Data-driven Tight Frame Research Articles

Data-Driven Tight Frame Learning Scheme Based on Local and Non-local Sparsity with Application to Image Recovery

Sparse representation has shown to be a successful technique in the fields of image restoration and other image processing tasks over the past few years. The key of the approach with application to...

Double-sparsity dictionary for seismic noise attenuation

A key step in sparsifying signals is the choice of a sparsity-promoting dictionary. There are two basic approaches to design such a dictionary: the analytic approach and the learning-based approach. Although the analytic approach enjoys the advantage of high efficiency, it lacks adaptivity to various data patterns. On the other hand, the learning-based approach can adaptively sparsify different data sets but has a heavier computational complexity and involves no prior-constraint pattern information for particular data. We have developed a double-sparsity dictionary (DSD) for seismic data to combine the benefits of both approaches. We have evaluated two models to learn the DSD: the synthesis model and the analysis model. The synthesis model learns DSD in the data domain, and the analysis model learns DSD in the model domain. We tested the analysis model and proposed to use the seislet transform and data-driven tight frame (DDTF) as the base transform and adaptive dictionary, respectively, in the DSD framework. The DDTF obtains an extra structure regularization by learning dictionaries, whereas the seislet transform obtains a compensation for the transformation error caused by slope dependency. The DSD aims to provide a sparser representation than the individual transform and dictionary and, therefore, can help to achieve better performance in denoising applications. Although for the purpose of compression, the DSD is less sparse than the seislet transform, it outperforms the seislet and DDTF in distinguishing the signal and noise. Two simulated synthetic examples and three field data examples confirm a better denoising performance of the proposed approach.

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

This paper proposes a spatial-Radon domain computed tomography (CT) image reconstruction model based on data-driven tight frames (SRD-DDTF). The proposed SRD-DDTF model combines the idea of the joint image and Radon domain inpainting model of Dong, Li, and Shen [J. Sci. Comput., 54 (2013), pp. 333--349] and that of the data-driven tight frames for image denoising [J.-F. Cai, H. Ji, Z. Shen, and G.-B. Ye, Appl. Comput. Harmon. Anal., 37 (2014), p. 89--105]. It is different from existing models in that both the CT image and its corresponding high quality projection image are reconstructed simultaneously using sparsity priors by tight frames that are adaptively learned from the data to provide optimal sparse approximations. An alternative minimization algorithm is designed to solve the proposed model, which is nonsmooth and nonconvex. Convergence analysis of the algorithm is provided. Numerical experiments show that the SRD-DDTF model is superior to the model of Dong, Li, and Shen [J. Sci. Comput., 54 (2013)...

Interpolation and denoising of high-dimensional seismic data by learning a tight frame

Sparse transforms play an important role in seismic signal processing steps, such as prestack noise attenuation and data reconstruction. Analytic sparse transforms (so-called implicit dictionaries), such as the Fourier, Radon, and curvelet transforms, are often used to represent seismic data. There are situations, however, in which the complexity of the data requires adaptive sparse transform methods, whose basis functions are determined via learning methods. We studied an application of the data-driven tight frame (DDTF) method to noise suppression and interpolation of high-dimensional seismic data. Rather than choosing a model beforehand (for example, a family of lines, parabolas, or curvelets) to fit the data, the DDTF derives the model from the data itself in an optimum manner. The process of estimating the basis function from the data can be summarized as follows: First, the input data are divided into small blocks to form training sets. Then, the DDTF algorithm is applied on the training sets to estimate the dictionary. The DDTF is typically embodied as an explicit dictionary, and a sparsity-promoting algorithm is used to obtain an optimized tight frame representation of the observed data. The computational time and redundancy is controlled by the block overlap of the training set. Finally, the learned dictionary is used to represent the observed data and to estimate data at unobserved spatial positions. Our numerical results showed that the proposed methodology is capable of recovering n-dimensional prestack seismic data under different signal-to-noise ratio scenarios. We determined that subtle features tend to be better preserved with the DDTF method in comparison with standard Fourier and directional transform reconstruction methods.

