ABSTRACT With advances in artificial intelligence, image processing has gained significant interest. Image super‐resolution is a vital technology closely related to real‐world applications, as it enhances the quality of existing images. Since enhancing fine details is crucial for the super‐resolution task, pixels that contribute to high‐frequency information should be emphasized. This paper proposes two methods to enhance high‐frequency details in super‐resolution images: a Laplacian pyramid‐based detail loss and a repeated upscaling and downscaling process. Total loss with our detail loss guides a model by separately generating and controlling super‐resolution and detail images. This approach allows the model to focus more effectively on high‐frequency components, resulting in improved super‐resolution images. Additionally, repeated upscaling and downscaling amplify the effectiveness of the detail loss by extracting diverse information from multiple low‐resolution features. We conduct two types of experiments. First, we design a CNN‐based model incorporating our methods. This model achieves state‐of‐the‐art results, surpassing all currently available CNN‐based and even some attention‐based models. Second, we apply our methods to existing attention‐based models on a small scale. In all our experiments, attention‐based models adding our detail loss show improvements compared to the originals. These results demonstrate our approaches effectively enhance super‐resolution images across different model structures.

Scattering network is a convolutional network, consisting of cascading convolutions using pre-defined wavelets followed by the modulus operator [1]. The scattering network is one of few mathematical tools explaining the convolutional neural networks (CNNs). However, a pooling operator, which is a main component of CNNs, is not considered in the original scattering network. We model a continuous max-pooling, apply it to the scattering network, and get a new network named scattering-maxp network. We show that the scattering-maxp network shares many useful properties of the scattering network including translation invariance, and we conduct numerical experiments showing the computational advantage of our network. We summarize our main findings below. More details about our results can be found in [2] and Python codes are available in https://github.com/TaekvungKi/Scattering_maxp.

As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D low-pass filters, with one of which being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation; 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D low-pass filters, and furthermore; 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.

A multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in multiresolution analysis setup. Some multivariate refinement masks may be decomposed into lower dimensional refinement masks. Tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks. We present an alternative method, which we call coset sum, for constructing multivariate refinement masks from univariate refinement masks. The coset sum shares many essential features of the tensor product that make it attractive in practice: 1) it preserves the biorthogonality of univariate refinement masks, 2) it preserves the accuracy number of the univariate refinement mask, and 3) the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients. The coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product. These features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems. Some experimental results using 2-D images are presented to illustrate our findings.

Given a lowpass filter, finding a dual lowpass filter is an essential step in constructing non-redundant wavelet filter banks. Obtaining dual lowpass filters is not an easy task. In this paper, we introduce a new method called committee algorithm that builds a dual filter straightforwardly from two easily-constructible lowpass filters. It allows to design a wide range of new wavelet filter banks. An example based on the family of Burt-Adelson's 1-D Laplacian filters is given.

BACKGROUND: Copy number aberrations (CNAs) are an important molecular signature in cancer initiation, development, and progression. However, these aberrations span a wide range of chromosomes, making it hard to distinguish cancer related genes from other genes that are not closely related to cancer but are located in broadly aberrant regions. With the current availability of high-resolution data sets such as single nucleotide polymorphism (SNP) microarrays, it has become an important issue to develop a computational method to detect driving genes related to cancer development located in the focal regions of CNAs. RESULTS: In this study, we introduce a novel method referred to as the wavelet-based identification of focal genomic aberrations (WIFA). The use of the wavelet analysis, because it is a multi-resolution approach, makes it possible to effectively identify focal genomic aberrations in broadly aberrant regions. The proposed method integrates multiple cancer samples so that it enables the detection of the consistent aberrations across multiple samples. We then apply this method to glioblastoma multiforme and lung cancer data sets from the SNP microarray platform. Through this process, we confirm the ability to detect previously known cancer related genes from both cancer types with high accuracy. Also, the application of this approach to a lung cancer data set identifies focal amplification regions that contain known oncogenes, though these regions are not reported using a recent CNAs detecting algorithm GISTIC: SMAD7 (chr18q21.1) and FGF10 (chr5p12). CONCLUSIONS: Our results suggest that WIFA can be used to reveal cancer related genes in various cancer data sets.

A new wavelet-based methodology for representing data on regular grids is introduced and studied. The main attraction of this new “Local Compression-Alignment-Modified- Prediction (L-CAMP)” methodology is in the way it scales with the spatial dimension, making it, thus, highly suitable for the representation of high dimensional data. The specific highlights of the L-CAMP methodology are three. First, it is computed and inverted by fast algorithms with linear complexity and very small constants; moreover, the constants in the complexity bound decay, rather than grow, with the spatial dimension. Second, the representation is accompanied by solid mathematical theory that reveals its performance in terms of the maximal level of smoothness that is accurately encoded by the representation. Third, the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme. An illustration of this last property is done by comparing the L-CAMP system that is marked in this paper as V with the widely used tensor-product biorthogonal 9/7. Both are essentially equivalent in terms of performance. However, the L-CAMP V has in 10D localness score${≪ 29}$. The localness score of the 9/7 is, in that same dimension,${> 575{\,}000{\,}000{\,}000}$.