Youngmi Hur

Youngmi Hur, Kasso A. Okoudjou

Laplacian pyramid based Laurent polynomial (LP$^2$) matrices are generated by Laurent polynomial column vectors and have long been studied in connection with Laplacian pyramidal algorithms in Signal Processing. In this paper, we investigate when such matrices are scalable, that is when right multiplication by Laurent polynomial diagonal matrices results in paraunitary matrices. The notion of scalability has recently been introduced in the context of finite frame theory and can be considered as a preconditioning method for frames. This paper significantly extends the current research on scalable frames to the setting of polyphase representations of filter banks. Furthermore, as applications of our main results we propose new construction methods for tight wavelet filter banks and tight wavelet frames.

Youngmi Hur, Fang Zheng

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.

Youngmi Hur, Fang Zheng

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 lowpass filters, with one of whom 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 lowpass 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.

Sangjun Han, Taeil Hur, Youngmi Hur

In this paper, we develop the Laplacian pyramid-like autoencoder (LPAE) by adding the Laplacian pyramid (LP) concept widely used to analyze images in Signal Processing. LPAE decomposes an image into the approximation image and the detail image in the encoder part and then tries to reconstruct the original image in the decoder part using the two components. We use LPAE for experiments on classifications and super-resolution areas. Using the detail image and the smaller-sized approximation image as inputs of a classification network, our LPAE makes the model lighter. Moreover, we show that the performance of the connected classification networks has remained substantially high. In a super-resolution area, we show that the decoder part gets a high-quality reconstruction image by setting to resemble the structure of LP. Consequently, LPAE improves the original results by combining the decoder part of the autoencoder and the super-resolution network.

Youngmi Hur

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.

Sangjun Han, Youngmi Hur

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.

Youngmi Hur, Hyojae Lim, Mikyoung Lim

In this paper, we develop a wavelet-based theoretical framework for analyzing the universal approximation capabilities of neural networks over a wide range of activation functions. Leveraging wavelet frame theory on the spaces of homogeneous type, we derive sufficient conditions on activation functions to ensure that the associated neural network approximates any functions in the given space, along with an error estimate. These sufficient conditions accommodate a variety of smooth activation functions, including those that exhibit oscillatory behavior. Furthermore, by considering the $L^2$-distance between smooth and non-smooth activation functions, we establish a generalized approximation result that is applicable to non-smooth activations, with the error explicitly controlled by this distance. This provides increased flexibility in the design of network architectures.

Sangjun Han, Taeil Hur, Youngmi Hur et al.

The challenge of formal proof generation has a rich history, but with modern techniques, we may finally be at the stage of making actual progress in real-life mathematical problems. This paper explores the integration of ChatGPT and basic searching techniques to simplify generating formal proofs, with a particular focus on the miniF2F dataset. We demonstrate how combining a large language model like ChatGPT with a formal language such as Lean, which has the added advantage of being verifiable, enhances the efficiency and accessibility of formal proof generation. Despite its simplicity, our best-performing Lean-based model surpasses all known benchmarks with a 31.15% pass rate. We extend our experiments to include other datasets and employ alternative language models, showcasing our models' comparable performance in diverse settings and allowing for a more nuanced analysis of our results. Our findings offer insights into AI-assisted formal proof generation, suggesting a promising direction for future research in formal mathematical proof.