60-520 Presentation Image Filters Student: Xiaoliu Chen Instructor: Dr. I. Ahmad School of Computer Science University of Windsor November 2003

Outline Introduction Spatial Filtering Frequency-Domain Filtering Smoothing Sharpening Frequency-Domain Filtering Low pass High pass Summary Image Filters

Introduction Filtering is the process of replacing a pixel with a value based on some operations or functions. The operations/functions used on the original image are called filters. or masks, kernels, templates, windows… Image Filters

Introduction In digital image processing, filters are usually used to suppress the high frequencies in an image i.e., smoothing the image suppress the low frequencies in an image i.e., enhancing or detecting edges in the image Image Filters

Introduction Image filters fall into two categories: Spatial domain Filters are based on direct manipulation of pixels on an image plane. Frequency domain Filters are based on modifying the Fourier transform (FT) of an image. Image Filters

Spatial Filters The general processes can be denoted by the expression: f(x,y) is the input image g(x,y) is the processed image T is an operator on f, defined over some neighborhood of (x,y) Image Filters

Spatial Filters The principal approach in defining a neighborhood about a point (x,y) use a subimage area centered at (x,y) shapes of the neighborhood circle square rectangular Image Filters

Spatial Filters Example: 3×3 neighborhood about a point (x,y) in an image x (x,y) Image f(x,y) (x+1,y+1) (x,y+1) (x-1,y+1) (x+1,y) (x-1,y) (x+1,y-1) (x,y-1) (x-1,y-1) y Image Filters

x Image f(x,y) y Mask Mask coefficients Pixels under mask f(x+1,y+1) w(1,1) w(0,1) w(-1,1) w(1,0) w(0,0) w(-1,0) w(1,-1) w(0,-1) w(-1,-1) Mask Mask coefficients x y Image Filters

Spatial Filters – linear filters For linear spatial filtering, the result, R, at a point (x,y) is R=w(-1,-1)f(x-1,y-1) + w(0,-1)f(x,y-1) + …+ w(0,0)f(x,y) +… + w(0,1)f(x,y+1) + w(1,1)f(x+1,y+1) Image Filters

Spatial Filters – convolution In general, linear filtering of an image is given by the expression: The image f is of size M×N The filter mask is of size m×n m=2a+1, n=2b+1 Image Filters

Spatial Filters – smoothing Smoothing filters are used for blurring and for noise reduction. Smoothing, linear spatial filter average filters reduce “sharp” transitions side effect Image Filters

Spatial Filters – smoothing, linear 1 Mean filters example: Original Gaussian noise 5×5 mean filter 3×3 mean filter Image Filters

Spatial Filters – smoothing, linear 1 Mean filters example: Salt and pepper 5×5 mean filter 3×3 mean filter Image Filters

Spatial Filters – smoothing, linear Weighted average filters example: general expression: 1 2 4 Image Filters

Spatial Filters – smoothing, nonlinear Order-statistic filters nonlinear spatial filters order/rank the pixels contained in the image area encompassed by the filter Image Filters

Spatial Filters – smoothing, nonlinear Median filters replace a pixel value with the median of its neighboring pixel values example: 23 25 26 30 40 22 24 27 35 18 20 50 34 19 15 33 11 16 10 Neighborhood values: 15, 19, 20, 23, 24, 25, 26, 27, 50 Median value: 24 Image Filters

Spatial Filters – smoothing, nonlinear Median filters have excellent noise-reduction capabilities V.S. Gaussian noise removed by 3×3 mean filter Gaussian noise removed By 3×3 median filter Image Filters

Spatial Filters – smoothing, nonlinear Median filters are particularly effective in salt & pepper V.S. Salt & pepper removed by 3×3 mean filter Salt & pepper removed By 3×3 median filter Image Filters

Spatial Filters – smoothing, nonlinear Max filters maximum of neighboring pixel values useful for finding the brightest points in an image Min filters minimum of neighboring pixel values useful for finding the darkest points in an image Image Filters

Spatial Filters – sharpening Principal objective highlight fine detail in an image enhance detail that has been blurred Sharpening can be accomplished by spatial differentiation Image Filters

Spatial Filters – sharpening For one dimensional function f(x) first order derivative second order derivative Image Filters

Spatial Filters – sharpening A sample (a) a scan line (b) image strip (c) first derivative (d) second derivative (a) (b) (c) (d) 7 1 3 6 2 4 5 -1 -2 -6 -7 -4 -12 Image Filters

Spatial Filters – sharpening The Laplacian second derivative of a two dimensional function f(x,y) = [f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)] -4f(x,y) Image Filters

Spatial Filters – sharpening The Laplacian use a convolution mask to approximate 1 -4 1 -8 -1 2 -4 Image Filters

Spatial Filters – sharpening The Laplacian example: Image Filters

Spatial Filters – sharpening The Laplacian example: Image Filters

Frequency Filters – Fourier transform Fourier transform (FT) decompose an image into its sine and cosine components transform real space images into Fourier or frequency space images In a frequency space image, each point represents a particular frequency contained in the real domain image. Image Filters

Frequency Filters – Fourier transform Discrete Fourier transform (DFT) Inverse DFT Image Filters

Frequency Filters – Fourier transform example: FT (log) Image Filters

Frequency Filters Basic steps for filtering in the frequency domain function DFT Inverse f(x,y) Input image g(x,y) Processed image F(u,v) H(u,v)F(u,v) Image Filters

Frequency Filters Frequencies in an image correspond to the rate of change in pixel values High frequencies rapid changes of gray level values Low frequencies slow changes of gray level values Image Filters

Frequency Filters Lowpass filters Highpass filters attenuate high frequencies while “passing” low frequencies Highpass filters attenuate low frequencies while “passing” high frequencies Image Filters

Frequency Filters – lowpass filters Ideal lowpass filters (ILPF) Image Filters

Frequency Filters – lowpass filters Butterworth lowpass filters (BLPF) Image Filters

Frequency Filters – lowpass filters Gaussian lowpass filters (GLPF) Image Filters

Frequency Filters – highpass filters Ideal higpass filters (IHPF) Butterworth highpass filters (BHPF) Gaussian highpass filters (GHPF) Image Filters

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Frequency Filters – bandpass filters attenuate very low frequencies and very high frequencies enhance edges while reducing the noise at the same time Image Filters

Frequency Filters Examples: (lowpass filters) BLPF with cut-off frequency of 1/3 BLPF with cut-off frequency of 1/2 ILPF with cut-off frequency of 1/2 ILPF with cut-off frequency of 1/3 Original Gaussian noise Image Filters

Frequency Filters Examples: (highpass filters) Image Filters

Frequency Filters Relationship and comparison with spatial filters spatial filtering frequency filtering Image Filters

Frequency Filters Comparison with spatial filters more computational efficient more intuitive Image Filters

Summary Filtering is the operation of applying a transform on an image in order to enhance it. Filtering techniques can be subdivided into two types Spatial domain filtering Frequency domain filtering Image Filters

Summary Filtering techniques are very useful in image analysis and processing Noise removal Edge detection Image Filters

Thank you & Questions ? The end Image Filters