimensional systems. While these systems are easy to implement and design, they are inadequate to represent multidimensional signals since they cannot capture the geometric structure. Therefore, “true” multidimensional systems are more suited to multidimensional signals, such as images and videos. This thesis focuses on the characterization, design, and applications of “true” multidimensional multirate systems. One key property of multidimensional multirate systems is perfect reconstruction, which guarantees the original input can be perfectly reconstructed from the outputs. The most popular multidimensional multirate systems are multidimensional filter banks, including critically sampled and oversampled ones. Characterizing and designing multidimensional perfect reconstruction filter banks have been challenging tasks. For critically sampled filter banks, previous one-dimensional theory cannot be extended to the multidimensional

Image registration is a process of matching images, which are taken at different times, from different sensors or from different view points. It is an important step for a great variety of applications such as computer vision, stereo navigation, medical image analysis, pattern recognition and watermarking applications. In this paper an improved feature point selection and matching technique for image registration is proposed. This technique is based on the ability of Nonsubsampled Contourlet Transform (NSCT) to extract significant features irrespective of feature orientation. Then the correspondence between the extracted feature points of reference image and sensed image is achieved using Zernike moments. Feature point pairs are used for estimating the transformation parameters mapping the sensed image to the reference image. Experimental results illustrate the registration accuracy over a wide range for panning and zooming movement and also the robustness of the proposed algorithm to noise. Apart from image registration proposed method can be used for shape matching and object classification.

The paper proposes a novel robust watermarking technique based on newly introduced Nonsubsampled contourlet transform(NSCT) and singular value decomposition(SVD) for multimedia copyright protection. The NSCT can give the asymptotic optimal representation of the edges and contours in image by virtue of the characteristics of good multi resolution shift invariance and multi directionality. After decomposing the host image into sub bands, we choose the low frequency directional sub band and apply singular value decomposition. The singular values of the original image are then modified by the singular values of nonsubsampled contourlet transformed visual grayscale logo watermark image. This hybrid approach improves the performance of the watermarking technique compared to earlier techniques. Experimental results shows that the hybrid technique is resilient to various linear and non linear filtering,JPEG