An Introduction to Wavelets Amara Graps Computer Vision

An Introduction to Wavelets (Amara Graps) Computer Vision & Pattern Recognition Lab. 김 태 철 2004. 10. 4 (월) Computer Vision & Pattern Recognition Lab.

Overview § 1. Wavelets Overview § 2. Historical Perspective § 3. Fourier Analysis § 4. Wavelet Transforms versus Fourier Transforms § 5. What do some wavelets look like? § 6. Wavelet Analysis § 7. Wavelet Applications Computer Vision & Pattern Recognition Lab. 2

2. Historical Perspective § Pre-1930 √ 1807: J. Fourier, Fourier Series 주파수 분석 (Fourier Transform) √ 1909: Haar, Orthonormal Function System 다중 해상도 방법과 웨이블릿의 긴밀한 변환 관계를 쉽게 설명 척도(scale) 분석 (Harr System) § The 1930 S scale-varying basis Function : Haar basis function √ Brownian motion - Paul Levy wavelet을 이해하기 위한 key √ Another researcher: energy function David Marr : Effective Algorithm for numerical image processing Computer Vision & Pattern Recognition Lab. 4

3. Fourier Analysis (Windowed Fourier Transforms) § WFT (Windowed Fourier Transform) § 일정한 크기를 가진 창함수와 Fourier 변환의 결합 § 시간 변화에 따른 주파수 특성을 분석 § 단점 √ 분석 영역이 시간-주파수에 대해 항상 일정 √ 비정상적인 신호(non-stationary signals)를 효율적 으로 분석할 수 없다. Computer Vision & Pattern Recognition Lab. 8

4. Wavelet Transforms Versus Fourier Transforms § Similarities between fourier and wavelet transforms √ Both Linear Operation √ FFT와 DWT의 inverse transform matrix는 original의 transpose √ fft는 sines과 cosines를 포함하는 basis function √ wavelet transform은 wavelets 혹은 mother wavelets or analyzing wavelets이라고 불리우는 더 복잡한 basis funtion Computer Vision & Pattern Recognition Lab. 11

4. Wavelet Transforms Versus Fourier Transforms § Dissimilarities between fourier and wavelet transforms √ wavelet funtions are localized in space but fourier sin and cosine functions are not. Computer Vision & Pattern Recognition Lab. 12

Continue… FT WT Computer Vision & Pattern Recognition Lab. 13

5. What do some wavelets look like? § Wavelets family는 상관계수의 수와 반복횟수 의 의해서 구별되는 웨이블릿의 부분집합 § wavelets family는 vanishing moments의 수에 의 해 구별 Computer Vision & Pattern Recognition Lab. 14

5. What do some wavelets look like? (continue) Computer Vision & Pattern Recognition Lab. 15

7. Wavelet Applications § Computer and Human Vision √ 1980’s 초 David Marr - 로봇에 대한 인공 시각 (artifical vision) 연구 시작 √ Human Visual System에서의 영상 처리 – vision filter ( Marr’s operator ‘Marr wavelet’ ) Computer Vision & Pattern Recognition Lab. 17

7. Wavelet Applications § FBI fingerprint Compression 원본 이미지 26: 1 압축 Computer Vision & Pattern Recognition Lab. 18

7. Wavelet Applications § Denoising Noisy Data Computer Vision & Pattern Recognition Lab. 19

7. Wavelet Applications § Musical Tones (Ex. 음성, 음악의 합성) Computer Vision & Pattern Recognition Lab. 20