Compactly supported tight framelets are of great interest and importance in both theory and application. In this paper we discuss how to construct directional compactly supported tensor product complex tight framelets having varied directionality and good performance for applications in image processing. Our construction algorithms employ optimization techniques and put extensive emphasis on frequency response and spatial localization of their underlying one-dimensional tight framelet filter banks. Several concrete examples of directional compactly supported tensor product complex tight framelet filter banks are provided in this paper. Our numerical experiments show that such constructed directional compactly supported tensor product complex tight framelets have good performance for applications such as image denoising and inpainting compared with several other state-of-the-art transform-based methods.

Tensor product real-valued wavelets have been employed in many applications such as image processing with impressive performance. Edge singularities are ubiquitous and play a fundamental role in image processing and many other two-dimensional problems. Tensor product real-valued wavelets are known to be only suboptimal since they can only capture edges well along the coordinate axis directions (that is, the horizontal and vertical directions in dimension two). Among several approaches in the literature to enhance the performance of tensor product real-valued wavelets, the dual tree complex wavelet transform ($DT-\mathbb{C}WT$), proposed by Kingsbury [Phil. Trans. R. Soc. Lond. A, 357 (1999), pp. 2543--2560] and further developed by Selesnick, Baraniuk, and Kingsbury [IEEE Signal Process. Mag., 22 (2005), pp. 123--151], is one of the most popular and successful enhancements of the classical tensor product real-valued wavelets by employing a correlated pair of orthogonal wavelet filter banks. The two-dimensional $DT-\mathbb{C}WT$ is obtained essentially via tensor product and offers improved directionality with six directions. In this paper we shall further enhance the performance of the $DT-\mathbb{C}WT$ for the problem of image denoising. Using a framelet-based approach and the notion of discrete affine systems, we shall propose a family of tensor product complex tight framelets $TP-\mathbb{C}TF_n$ for all integers $n\ge 3$ with increasing directionality, where $n$ refers to the number of filters in the underlying one-dimensional complex tight framelet filter bank. For dimension two, such a tensor product complex tight framelet $TP-\mathbb{C}TF_n$ offers $\frac{1}{2}(n-1)(n-3)+4$ directions when $n$ is odd and $\frac{1}{2}(n-4)(n+2)+6$ directions when $n$ is even. In particular, we shall show that $TP-\mathbb{C}TF_4$, which is different from the $DT-\mathbb{C}WT$ in both nature and design, provides an alternative to the $DT-\mathbb{C}WT$. Indeed, we shall see that $TP-\mathbb{C}TF_4$ behaves quite similar to the $DT-\mathbb{C}WT$ by offering six directions in dimension two, employing the tensor product structure, and enjoying slightly less redundancy than the $DT-\mathbb{C}WT$. Then we shall apply $TP-\mathbb{C}TF_n$ to the problem of image denoising. We shall see that the performance of $TP-\mathbb{C}TF_4$ for image denoising is comparable to that of the $DT-\mathbb{C}WT$. Better results on image denoising can be obtained by using other $TP-\mathbb{C}TF_n$, for example, n=6, which has 14 directions in dimension two. Moreover, $TP-\mathbb{C}TF_n$ allows us to further improve the $DT-\mathbb{C}WT$ by using $TP-\mathbb{C}TF_n$ as the first stage filter bank in the $DT-\mathbb{C}WT$. We shall also provide discussion and comparison of $TP-\mathbb{C}TF_n$ with several generalizations of the $DT-\mathbb{C}WT$, shearlets, and directional nonseparable tight framelets.

It has been found that the quantization error e for a conventional oversampled analog-to-digital (A/D) conversion behaves like /spl par/e/spl par//sup 2/=O(/spl tau//sup 2/) with respect to the sampling rate /spl tau/. Recently, conventional A/D conversion has been extended to A/D conversion based on shift-invariant spaces. As consequences of such extension, it offers rich choices to build a nonideal A/D conversion system of high accuracy and low computational complexity, as well as reduces the noise sensitivity and computational complexity in digital-to-analog (D/A) conversion. Therefore, it is necessary to establish the estimate of quantization error for the extended A/D conversion based on shift-invariant spaces. In this paper, we introduce a constructive method to establish an estimate of the quantization error as |e|/sup 2/=O(/spl tau//sup 2/) for oversampled A/D conversion in shift-invariant spaces. Meanwhile, we demonstrate that the bit rate required to encode the converted digital signal in such A/D conversion scheme only increases as the logarithm of the sampling ratio. Therefore, the quantization error is an exponentially decaying function of the bit rate. In order to establish such an estimate, we need the nonuniform sampling theorem for shift-invariant spaces, which, as the necessary preparation, is studied prior to introducing the constructive method.

A conventional A/D converter prefilters a signal by an ideal lowpass filter and performs sampling for bandlimited signals by the Nyquist sampling rate. Recent research reveals that A/D conversion in a shift invariant space provides more flexible choices for designing a practical A/D conversion system of high accuracy. This paper focuses on the maximal gap of a sampling set for the iterative algorithm in shift invariant spaces, which provides an explicit formula to calculate the maximal gap of a sampling set in terms of a generator of the undertaken shift invariant spaces.