Tensor Product Complex Tight Framelets Research Articles

Directional tensor product complex tight framelets for compressed sensing MRI reconstruction

Compressed sensing magnetic resonance imaging (CS-MRI) is an effective way of reducing the sampling data in the k-space and shortening the scanning time. Motivated by the high performance of directional tensor product complex tight framelets (TPCTFs) for the image denoising problem, the authors proposed a novel framework that integrated TPCTF for sparse representation and projected fast iterative soft-thresholding algorithm (pFISTA) for CS-MRI reconstruction. Furthermore, to take advantage of the cross-scale relations in the wavelet tree of frame coefficients, the bivariate shrinkage (BS) function with local variance estimation is proposed to shrink thresholding. Such TPCTFs can provide sparse directional representations very well for MR image. When compared with other the state-of-the-art CS-MRI algorithms in numerical experiments, the proposed TPCTF-BS method achieves a higher reconstruction quality with respect to image edge preservation and the artefact suppression.

Directional Compactly Supported Tensor Product Complex Tight Framelets with Applications to Image Denoising and Inpainting

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.

Image inpainting from partial noisy data by directional complex tight framelets

Image inpainting methods recover true images from partial noisy observations. Natural images usually have two layers consisting of cartoons and textures. Methods using simultaneous cartoon and texture inpainting are popular in the literature by using two combined tight frames: one (often built from wavelets, curvelets or shearlets) provides sparse representations for cartoons and the other (often built from discrete cosine transforms) offers sparse approximation for textures. Inspired by the recent development on directional tensor product complex tight framelets (TP-\(\mathbb{C}\)TFs) and their impressive performance for the image denoising problem, we propose an iterative thresholding algorithm using tight frames derived from TP-\(\mathbb{C}\)TFs for the image inpainting problem. The tight frame TP-\(\mathbb{C}\)TF\(_6\) contains two classes of framelets; one is good for cartoons and the other is good for textures. Therefore, it can handle both the cartoons and the textures well. For the image inpainting problem with additive zero-mean independent and identically distributed Gaussian noise, our proposed algorithm does not require us to tune parameters manually for reasonably good performance. Experimental results show that our proposed algorithm performs comparatively better than several well-known frame systems for the image inpainting problem. doi:10.1017/S1446181117000219

IMAGE INPAINTING FROM PARTIAL NOISY DATA BY DIRECTIONAL COMPLEX TIGHT FRAMELETS

Image inpainting methods recover true images from partial noisy observations. Natural images usually have two layers consisting of cartoons and textures. Methods using simultaneous cartoon and texture inpainting are popular in the literature by using two combined tight frames: one (often built from wavelets, curvelets or shearlets) provides sparse representations for cartoons and the other (often built from discrete cosine transforms) offers sparse approximation for textures. Inspired by the recent development on directional tensor product complex tight framelets ($\text{TP}\text{-}\mathbb{C}\text{TF}$s) and their impressive performance for the image denoising problem, we propose an iterative thresholding algorithm using tight frames derived from $\text{TP}\text{-}\mathbb{C}\text{TF}$s for the image inpainting problem. The tight frame $\text{TP}\text{-}\mathbb{C}\text{TF}_{6}$ contains two classes of framelets; one is good for cartoons and the other is good for textures. Therefore, it can handle both the cartoons and the textures well. For the image inpainting problem with additive zero-mean independent and identically distributed Gaussian noise, our proposed algorithm does not require us to tune parameters manually for reasonably good performance. Experimental results show that our proposed algorithm performs comparatively better than several well-known frame systems for the image inpainting problem.

