PDF《Filters》

Page ?#? Filters Basic Signal Processing CS148 Lecture

14 Pat Hanrahan, Winter

2007 Topics Filters in the spatial domain Spectral representation Fourier transforms The convolution theorem Filters in the frequency domain Next week: Sampling, aliasing and antialiasing Compression Page ?#? Filter = Convolution CS148 Lecture

14 Pat Hanrahan, Winter

2007 Convolution Signal/Image Filter

1 3

0 4

2 1 …

1 2 Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Convolution

1 *

1 +

2 *

3 =

7 1

3 0

4 2

1 …

1 2

7 … CS148 Lecture

14 Pat Hanrahan, Winter

2007 Convolution

1 *

3 +

2 *

0 =

3 1

3 0

4 2

1 …

1 2

7 3 … Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Convolution

1 *

0 +

2 *

4 =

8 1

3 0

4 2

1 …

1 2

7 3

8 … CS148 Lecture

14 Pat Hanrahan, Winter

2007 Convolution of a "Spike" Signal/Image Filter Result: copy of the filter centered at the spike

0 0

1 0

0 0 …

0 1

2 1

0 0 …

1 2

1 Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Convolution of a Two Boxes Signal/Image Filter Result: Convolution of two boxes is a triangle

0 0

1 1

0 0 …

0 1

2 1

0 0 …

1 1 CS148 Lecture

14 Pat Hanrahan, Winter

2007 Mathematical Definition: Convolution Signal Filter Convolution Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Low-Pass Filter Original Blurred CS148 Lecture

14 Pat Hanrahan, Winter

2007 High-Pass Filter Original Edge Enhancement Page ?#? Basic Signal Processing CS148 Lecture

14 Pat Hanrahan, Winter

2007 Sines and Cosines Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Frequency Frequency (cycles per interval) Angular frequency CS148 Lecture

14 Pat Hanrahan, Winter

2007 Spectral Representation Functions may be represented as a sum of sin/cos Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Cosines are Orthogonal CS148 Lecture

14 Pat Hanrahan, Winter

2007 Fourier Coefficients Power spectra: Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Two Domains Two representations of a function Spatial domain - normal representation Frequency domain - spectral representation The Fourier transform converts between domains Spatial Domain Frequency Domain Fourier Transform Inverse Fourier Transform CS148 Lecture

14 Pat Hanrahan, Winter

2007 Box Function and Sinc Function Spatial Domain Frequency Domain Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 The Box Function This EPS image does not contain a screen preview.

It will print correctly to a PostScript printer. File Name : sinc.eps box(x) =

0 1

0 x < 1/2 1/2 < x 1/2 CS148 Lecture

14 Pat Hanrahan, Winter

2007 The "Sinc" Function This EPS image does not contain a screen preview. It will print correctly to a PostScript printer. File Name : sinc.eps box(x)cos2fxdx = cos2fxdx

1 2

1 2 = sin2fx 2f

1 2

1 2 = sin

1 2 2f sin

1 2 2f 2f = sinf + sinf 2f = sinf f Page ?#? Gallery of Properties CS148 Lecture

14 Pat Hanrahan, Winter

2007 Pat's Frequencies Spatial Domain Frequency Domain Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Product Property Spatial Domain Frequency Domain CS148 Lecture

14 Pat Hanrahan, Winter

2007 Product Property Spatial Domain Frequency Domain Page ?#? CS148 Lecture

14 Pat Hanrahan, Winter

2007 Product Property Spatial Domain Frequency Domain CS148 Lecture

14 Pat Hanrahan, Winter

2007 Scaling Property Spatial Domain Frequency Domain Page ?#? CS148 Lecture