Directional Vanishing Moments Research Articles

On the design of multi-dimensional compactly supported Parseval framelets with directional characteristics

In this paper, we propose a new method for the construction of multi-dimensional, wavelet-like families of affine frames, commonly referred to as framelets, with specific directional characteristics, small and compact support in space, directional vanishing moments (DVM), and axial symmetries or anti-symmetries. The framelets we construct arise from readily available refinable functions. The filters defining these framelets have few non-zero coefficients, custom-selected orientations and can act as finite-difference operators.

Image denoising based on adaptive spatial segmentation and multi-scale correlation in directionlet domain

Directionlet transform DT has become popular over the last few years as an efficient image representation tool due to its fine frequency tiling and directional vanishing moments along any two directions. A novel denoising algorithm based on DT is proposed here for images corrupted with Gaussian noise. The image is first spatially segmented based on the content directionality. Then an undecimated version of DT is applied to effectively capture the directional features and edge information of these segments. The DT coefficients so obtained are then modelled using a bivariate heavy tailed Gaussian distribution and the noise free coefficients are computed using MAP estimator. By employing bivariate probability distribution, the heavy-tail behaviour of natural images is accurately modelled and the interscale properties of DT coefficients are properly exploited. In addition, the local variance parameter of the model is estimated based on classification of DT coefficients within a particular scale using context modelling. Due to this the intrascale dependency of the directionlet coefficients is also well exploited in the enhancement process. The proposed algorithm is competitive with the existing algorithms with better results in terms of output peak signal-to-noise ratio while having comparable computational complexity. It exhibits good capability to preserve edges, contours and textures especially in images with abundant high frequency contents.

Spatially adaptive image denoising using inter‐scale dependence in directionlet domain

The performance of image processing algorithms can be significantly improved by the application of multi-resolution image representation with directional features. Directionlet transform (DT) is one such representation which has gained popularity over the past few years as an anisotropic, perfect reconstruction and critically sampled basis function with directional vanishing moments along any two directions. In this study, the authors propose a spatially adaptive image denoising scheme for Gaussian noise based on DT by considering the dependences of the directionlet coefficients across different scales. The image is first decomposed using DT and the coefficients so obtained are modelled using a bivariate heavy tailed ‘pdf’ with a local variance parameter to account for inter- and intra-scale dependencies of the coefficients. The DT is made adaptive to the local dominant directions in the image by identifying the dominant directions in the spatially segmented image through the computation of a parameter called ‘directional variance’. Bayesian ‘maximum a posteriori’ estimator is then used to compute the noise free coefficients from the bivariate models of the signal and noise. The denoised image is obtained from the transform coefficients, which were modified using the bivariate shrinkage function, using directional information and inverse DT. Experimental results show that the bivariate shrinkage in directionlet domain achieves better performance than that in wavelet domain, in terms of numerical and perceptual quality.

SAR image despeckling using adaptive multiscale products thresholding in directionlet domain

A directionlet transform (DT)-based multiscale products thresholding is presented which uses a generalised cross-validation (GCV) technique for synthetic aperture radar image despeckling. DT has gained popularity over the past few years as an anisotropic, critically sampled and perfect reconstruction transform with directional vanishing moments along any two directions. In this reported work, the multiscale correlation of DT is exploited by multiplying the adjacent scale coefficients to enhance edge structures while weakening noise. An optimal subband adaptive threshold based on GCV is then computed using this multiscale product. The despeckled image is finally synthesised by using the threshold applied DT coefficients. The employment of DT results in the significant features in images evolving with high magnitudes across scales, while the noise decays rapidly. The proposed scheme outperforms many of the traditional despeckling schemes in terms of speckle reduction and edge preservation.

Space-Frequency Quantization for Image Compression With Directionlets

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.

On Two-Channel Filter Banks With Directional Vanishing Moments

The contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform. One way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response. This requires filters with large support size. We seek to isolate the key filter property that ensures good approximation. In this direction, we propose filters with directional vanishing moments (DVM). These filters, we show, annihilate information along a given direction. We study two-channel filter banks with DVM filters. We provide conditions under which the design of DVM filter banks is possible. A complete characterization of the product filter is, thus, obtained. We propose a design framework that avoids 2-D factorization using the mapping technique. The filters designed, when used in the contourlet transform, exhibit nonlinear approximation comparable to the conventional filters while being shorter and, therefore, providing better visual quality with less ringing artifacts. Furthermore, experiments show that the proposed filters outperform the conventional ones in image approximation and denoising.

The contourlet transform: an efficient directional multiresolution image representation

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. Index Terms-Contourlets, contours, filter banks, geometric image processing, multidirection, multiresolution, sparse representation, wavelets.