We propose a direction-adaptive DWT (DA-DWT) that locally adapts the filtering directions to image content based on directional lifting. With the adaptive transform, energy compaction is improved for sharp image features. A mathematical analysis based on an anisotropic statistical image model is presented to quantify the theoretical gain achieved by adapting the filtering directions. The analysis indicates that the proposed DA-DWT is more effective than other lifting-based approaches. Experimental results report a gain of up to 2.5 dB in PSNR over the conventional DWT for typical test images. Subjectively, the reconstruction from the DA-DWT better represents the structure in the image and is visually more pleasing.

Consider the problem of sampling signals that are nonbandlimited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI) [3]. We know that the classical (‘bandlimited-sinc’) sampling theory does not enable perfect reconstruction of such signals from their samples since they are not bandlimited. However, the recent results on FRI sampling [3], [4] suggest that it is possible to sample and perfectly reconstruct such nonbandlimited signals using a rich class of kernels. In this paper, we extend the results of [4] in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous-moments of the signal from its samples. Using these moments and the annihilating filter method (Prony’s method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional derivatives based) and global (complex-moments, Radon transform based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel and planar polygons, quadrature domains (e.g. circles, ellipses, cardioids), 2-D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This reaearch has

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.

Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr’s theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of, irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation-invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr’s theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative

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—The standard separable 2-D wavelet transform (WT) has recently achieved a great success in image processing because it provides a sparse representation of smooth images. However, it fails to efficiently capture 1-D discontinuities, like edges or contours. These features, being elongated and characterized by geometrical regularity along different directions, intersect and generate many large magnitude wavelet coefficients. Since contours are very important elements in the visual perception of images, to provide a good visual quality of compressed images, it is fundamental to preserve good reconstruction of these directional features. In our previous work, we proposed a construction of critically sampled perfect reconstruction transforms with directional vanishing moments imposed in the corresponding basis functions along different directions, called directionlets. In this paper, we show how to design and implement a novel efficient space-frequency quantization (SFQ) compression algorithm using directionlets. Our new compression method outperforms the standard SFQ in a rate-distortion sense, both in terms of mean-square error and visual quality, especially in the low-rate compression regime. We also show that our compression method, does not increase the order of computational complexity as compared to the standard SFQ algorithm. Index Terms—Directional transforms, directional vanishing moments (DVMs), image coding, image orientation analysis, image segmentation, multiresolution analysis, nonseparable transforms, wavelet transforms (WTs). 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”.

The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal N-term approximations for piecewise Hölder continuous functions with singularities along curves.

Abstract—In this paper, we present the design of directional lapped transforms for image coding. A lapped transform, which can be implemented by a prefilter followed by a discrete cosine transform (DCT), can be factorized into elementary operators. The corresponding directional lapped transform is generated by applying each elementary operator along a given direction. The proposed directional lapped transforms are not only nonredundant and perfectly reconstructed, but they can also provide a basis along an arbitrary direction. These properties, along with the advantages of lapped transforms, make the proposed transforms appealing for image coding. A block-based directional transform scheme is also presented and integrated into HD Phtoto, one of the state-of-the-art image coding systems, to verify the effectiveness of the proposed transforms. Index Terms—Directional transform, image coding, lapped transform.

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C2-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ’shear ’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.