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

Visual Perception Modeling Source Separation Mixed signal Music and speech Separated signals Music Speech 11/29/2018 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? 11/29/2018 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 11/29/2018 Visual Perception Modeling

Inverse Fourier Transform It recovers a signal from its Fourier transform 11/29/2018 Visual Perception Modeling

Some Fourier Transform Pairs Step function Window function sinc function Gaussian function 11/29/2018 Visual Perception Modeling

Properties of Fourier Transform There are nice properties of Fourier transforms Convolution theorem F(f(x,y) * g(x,y)) = F(f(x,y)) F(g(x,y)) Can be used to speed up convolution especially for large filters 11/29/2018 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 11/29/2018 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 11/29/2018 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? 11/29/2018 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 11/29/2018 Visual Perception Modeling

PCA for Face Recognition 11/29/2018 Visual Perception Modeling

PCA for Face Recognition – cont. First 20 principal components 11/29/2018 Visual Perception Modeling

PCA for Face Recognition – cont. Components with low eigenvalues 11/29/2018 Visual Perception Modeling

PCA for Face Recognition – cont. 11/29/2018 Visual Perception Modeling

Wavelet Transformations Transient signal components Nonzero only during a short interval Many important features in images are highly localized Wavelets Given a real-valued function (s) 11/29/2018 Visual Perception Modeling