Outline Linear Shift-invariant system Linear filters Fourier transformation Time and frequency representation Filter Design

Visual Perception Modeling Linear System Theory What is a system? A system is anything that accepts an input and produces an output in response y[n] = T{x[n]} where x[n] is the input sequence and y[n] is the output sequence in responses to x[n] How to represent a sequence? 11/16/2018 Visual Perception Modeling

Visual Perception Modeling Linear System Linearity y1[n] = T{x1[n]} y2[n] = T{x2[n]} Then y1[n]+y2[n] = T{x1[n]+x2[n]} 11/16/2018 Visual Perception Modeling

Shift-Invariant System Shift invariance y[n] = T{x[n]} y[n-T] = T{x[n-T]} LSI system A LSI system is completely characterized by its impulse response h[n] For any other input, we can obtain the response through convolution 11/16/2018 Visual Perception Modeling

Visual Perception Modeling Filtering Closely related to convolution Filter examples Smoothing by averaging Smoothing by Gaussian 11/16/2018 Visual Perception Modeling

Multi-scale Representation Scale in the Gaussian function  is the standard deviation of the Gaussian distribution When  is small, no smoothing or very little When  is large, the noise will be largely disappear. However, the image detail will disappear along with the noise 11/16/2018 Visual Perception Modeling

Visual Perception Modeling Gaussian Pyramid 11/16/2018 Visual Perception Modeling

Why Gaussian Smoothing? Scale space If we convolve a Gaussian with a Gaussian, it will also be a Gaussian Efficiency A small kernel is generally enough Separable Central limit theorem 11/16/2018 Visual Perception Modeling

Spatial Frequency Analysis Filter response analysis For example, why does smoothing reduce noise? What is the difference between the discrete image representation and a continuous surface representation? Is there any way we can design the best filter for a certain task? For smoothing, how can we have the best smoothing kernel? 11/16/2018 Visual Perception Modeling

Visual Perception Modeling Fourier Transforms Fourier transform The transformation takes a complex valued function x, y and returns a complex valued function of u, v U and v determine the spatial frequency and orientation of the sinusoidal component 11/16/2018 Visual Perception Modeling

Inverse Fourier Transform It recovers a signal from its Fourier transform 11/16/2018 Visual Perception Modeling

Some Fourier Transform Pairs Step function Window function sinc function Gaussian function 11/16/2018 Visual Perception Modeling

Visual Perception Modeling Filter Design Design filters to accomplish particular goals Lowpass filters Reduce the amplitude of high-frequency components Can reduce the visible effects of noise Box filter Triangle filter High-frequency cutoff Gaussian lowpass filter 11/16/2018 Visual Perception Modeling

Visual Perception Modeling Filter Design – cont. Bandpass and bandstop filters Highpass filters Optimal filter design In some sense, optimal of doing a particular job Establish a criterion of performance and then maximize the criterion by proper selection of the impulse response Wiener estimator Wiener deconvolution 11/16/2018 Visual Perception Modeling

Other Transformations Fourier transform is one of a number of linear transformations that are useful in image processing Basis functions How to represent an image by weighted sum of some functions of our choice? 11/16/2018 Visual Perception Modeling

Principal Component Analysis Optimal representation with fewer basis functions We want to design a set of basis functions such that we can reconstruct the original image with smallest possible error with a given number of basis functions 11/16/2018 Visual Perception Modeling

PCA for Face Recognition 11/16/2018 Visual Perception Modeling

PCA for Face Recognition – cont. First 20 principal components 11/16/2018 Visual Perception Modeling

PCA for Face Recognition – cont. Components with low eigenvalues 11/16/2018 Visual Perception Modeling

PCA for Face Recognition – cont. 11/16/2018 Visual Perception Modeling