Directional tensor product complex tight framelets with low redundancy

Having the advantages of redundancy and flexibility, various types of tight frames have already shown impressive performance in applications such as image and video processing. For example, the undecimated wavelet transform, which is a particular case of tight frames, is known to have good performance for the denoising problem. Empirically, it is widely known that higher redundancy rate of a tight frame often leads to better performance in applications. The wavelet/framelet transform is often implemented in an undecimated fashion for the purpose of better performance in practice. Though high redundancy rate of a tight frame can improve performance in applications, as the dimension increases, it also makes the computational cost skyrocket and the storage of frame coefficients increase exponentially. This seriously restricts the usefulness of such tight frames for problems in moderately high dimensions such as video processing in dimension three. Inspired by the directional tensor product complex tight framelets TP-CTFm with m≥3 in [15,20] and their impressive performance for image processing in [20,33], in this paper we introduce directional tensor product complex tight framelets TP-CTFm↓ (called reduced TP-CTFm) with low redundancy. Such TP-CTFm↓ are particular examples of tight framelet filter banks with mixed sampling factors. In particular, we shall develop a directional tensor product complex tight framelet TP-CTF6↓ such that it performs nearly as well as the original TP-CTF6 in [20] for image/video denoising/inpainting but it has significantly lower redundancy rates than TP-CTF6 in every dimension. The TP-CTF6↓ in d dimensions not only offers good directionality as the original TP-CTF6 does but also has the low redundancy rate 3d−12d−1 (e.g., the redundancy rates are 2,223,357,513 and 72531 for dimension d=1,…,5, respectively), in comparison with the redundancy rate 2d×3d−12d−1 of TP-CTF6 in dimension d. Moreover, our numerical experiments on image/video denoising and inpainting show that the performance using our proposed TP-CTF6↓ is often comparable with or sometimes better than several state-of-the-art frame-based methods which have much higher redundancy rates than that of TP-CTF6↓.

Removal of Mixed Gaussian and Impulse Noise Using Directional Tensor Product Complex Tight Framelets

In this paper, we propose a frame-based iterative algorithm to restore images which are corrupted by mixed Gaussian and impulse noise, under the assumption that the image region corrupted by impulse noise is unknown. The removal of mixed Gaussian and impulse noise by our proposed algorithm is split into two subproblems which are solved alternatively and iteratively. With an initial guessed region of location for impulse noise, the first subproblem is to inpaint a corrupted image by solving a frame-based convex minimization scheme using the balanced approach, where sparse and redundant directional representations play a key role. Motivated by our recent work on frame-based image denoising and image inpainting, we shall employ the tight frame generated from the directional tensor product complex tight framelets in our balanced approach to remove the mixed Gaussian and impulse noise. Such tensor product complex tight framelets provide sparse directional representations for natural images and can capture the cartoon and texture parts of images very well. The second subproblem is to estimate the image region of locations where the pixels are corrupted by impulse noise. We solve the second subproblem using an $$l_0$$l0-minimization scheme. We consider both salt-and-pepper impulse noise and random-valued impulse noise. Numerical experiments show that our proposed algorithm compares favorably or often outperforms three well-known recent image-restoration methods employed for removing the mixed Gaussian and impulse noise.

Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can capture well only edge singularities along the coordinate axis directions. As an alternative and improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensor product complex tight framelets with increasing directionality have been introduced in [B. Han, Math. Model. Nat. Phenom., 8 (2013), pp. 18--47] and applied to image denoising in [B. Han and Z. Zhao, SIAM J. Imaging Sci., 7 (2014), pp. 997--1034]. Despite several desirable properties, those directional tensor product complex tight framelets are bandlimited and do not have compact support in the space/time domain. Since compactly supported wavelets and framelets are of great interest and importance in both theory and application, it remains as an unsolved problem whether there exist compactly supported tensor product complex tight framelets with directionality. In this paper, we shall answer this question by proving a theoretical result on directionality of tight framelets and by introducing an algorithm to construct compactly supported complex tight framelets with directionality. Our examples show that compactly supported complex tight framelets with directionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examples of compactly supported tensor product complex tight framelets with directionality have been presented.

Tensor Product Complex Tight Framelets with Increasing Directionality

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.