The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.

/pdf/the-contourlet-transform-an-efficient-directional-bbl3m3gyb1.pdf

The contourlet transform: an efficient directional multiresolution image representation

In this paper, we develop the nonsubsampled contourlet transform (NSCT) and study its applications. The construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks. The result is a flexible multiscale, multidirection, and shift-invariant image decomposition that can be efficiently implemented via the a trous algorithm. At the core of the proposed scheme is the nonseparable two-channel nonsubsampled filter bank (NSFB). We exploit the less stringent design condition of the NSFB to design filters that lead to a NSCT with better frequency selectivity and regularity when compared to the contourlet transform. We propose a design framework based on the mapping approach, that allows for a fast implementation based on a lifting or ladder structure, and only uses one-dimensional filtering in some cases. In addition, our design ensures that the corresponding frame elements are regular, symmetric, and the frame is close to a tight one. We assess the performance of the NSCT in image denoising and enhancement applications. In both applications the NSCT compares favorably to other existing methods in the literature

/pdf/the-nonsubsampled-contourlet-transform-theory-design-and-3a4lsit6lx.pdf

The Nonsubsampled Contourlet Transform: Theory, Design, and Applications

https://www.math.uh.edu/~dlabate/shear_ELL.pdf

Sparse directional image representations using the discrete shearlet transform

In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f which are $C^2$ except for discontinuities along $C^2$ curves. More specifically, if $f_N^S$ is the N-term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as $\norm{f-f_N^S}_2^2 \asymp N^{-2} (\log N)^3, N \to \infty,$ which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate $N^{-1}$ associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.

/pdf/optimally-sparse-multidimensional-representation-using-2wf5uxgo8h.pdf

Optimally sparse multidimensional representation using shearlets

In this paper we describe a new class of multidimensional
representation systems, called shearlets. They are obtained by
applying the actions of dilation, shear transformation and
translation to a fixed function, and exhibit the geometric and
mathematical properties, e.g., directionality, elongated shapes,
scales, oscillations, recently advocated by many authors for
sparse image processing applications. These systems can be studied
within the framework of a generalized multiresolution analysis.
This approach leads to a recursive algorithm for the
implementation of these systems, that generalizes the classical
cascade algorithm.

/pdf/sparse-multidimensional-representation-using-shearlets-25imuocs4d.pdf

Sparse multidimensional representation using shearlets

The contourlet transform is a new two-dimensional extension of the wavelet transform using multiscale and directional filter banks. The contourlet expansion is composed of basis images oriented at various directions in multiple scales, with flexible aspect ratios. Given this rich set of basis images, the contourlet transform effectively captures smooth contours that are the dominant feature in natural images. We begin with a detailed study on the statistics of the contourlet coefficients of natural images: using histograms to estimate the marginal and joint distributions and mutual information to measure the dependencies between coefficients. This study reveals the highly non-Gaussian marginal statistics and strong interlocation, interscale, and interdirection dependencies of contourlet coefficients. We also find that conditioned on the magnitudes of their generalized neighborhood coefficients, contourlet coefficients can be approximately modeled as Gaussian random variables. Based on these findings, we model contourlet coefficients using a hidden Markov tree (HMT) model with Gaussian mixtures that can capture all interscale, interdirection, and interlocation dependencies. We present experimental results using this model in image denoising and texture retrieval applications. In denoising, the contourlet HMT outperforms other wavelet methods in terms of visual quality, especially around edges. In texture retrieval, it shows improvements in performance for various oriented textures.

/pdf/directional-multiscale-modeling-of-images-using-the-1ymjuz18a3.pdf

Directional multiscale modeling of images using the contourlet transform

The contourlet transform is a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks. Because of its multiscale and directional properties, it can effectively capture the image edges along one-dimensional contours with few coefficients. This paper investigates image modeling in the contourlet transform domain and its applications. We begin with a detail study of the statistics of the contourlet coefficients, which reveals their non-Gaussian marginal statistics and strong dependencies. Conditioned on neighboring coefficient magnitudes, contourlet coefficients are found to be approximately Gaussian. Based on these statistics, we constructed a contourlet hidden Markov tree (HMT) model that can capture all of contourlets' inter-scale, inter-orientation, and intra-subband dependencies. We experiment using this model in image denoising and texture retrieval. In denoising, contourlet HMT outperforms wavelet HMT and other classical methods in terms of both visual quality and peak signal-to-noise ratio (PSNR). In texture retrieval, it shows improvements in performance over wavelet methods for various oriented textures.

/pdf/directional-multiscale-modeling-of-images-using-the-2bekn3e4ak.pdf

D.D.-Y. Po

Papers

Directional multiscale modeling of images using the contourlet transform

Directional multiscale modeling of images using the contourlet transform