Digital image processing - Image transforms

Nguyễn Công Phương DIGITAL IMAGE PROCESSING Image Transforms Contents I. Introduction to Image Processing & Matlab II. Image Acquisition, Types, & File I/O III. Image Arithmetic IV. Affine & Logical Operations, Distortions, & Noise in Images V. Image Transform VI. Spatial & Frequency Domain Filter Design VII. Image Restoration & Blind Deconvolution VIII. Image Compression IX. Edge Detection X. Binary Image Processing XI. Image Encryption & Watermarking XII. Image Classification & Segmentation XIII. Image – Based Object Tracking XIV. Face Recognition XV. Soft Computing in Image Processing sites.google.com/site/ncpdhbkhn 2 Image Transforms 1. Discrete Fourier Transform (DFT) in 2D 2. Wavelet Transform 3. Hough Transform sites.google.com/site/ncpdhbkhn 3 DFT (1) • For an image of size M×N. • f(m,n): the image in the spatial domain. • F(u,v): in the Fourier space. sites.google.com/site/ncpdhbkhn 4 1 1 2 2 0 0 1( , ) ( , ) um vnM N j M N m n F u v f m n e N             1 1 2 2 0 0 1( , ) ( , ) um vnM N j M N m n f m n F u v e N            DFT (2) • DFT  FFT (Fast Fourier Transform). • F(0,0) represents the DC component of the image, which corresponds to the average brightness. • F(N – 1, N – 1) represents the highest frequency. • DFT is used to access the geometric characteristics of a spatial domain image. • In most implementation, the Fourier image is shifted in such a way that the DC value (the image mean), F(0,0), is displayed in the center of the image. The further away from the center an image point is, the higher is its corresponding frequency. sites.google.com/site/ncpdhbkhn 5 Image Transforms 1. Discrete Fourier Transform (DFT) in 2D 2. Wavelet Transform 3. Hough Transform sites.google.com/site/ncpdhbkhn 6 Wavelet Transform (1) • a specific type of wavelet with scaling & shifting in x & y axes as (a1,b1) & (a2,b2), respectively. • Haar, Daubechies, Gaussian, Mexican hat, etc. sites.google.com/site/ncpdhbkhn 7 1 1 1 2 1 1 2 2 0 0 1 21 2 1( , , , ) ( , ) , M N m n m a n aF a b a b f m n b bb b           1 1 2 2 1 1 2 2 1 1 1 1 1 1 2 1 1 2 2 0 0 0 0 1 21 2 1( , ) ( , , , ) , A B A B a b a b m a n af m n F a b a b b bb b                 1 2 1 21 2 1 , :m a n a b bb b       Wavelet Transform (2) • Wavelet transform takes a mother wavelet (e.g., Haar), and the signal is translated into shifted & scaled versions of this mother wavelet. • Used to divide the information of an image into approximation & detail subsignals: – Approximation subsignal: shows the general trend of pixel value, – Detail subsignals: show the vertical, horizontal, & diagonal details or changes in the image. • Applied in image filtering & image compression. sites.google.com/site/ncpdhbkhn 8 Image Transforms 1. Discrete Fourier Transform (DFT) in 2D 2. Wavelet Transform 3. Hough Transform sites.google.com/site/ncpdhbkhn 9 Hough Transform • From the rectangular coordinate system to the polar coordinate system. • m, n: in the rectangular system. • ρ, θ: in the polar system. • Can be commonly used to detect regular curves such as lines, circles, ellipses, etc. sites.google.com/site/ncpdhbkhn 10 cos sinm n   