# Data-Driven Redundant Transform Based on Parseval Frames
## 1 Introduction
Compared to a general frame, a tight frame (e.g., wavelet tight frames [26], ridgelets [27], curvelets [28], shearlets [29], and others) can achieve wider use, as the lower and upper frame bounds are equal. A tight frame inherits the good characteristics of an orthogonal basis in signal processing, as its rows are orthogonal [30]. In a sparse representation, a redundant frame serves as an overcomplete dictionary to represent the signal [23]. With the development of data-driven approaches, learning-based tight frames have recently been researched [31; 32; 33]. In [31], redundant tight frames were used in compressed sensing. In [32], tight frames were applied to few-view image reconstruction. In [33], a data-driven method was presented, in which the dictionary atoms associated with a tight frame are generated by filters. In general, these studies model the frame learning problem in the dictionary learning form with tight frame constraints. These methods focus on tight frames, as the singular values of a tight frame are equal, which leads to simple optimization. A tight frame is a Parseval frame if the frame bounds are equal to 1. In fact, a Parseval frame is a redundant extension of the concept of a standard orthogonal basis. Due to its super-performance in linear signal representation, it can be well-used in sparse signal representation and optimization. In this paper, we propose a data-driven redundant transform model based on Parseval frames (DRTPF for short), and present a model for learning DRTPF as well as a corresponding algorithm for solving the model. The algorithm consists of a sparse coding phase and a transform learning phase. The sparse coding phase updates the sparse coefficients and a threshold value using a conventional Batch Orthogonal Matching Pursuit (BUOMP) and pointwise thresholding. The transform learning phase performs the update of the frame using Gradient Descent and a relaxation or contraction singular values mapping, as well as updating the dual frame, in an atom-wise manner, using Least Squares. The advantages of the proposed DRTPF model (as well as the algorithm) are demonstrated with natural image denoising. To summarize, this paper makes the following contributions:  
1. We propose the DRTPF method by integrating redundant Parseval frames with sparse constraints. The DRTPF method consists of a forward transform and a backward transform, which correspond to a frame and its dual frame, respectively. In other words, DRTPF bridges synthesis and analysis models by assuming that two models share almost the same sparse coefficients. 2. DRTPF outperforms traditional transforms and frames by learning from data which exploits the features of natural images, whereas traditional transforms and frames admit a uniform representation of various images, which tend to fail to characterize the intrinsic individual-specific features. 3. Traditional transforms are usually orthogonal transforms and the signals remain isometric, yet they suffer from weak robustness due to their strict properties. In contrast, DRTPF preserves the signals in a bounded fashion, which admits higher robustness and flexibility. 4. We propose a model for learning DRTPF and compare DRTPF with traditional transforms and sparse models in robustness analysis and image denoising experiments. Both qualitative and quantitative results demonstrate that DRTPF outperforms traditional transforms and sparse models. The rest of this paper is organized as follows. Section 2 reviews the related work on frames. Section 3 proposes the framework of DRTPF, including the form of DRTPF (Section 3.1) and the learning model and corresponding algorithm for DRTPF (Section 3.2). In Section 4, we demonstrate the effectiveness of our DRTPF model by analyzing the convergence of the corresponding algorithm and give experimental results on robustness analysis and image denoising, as well as evaluating the effectiveness of DRTPF compared with traditional transforms and sparse models.