filter bank (DFB) for an efficient directional decomposition of 2-D signals. Due to the nonseparable nature of the system, extending the DFB to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem. We propose a new family of filter banks, named NDFB, that can achieve the directional decomposition of arbitrary-dimensional ( 2) signals with a simple and efficient tree-structured construction. In 3-D, the ideal passbands of the proposed NDFB are rectangular-based pyramids radiating out from the origin at different orientations and tiling the entire frequency space. The proposed NDFB achieves perfect reconstruction via an iterated filter bank with a redundancy factor of in-D. The angular resolution of the proposed NDFB can be iteratively refined by invoking more levels of decomposition through a simple expansion rule. By combining the NDFB with a new multiscale pyramid, we propose the surfacelet transform, which can be used to efficiently capture and represent surface-like singularities in multidimensional data. Index Terms—Directional decomposition, directional filter banks (DFBs), filter design, high-dimensional transforms, surfacelets. I.

Directional information is an important and unique feature of multidimensional signals. As a result of a separable extension from one-dimensional (1-D) bases, multidimensional wavelet transforms have very limited directionality. Furthermore, different directions are mixed in certain wavelet subbands. In this paper, we propose a simple directional extension for wavelets (DEW) that fixes this subband mixing problem and improves the directionality. The building block of the DEW is a two-channel 2-D filter bank with a checkerboard-shaped frequency partition. The DEW works with both the critically-sampled wavelet transform as well as the undecimated wavelet transform. In the 2-D case, it further divides the three wavelet subbands (i.e. horizontal, vertical, and diagonal) at each scale into six finer directional subbands. The DEW itself is critically-sampled, and hence will not increase the redundancy of the overall transform. Though nonseparable in essence, the proposed DEW has an efficient implementation that only requires 1-D filtering. Meanwhile, the DEW can be easily generalized to higher dimensions. In a nutshell, the proposed directional extension provides an optional tool to efficiently enhance the directionality of multidimensional wavelet transforms. Numerical experiments show that certain wavelet-based image processing applications will benefit from this improved directionality.

We reconsider the problem of generating a dual filterbank to a given one with N channels such that its N − 1 mother wavelets will be related to those of the original one by the Hilbert-transform. Known solutions treat the case of unitary filterbanks [1, 2], while [3, 4] treats the biorthogonal case and [5] the overcomplete case. We present a solution which is valid for arbitrary filterbanks and obtain identical results. This fact indicates that the proposition of unitarity is not necessary at all. Moreover, the phase condition is obtained in a constructive way and is given by a closed form expression. 1.1 Notation 1.

Abstract—The identification of image acquisition source is an important problem in digital image forensics. In this work, we focus on building a classifier to effectively distinguish between digital images taken from digital single lens reflex (DSLR) and compact cameras. Based on the architecture and the imaging features of DSLR and compact cameras, the images taken from different sources may have different statistical properties in both spatial and transform domains. In this work, we utilized wavelet coefficients and pixel noise statistics to model these two different source classes over 20 different digital cameras. The efficacy of the digital source class identifier, introduced in the paper, has been tested over 1000 high quality camera outputs and postprocessed images (resized, re-compressed). Experimental analysis shows that the proposed method has good potential to distinguish DSLR and compact source classes. I.

One of the key differences between one-dimensional (1-D) and N-D (N ≥ 2) signals is the geometrical information, an important and unique feature of multidimensional signals. The goal of this research is to develop a new set of theories and techniques in signal processing that can make better use of the intrinsic geometrical information in multidimensional data in a robust and efficient way. The primary technique we employ to approach this problem is multidimensional filter banks, due to their computational advantages, their design flexibilities, and very importantly, their direct connection with the theory of basis (nonredundant) and frame (redundant) decomposition of multidimensional signals. Directional information is an important geometric feature of multidimensional signals. As a result of a separable extension from 1-D bases, multidimensional wavelet transforms have very limited directionality. Furthermore, different directions are mixed in certain wavelet subbands. To solve this problem, we propose a simple Directional Extension for Wavelets (DEW) that fixes this subband mixing problem and improves the directionality. In a nutshell, the proposed

Abstract—Multidimensional hourglass filter banks decompose the frequency spectrum of input signals into hourglass-shaped directional subbands, each aligned with one of the frequency axes. The directionality of the spectral partitioning makes these filter banks useful in separating the directional information in multidimensional signals. Despite the existence of various design techniques proposed for the 2-D case, to our best knowledge, the design of hourglass filter banks in 3-D and higher dimensions with finite impulse response (FIR) filters and perfect reconstruction has not been previously reported. In this paper, we propose a novel mapping-based design for the hourglass filter banks in arbitrary dimensions, featuring perfect reconstruction, FIR filters, efficient implementation using lifting/ladder structures, and a near-tight frame construction. The effectiveness of the proposed mapping-based design depends on the study of a set of conditions on the frequency supports of the mapping kernels. These conditions ensure that we can still get good frequency responses when the component filters used are nonideal. Among all feasible choices, we then propose an optimal specification for the mapping kernels, which leads to the simplest passband shapes and involves the fewest number of frequency variables. Finally, we illustrate the proposed techniques by a design example in 3-D, and an application in video denoising. Index Terms—multidimensional transforms, directional filter banks, hourglass filter banks, filter design, directional decomposition. I.

Abstract—In this paper, we introduce a digital implementation of the 3D shearlet transform and illustrate its application to problems of video denoising and enhancement. The shearlet representation is a multiscale pyramid of well-localized waveforms defined at various locations and orientations, which was introduced to overcome the limitations of traditional multiscale systems in dealing with multidimensional data. While the shearlet approach shares the general philosophy of curvelets and surfacelets, it is based on a very different mathematical framework which is derived from the theory of affine systems and uses shearing matrices rather than rotations. This allows a natural transition from the continuous to the digital setting and a more flexible mathematical structure. The 3D digital shearlet transform algorithm presented in this paper consists in a cascade of a multiscale decomposition and a directional filtering stage. The filters employed in this decomposition are implemented as finite-length filters and this ensures that the transform is local and numerically efficient. To illustrate its performance, the 3D Discrete Shearlet Transform is applied to problems of video denoising and enhancement, and compared against other state-ofthe-art multiscale techniques, including curvelets and surfacelets. Index Terms—Affine systems, curvelets, denoising, shearlets, sparsity, video processing, wavelets.