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This paper proposes an epigraphical relaxation (ERx) technique for non-proximable mixed norm minimization. Mixed norm regularization methods play a central role in signal reconstruction and processing, where their optimization relies on the fact that the proximity operators of the mixed norms can be computed efficiently. To bring out the power of regularization, sophisticated layered modeling of mixed norms that can capture inherent signal structure is a key ingredient, but the proximity operator of such a mixed norm is often unavailable (non-proximable). Our ERx decouples a layered non-proximable mixed norm into a norm and multiple epigraphical constraints. This enables us to handle a wide range of non-proximable mixed norms in optimization, as long as both the proximal operator of the outermost norm and the projection onto each epigraphical constraint are efficiently computable. Moreover, under mild conditions, we prove that ERx does not change the minimizer of the original problem despite relaxing equality constraints into inequality ones. We also develop new regularizers based on ERx: one is decorrelated structure-tensor total variation for color image restoration, and the other is amplitude-spectrum nuclear norm for low-rank amplitude recovery. We examine the power of these regularizers through experiments, which illustrates the utility of ERx., accepted to IEEE Transactions on Signal Processing

A raw image captured by a color filter array (CFA), such as a Bayer pattern, is usually compressed after demosaicing with some processings (denoising, deblurring, tone-mapping, and so on). However, since photographers, designers, and high-end users prefer to work with the raw image sampled by CFA (referred to as “raw image”) directly, a raw image should be compressed before demosaicing. For effective raw image compression, this study introduces variable macropixel spectral-spatial transforms (VMSSTs), that can successfully decorrelate not only Bayer raw images but any other pure-color (RGB) ones. The proposed VMSSTs are designed by the following two steps: 1) intra-color decorrelation and 2) inter-color decorrelation. In lossless compression with JPEG 2000, compared with methods which do not use transforms, the VMSSTs reduced the average bitrates of three types of CFAs: from approximately 0.09 to 0.12 bpp for the modified Bayer CFA, from 0.25 to 0.65 bpp for the diagonal stripe CFA, and from 0.33 to 0.70 bpp for the Fujifilm X-Trans CFA due to their high color decorrelation efficiency. In addition, in lossy compression with JPEG 2000, compared with a rearranged method, the VMSSTs improved the average bitrates of the Bjontegaard delta by around 3.97%, 14.95%, and 18.65% for each CFA model, respectively. Although a data-dependent adaptive transformation, the Karhunen-Loeve transform (KLT), showed the best performance in lossy compression, the introduced VMSSTs have shown performances comparable to those of the KLT in lossless compression, despite their simple structures.

We present a novel image boundary extension and mean value extension (MVE) for directional lapped transforms, particularly cosine–sine modulated lapped transforms (CSMLTs). Lapped transforms are usually used with an extension technique, such as periodic extension (PE) or symmetric extension (SE), for nonexpansive convolution at signal boundaries. When directional textures (oblique lines and curves) appear at the 2D signal (image) boundaries, both PE and SE produce directional discontinuities, which degrade the sparsity of the transformed coefficients, especially in the case of directional lapped transforms. MVE reduces the discontinuities for directional textures better than PE or SE do; it thus improves the efficiency of the sparse representation based on directional lapped transforms. Moreover, to reduce computational costs compared with those of directional lapped transforms, we introduce new directional block transforms called cosine–sine modulated block transforms. These new transforms are derived from a minimum tile processing (tiling) of $M$ -band CSMLTs with $2M$ filter lengths and nonexpansive convolutions. The resulting directional block transforms, particularly in the case of the MVE, have richer directional selectivity and have better performance compared with a discrete Fourier transform as shown in experiments.

Complex-valued conjugate-symmetric Hadamard transforms ( $\BBC $ -CSHT) are variants of complex Hadamard transforms and found applications in signal processing. In addition, their real-valued transform counterparts ( $\BBR $ -CSHTs) perform comparably with Hadamard transforms (HTs) despite their lower computational complexity. Closed-form factorizations of $\BBC $ -CSHTs and $\BBR $ -CSHTs have recently been proposed to make calculations more efficient. However, there is still room to find effective and general factorizations. This paper presents a simple closed-form complete factorization of $\BBC $ -CSHTs based on that of $\BBR $ -CSHTs. The proposed factorization can be applied to both $\BBC $ - and $\BBR $ -CSHTs with one factorization and it provides several benefits: 1) It can save total implementation costs for both $\BBC $ -CSHTs and $\BBR $ -CSHTs; 2) the generalized $\BBR $ -CSHT factorization significantly reduces its computational cost; 3) memory-saved local orientation detection of images can be achieved; 4) a fast direction-aware transform can be attained; 5) it clarifies that $\BBC $ - and $\BBR $ -CSHTs are closely related to common block transforms, such as the discrete Fourier transform (DFT), binDCT, and HT; and 6) it achieves a new integer complex-valued transform, which can approximate the DFT better than the original $\BBC $ -CSHT. The image orientation estimation and performance in image coding of our $\BBR $ -CSHTs were evaluated through examples of practical applications based on the proposed factorization.