Data-Driven Tight Frame for Multi-channel Images and Its Application to Joint Color-Depth Image Reconstruction

In image restoration, we usually assume that the underlying image has a good sparse approximation under a certain system. Wavelet tight frame system has been proven to be such an efficient system to sparsely approximate piecewise smooth images. Thus, it has been widely used in many practical image restoration problems. However, images from different scenarios are so diverse that no static wavelet tight frame system can sparsely approximate all of them well. To overcome this, recently, Cai et. al. (Appl Comput Harmon Anal 37:89–105, 2014) proposed a method that derives a data-driven tight frame adapted to the specific input image, leading to a better sparse approximation. The data-driven tight frame has been applied successfully to image denoising and CT image reconstruction. In this paper, we extend this data-driven tight frame construction method to multi-channel images. We construct a discrete tight frame system for each channel and assume their sparse coefficients have a joint sparsity. The multi-channel data-driven tight frame construction scheme is applied to joint color and depth image reconstruction. Experimental results show that the proposed approach has a better performance than state-of-the-art joint color and depth image reconstruction approaches.

Undersampled MR Image Reconstruction with Data-Driven Tight Frame.

Undersampled magnetic resonance image reconstruction employing sparsity regularization has fascinated many researchers in recent years under the support of compressed sensing theory. Nevertheless, most existing sparsity-regularized reconstruction methods either lack adaptability to capture the structure information or suffer from high computational load. With the aim of further improving image reconstruction accuracy without introducing too much computation, this paper proposes a data-driven tight frame magnetic image reconstruction (DDTF-MRI) method. By taking advantage of the efficiency and effectiveness of data-driven tight frame, DDTF-MRI trains an adaptive tight frame to sparsify the to-be-reconstructed MR image. Furthermore, a two-level Bregman iteration algorithm has been developed to solve the proposed model. The proposed method has been compared to two state-of-the-art methods on four datasets and encouraging performances have been achieved by DDTF-MRI.

SAR Image Despeckling Using Data-Driven Tight Frame

Data-driven sparse representation of images has been shown to be efficient approaches to image denoising. Following this observation, a novel technique is developed in this letter for reducing the speckle in synthetic aperture radar (SAR). This technique is conducted first by converting the multiplicative noisy image into a tractable additive one by taking logarithms. Then, a discrete data-driven tight frame is derived from the logarithmic transformed image and used for noise removal. Numerical experiments suggest that the proposed scheme not only runs relatively fast owing to the adoption of tight frame but also achieves comparable performance in comparison with some state-of-the-art methods.

Seismic data restoration via data-driven tight frame

Restoration/interpolation of missing traces plays a crucial role in the seismic data processing pipeline. Efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as Fourier, wavelet, curvelet, and shearlet transforms. Most existing methods are based on transforms with a fixed basis. We considered an adaptive sparse transform for restoration of data with complex structures. In particular, we evaluated a data-driven tight-frame-based sparse regularization method for seismic data restoration. The main idea of the data-driven tight frame (TF) is to adaptively learn a set of framelet filters from the currently interpolated data, under which the data can be more sparsely represented; hence, the sparsity-promoting [Formula: see text]-norm (SPL1) minimization methods can produce better restoration results by using the learned filters. A split inexact Uzawa algorithm, which can be viewed as a generalization of the alternating direction of multiplier method (ADMM), was applied to solve the presented SPL1 model. Numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid. Our experiments showed that our proposed method obtains the state-of-the-art restoration results in comparison with the traditional Fourier-based projection onto convex sets, the tight-frame-based method, and the recent shearlet regularization ADMM method.

Data-driven tight frame construction and image denoising

Sparsity-based regularization methods for image restoration assume that the underlying image has a good sparse approximation under a certain system. Such a system can be a basis, a frame, or a general over-complete dictionary. One widely used class of such systems in image restoration are wavelet tight frames. There have been enduring efforts on seeking wavelet tight frames under which a certain class of functions or images can have a good sparse approximation. However, the structure of images varies greatly in practice and a system working well for one type of images may not work for another. This paper presents a method that derives a discrete tight frame system from the input image itself to provide a better sparse approximation to the input image. Such an adaptive tight frame construction scheme is applied to image denoising by constructing a tight frame tailored to the given noisy data. The experiments showed that the proposed approach performs better in image denoising than those wavelet tight frames designed for a class of images. Moreover, by ensuring that the system derived from our approach is always a tight frame, our approach also runs much faster than other over-complete dictionary based approaches with comparable performance on denoising.