# Data-Driven Redundant Transform Based on Parseval Frames
## 1 Introduction  
A transform is a classical technique in signal processing, such as compression, classification, and recognition [1; 2; 3; 4; 5]. Traditional transforms, based on analytic orthogonal bases such as DCT, DFT, and Wavelets [1; 6], suffer from two shortcomings: they do not depend on the data, and they reconstruct each image by approximation in the same subspace spanned by a non-redundant basis of the transforms, which limits the compact representation of natural signals. Various models for sparse approximation have appeared in recent decades and play a fundamental role in modeling natural signals, with applications of denoising [7; 8; 9; 10], super-resolution [11; 12; 13], and compression [1]. Such techniques exploit the sparsity of natural signals in analytic transform domains such as DCT, DFT, and various learning-based dictionaries [14; 15; 16]. There are two typical models for sparse representation: synthesis [10; 14; 15] and analysis [16; 17; 18; 19] models. So far, most sparse models rely on the concept of synthesis, which represents the underlying signal as a sparse combination of atoms from a given dictionary. Specifically, \(\mathbf{x}=\mathbf{D}\boldsymbol{\alpha}\), where \(\mathbf{x}\in\mathbb{R}^{N}\) is the original signal, \(\mathbf{D}\in\mathbb{R}^{N\times M}\) is the given dictionary whose columns are the atoms, and \(\boldsymbol{\alpha}\in\mathbb{R}^{M}\) is thesparse coefficient, which is usually measured by the \(\ell_{0}\)-norm \(\|\cdot\|_{0}\). A learning analysis sparse model was proposed by [PERSON] [14; 19], formulated as \(\|\mathbf{\Omega}\mathbf{x}\|_{0}=r\) with notation similar to that of the synthesis one. Instead of reconstructing the signal using a few atoms in dictionary (like in the synthesis model), an analysis model decomposes a signal in a sparse fashion, based on an assumption that the signal lies in a sparse subset of the dictionary. An analysis model can be straightforwardly regarded as a forward transform if its corresponding backward transform \(\mathbf{\Omega}^{*}\) is available. Recent research on transforms [2; 4; 5; 20; 21] has demonstrated the advantages of applying sparse constraints in transform learning. Motivated by this idea, many studies have been devoted to image denoising [5; 20], classification [3; 4], and other signal processing methods [21]. Learning-based transforms with sparse constraints measure the transform error, called sparsificaiton error, in the analysis or frequency domain, rather than in the temporal domain. Given training data \(\mathbf{X}\in\mathbb{R}^{N\times L}\) with signal vectors \(\mathbf{x}_{i}\in\mathbb{R}^{N},i=1,\ldots,L\) as its columns, the problem of training a square sparsifying transform \(\mathbf{W}\in\mathbb{R}^{N\times N}\)[21] is formulated as  
\[\min_{\mathbf{W},\mathbf{Y}}\|\mathbf{W}\mathbf{X}-\mathbf{Y}\|_ {F}^{2}+\mu\|\mathbf{W}\|_{F}^{2}-\lambda\log\det(\mathbf{W}) \tag{1}\] \[\text{s.t.}\ \|\mathbf{y}_{i}\|_{0}\leq s,\]  
where \(\mathbf{y}_{i},i=1,2,\ldots,L\) are the columns of \(\mathbf{Y}\) satisfying a sparse constraint and \(\mu\|\mathbf{W}\|_{F}^{2}-\lambda\log\det(\mathbf{W})\) is a regularizer, which keeps \(\mathbf{W}\) non-singular. As we can see, learning-based models effectively reveal the relationship between the transform and the data. The square transform, which consists of a non-redundant basis, cannot express complicated images. In 2014, an overcomplete transform learning model called OCTOBOS [20] was proposed, which consists of a series of square transforms to represent different features of natural images. However, the number of transforms must be pre-defined, which admits limited flexibility in applications. In recent years, frames, as an overcomplete system, have been applied in image processing such as denoising [22; 23], image compressive [24] and high resolution image reconstruction [25]. A frame can be regarded as an extension of an orthogonal basis, as a frame \(\mathbf{\Phi}\in\mathbb{R}^{N\times M}(N<M)\) also spans an \(N\)-dimensional space.