Outline Linear Shiftinvariant system Linear filters Fourier transformation

Outline • Linear Shift-invariant system • Linear filters • Fourier transformation – Time and frequency representation • Filter Design 1/19/2022 Visual Perception Modeling 1

Source Separation • Mixed signal – Music and speech • Separated signals – Music – Speech 1/19/2022 Visual Perception Modeling 2

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? 1/19/2022 Visual Perception Modeling 3

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 1/19/2022 Visual Perception Modeling 4

Inverse Fourier Transform • Inverse Fourier transform – It recovers a signal from its Fourier transform 1/19/2022 Visual Perception Modeling 5

Some Fourier Transform Pairs • Step function • Window function • sinc function • Gaussian function 1/19/2022 Visual Perception Modeling 6

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 1/19/2022 Visual Perception Modeling 7

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 1/19/2022 Visual Perception Modeling 8

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 1/19/2022 Visual Perception Modeling 9

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? 1/19/2022 Visual Perception Modeling 10

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 1/19/2022 Visual Perception Modeling 11

PCA for Face Recognition 1/19/2022 Visual Perception Modeling 12

PCA for Face Recognition – cont. First 20 principal components 1/19/2022 Visual Perception Modeling 13

PCA for Face Recognition – cont. Components with low eigenvalues 1/19/2022 Visual Perception Modeling 14

PCA for Face Recognition – cont. 1/19/2022 Visual Perception Modeling 15

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) 1/19/2022 Visual Perception Modeling 16