No abstract

An important capability for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling. In this regards, this study first provides necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. Then, the approximation order of such non-stationary schemes is discussed to quantify their approximation power. Based on these results, we see that the exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present normalized exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that the set of exponential polynomials to be reproduced is varied depending on the normalization factor. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.

Patch-based low rank matrix approximation has shown great potential in image denoising. Among state-of-the-art methods in this topic, the weighted nuclear norm minimization (WNNM) has been attracting significant attention due to its competitive denoising performance. For each local patch in an image, the WNNM method groups nonlocal similar patches by block matching to formulate a low-rank matrix. However, the WNNM often chooses irrelevant patches such that it may lose fine details of the image, resulting in undesirable artifacts in the final reconstruction. In this regards, this paper aims to provide a denoising algorithm which further improves the performance of the WNNM method. For this purpose, we develop a new nonlocal similarity measure by exploiting both pixel intensities and gradients and present a filter that enhances edge information in a patch to improve the performance of low rank approximation. The experimental results on widely used test images demonstrate that the proposed denoising algorithm performs better than other state-of-the-art denoising algorithms in terms of PSNR and SSIM indices as well as visual quality.

Over the past decade, deep learning-based computer vision methods have been shown to surpass previous state-of-the-art computer vision techniques in various fields, and have made great progress in various computer vision problems, including object detection, object segmentation, face recognition, etc. Nowadays, major IT companies are adding new deep-learning-based computer technologies to edge devices such as smartphones. However, since the computational cost of deep learning-based models is still high for edge devices, research is being actively carried out to compress deep learning-based models while not sacrificing high performance. Recently, many lightweight architectures have been proposed for deep learning-based models which are based on low-rank approximation. In this paper, we propose an alternating tensor compose-decompose (ATCD) method for the training of low-rank convolutional neural networks. The proposed training method can better train a compressed low-rank deep learning model than the conventional fixed-structure based training method, so that a compressed deep learning model with higher performance can be obtained in the end of the training. As a representative and exemplary model to which the proposed training method can be applied, we propose a rank-1 convolutional neural network (CNN) which has a structure alternatively containing 3-D rank-1 filters and 1-D filters in the training stage and a 1-D structure in the testing stage. After being trained, the 3-D rank-1 filters can be permanently decomposed into 1-D filters to achieve a fast inference in the test time. The reason that the 1-D filters are not being trained directly in 1-D form in the training stage is that the training of the 3-D rank-1 filters is easier due to the better gradient flow, which makes the training possible even in the case when the fixed structured network with fixed consecutive 1-D filters cannot be trained at all. We also show that the same training method can be applied to the well-known MobileNet architecture so that better parameters can be obtained than with the conventional fixed-structure training method. Furthermore, we show that the 1-D filters in a ResNet like structure can also be trained with the proposed method, which shows the fact that the proposed method can be applied to various structures of networks.