Study of digital filters and nonlinear image processing by harmonic analysis methods and their applications

2023 Fiscal Year Final Research Report

• Project/Area Number: 19H01801
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Waseda University
• Principal Investigator: Arai Hitoshi 早稲田大学, 教育・総合科学学術院, 教授 (10175953)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: フレームレット / 単純かざぐるまフレームレット / かざぐるまフレームレット / 窓関数 / 調和解析 / ディジタル・フィルタ / スーパーハイブリッド画像

Outline of Final Research Achievements

The research results obtained during the research period (including carry-over extensions) are as follows: By using a harmonic analysis method, we constructed filters (for convenience, we call them atomic filters) which are the basis for image processing (H. Arai and S. Arai, 2013). In this study, I used deep learning to deform them according to purposes, making it possible to deform it without significantly changing their shape. I believe that this will be useful in studying image processing and the mechanisms of classification of images by deep learning. I also created a new super-hybrid image by manipulating atomic filters. In addition, I obtained a discretization of the　window of Bochner-Riesz summation, which is being studied in harmonic analysis, investigated the spatial-frequency state, and applied it to the spectrogram of signals, for example a speech signal. I also obtained future new themes based on these results.

Free Research Field

数学，解析学

Academic Significance and Societal Importance of the Research Achievements

本研究成果の学術的意義は，本研究において調和解析的方法（具体的には（単純）かざぐるまフレームレット（新井・新井，2009, 2011））と深層学習を融合して，新しい原子フィルタの変形の方法と実例を与えたこと，及び画像処理への応用の可能性と深層学習への応用の可能性も得られたことである．このほか調和解析から窓関数の応用を得たことも挙げられる．なお原子フィルタを調整してスーパーハイブリッド画像の新作を作成し，NHK総合のテレビ番組で社会に広く紹介され，またハイブリッド画像の新作も絵画の教本で紹介されるなど，エンターテイメントやアートとしての社会的意義もあった．