• 한국수학교육학회지시리즈B:순수및응용수학
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• 제31권2호
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• pp.235-249
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• 2024

We obtain a structured class of frames in separable Hilbert spaces which are generalizations of Gabor frames in L²(ℝ) in their construction aspects. For this, the concept of Gabor type unitary systems in [13] is generalized by considering a system of invertible operators in place of unitary systems. Pseudo Gabor like frames and pseudo Gabor frames are introduced and the corresponding frame operators are characterized.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제28권4호
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• pp.315-328
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• 2021

A characterization of frame operators of finite Gabor frames is presented here. Regularity aspects of Gabor frames in 𝑙²(ℤ_N) are discussed by introducing associated semi-frame operators. Gabor type frames in finite dimensional Hilbert spaces are also introduced and discussed.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제29권4호
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• pp.321-334
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• 2022

We introduce a special class of structured frames having single generators in finite dimensional Hilbert spaces. We call them as pseudo B-Gabor like frames and present a characterisation of the frame operators associated with these frames. The concept of Gabor semi-frames is also introduced and some significant properties of the associated semi-frame operators are discussed.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1247-1256
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• 2008

Based on two real and invertible $d{\times}d$ matrices Band C such that the norm $||C^T\;B||$ is sufficiently small, we provide a construction of tight Gabor frames $\{E_{Bm}T_{Cn}g\}_{m,n{\in}{\mathbb{Z}^d}$ with explicitly given and compactly supported generators. The generators can be chosen with arbitrary polynomial decay in the frequency domain.

• Korean Journal of Mathematics
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• 제28권4호
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• pp.877-888
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• 2020

In this paper we discuss about the image of Gabor frame under a unitary operator and derive a sufficient condition under which a unitary operator from L²(ℝ) to 𝑙²(ℤ) maps Gabor frame in L²(ℝ) to a Gabor frame in 𝑙²(ℤ).

• 대한수학회지
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• 제51권5호
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• pp.897-918
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• 2014

This paper addresses multi-window Gabor frames with rational time-frequency product. Such issue was considered by Zibulski and Zeevi (Appl. Comput. Harmonic Anal. 4 (1997), 188-221) in terms of Zak transform matrix (so-called Zibuski-Zeevi matrix), and by many others. In this paper, we introduce of a new Zak transform matrix. It is different from Zibulski-Zeevi matrix, but more direct and convenient for our purpose. Using such Zak transform matrix we characterize rational time-frequency multi-window Gabor frames (Riesz bases and orthonormal bases), and Gabor duals for a Gabor frame. Some examples are also provided, which show that our Zak transform matrix method is efficient.

• Korean Journal of Mathematics
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• 제32권1호
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• pp.1-14
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• 2024

We seek for an invertible map B from L²(Γ) to L²(G), where G is a finite abelian group and Γ is the direct product of finite cyclic groups which is isomorphic to G, so that any Gabor frame in L²(G), is a generalized pseudo B-Gabor frame.

• Journal of Communications and Networks
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• 제12권4호
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• pp.289-307
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• 2010

The problem of model selection arises in a number of contexts, such as subset selection in linear regression, estimation of structures in graphical models, and signal denoising. This paper studies non-asymptotic model selection for the general case of arbitrary (random or deterministic) design matrices and arbitrary nonzero entries of the signal. In this regard, it generalizes the notion of incoherence in the existing literature on model selection and introduces two fundamental measures of coherence-termed as the worst-case coherence and the average coherence-among the columns of a design matrix. It utilizes these two measures of coherence to provide an in-depth analysis of a simple, model-order agnostic one-step thresholding (OST) algorithm for model selection and proves that OST is feasible for exact as well as partial model selection as long as the design matrix obeys an easily verifiable property, which is termed as the coherence property. One of the key insights offered by the ensuing analysis in this regard is that OST can successfully carry out model selection even when methods based on convex optimization such as the lasso fail due to the rank deficiency of the submatrices of the design matrix. In addition, the paper establishes that if the design matrix has reasonably small worst-case and average coherence then OST performs near-optimally when either (i) the energy of any nonzero entry of the signal is close to the average signal energy per nonzero entry or (ii) the signal-to-noise ratio in the measurement system is not too high. Finally, two other key contributions of the paper are that (i) it provides bounds on the average coherence of Gaussian matrices and Gabor frames, and (ii) it extends the results on model selection using OST to low-complexity, model-order agnostic recovery of sparse signals with arbitrary nonzero entries. In particular, this part of the analysis in the paper implies that an Alltop Gabor frame together with OST can successfully carry out model selection and recovery of sparse signals irrespective of the phases of the nonzero entries even if the number of nonzero entries scales almost linearly with the number of rows of the Alltop Gabor frame.

• 대한수학회보
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• 제50권6호
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• pp.1841-1846
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• 2013

In this paper we establish some new results on sums of Hilbert space frames and Riesz bases. We also provide a correction to some recently established results in [2].

• 한국멀티미디어학회논문지
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• 제16권6호
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• pp.764-781
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• 2013

본 논문에서는 시차영상 생성과 레이블링(labeling)을 동시에 수행하는 빌보드 스윕 스테레오 시차정합 알고리즘을 적용하고, 두 단계로 구성된 복합 가설생성(hypothesis generation) 단계를 적용함으로서 거짓알림(false alarm)을 줄이고, 차량 검출의 정확도를 높이는 방법을 제안한다. 먼저 차량의 정면에 장착된 두 대의 카메라를 이용하여 영상을 획득하고, 이 영상을 사용하여 빌보드 스윕 스테레오 시차정합 알고리즘을 수행하여 지면과 배경이 제거된 장애물(obstacle)만이 존재하는 특수한 형태의 시차영상을 생성한다. 이렇게 생성된 지면과 배경이 제거된 레이블링된 시차영상을 이용하여 차량 검출 및 추적을 수행한다. 차량 검출 및 추적단계는 크게 세 단계로 나눠진다. 첫 번째 단계는 학습 단계로서 학습데이터로부터 Gabor필터를 사용해서 특징점을 추출하고, 추출된 특징점을 학습한 뒤 서포트 벡터머신 분류기를 생성하는 단계이다. 두 번째 단계는 스테레오 카메라의 영상 중 주 카메라의 영상으로부터 에지 정보를 추출하고, 지면과 배경이 제거된 시차 영상으로부터 얻어진 시차정보를 이용해서 차량이 존재하는 후보영역을 뽑은 뒤 서포트 벡터머신 분류기를 사용하여 차량을 검출하는 단계이다. 마지막 단계는 차량 추적단계로서 검출이 완료된 차량들은 다음 프레임에서 템플릿 매칭을 수행하여 추적한다. 이는 추적에 성공할 경우 다음 프레임의 차량 검출시 후보영역에서 배제함으로서 전체적인 차량 검출 성능을 향상시킨다.