COMP 9517 Computer Vision

Transcription

1 COMP 9517 Computer Vision Frequency Domain Techniques 1

2 Frequency Versus Spa<al Domain Spa<al domain The image plane itself Direct manipula<on of pixels Changes in pixel posi<on correspond to changes in the scene Frequency domain Fourier transform of an image Directly related to rate of changes in the image Changes in pixel posi<on correspond to changes in the spa<al frequency

3 Frequency Domain Overview Frequency in image High frequencies correspond to pixel values that change rapidly across the image Low frequency components correspond to large scale features in the image Frequency domain Defined by values of the Fourier transform and its frequency variables (u, v) 3

4 Frequency Domain Overview Frequency domain processing 4

5 Fourier Series Periodic func<on can be represented as a weighted sum of sines and cosines Even func<ons that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines and/or cosines mul<plied by a weight func<on sum = 5

6 One- Dim Fourier Transform and Inverse For a single variable con<nuous func<on f(x), the Fourier transform F(u) is defined by: F(u) = where j = 1. f(x) exp(-jπ ux) dx Given F(u), we recover f(x) using the inverse Fourier transform: f(x) = F(u) exp(jπ ux) du (1) and () cons<tute a Fourier transform pair (1) () 6

7 Two- Dim Fourier Transform and Inverse In two dimensions, we have: F(u, v) = - f(x, y) exp(-jπ (ux + vy)) dxdy (3) f(x, y) = - F(u, v) exp(jπ (ux + vy)) dudv (4) 7

8 Discrete Fourier Transform In one dimension, 1 F(u) = M M 1 x= 0 f(x) exp(-jπ ux/m) for u = 0,1,,,M -1. (5) f(x) = M 1 x= 0 F(u) exp(jπux/m) for x = 0,1,,,M -1. Note that the loca<on of 1/M does not ma\er, so long as the product of the two mul<pliers is 1/M Also in the discrete case, the Fourier transform and its inverse always exist (6) 8

9 Discrete Fourier Transform Consider Euler s formula: exp j Subs<tu<ng this expression into (5), and no<ng cos(- θ)=cos(θ), we obtain 1 F(u) = M M 1 x= 0 jsin( θ) Each term of F depends on all values of f(x), and values of f(x) are mul<plied by sines and cosines of different frequencies. The domain over which values of F(u) range is called the frequency domain, as u determines the frequency of the components of the transform. θ = cos( θi) + f(x)[cosπux/m - jsinπux/m], (7) for u = 0,1,,,M -1. (8) 9

10 - D Discrete Fourier Transform Digital images are - D discrete func<ons: F(u, v) = for u 1 MN M 1 N-1 x= 0 y= 0 f(x, y) exp(-jπ (ux/m + vy/n)), = 0,1,,,M -1and v = 0,1,,, N -1. (9) f(x, y) = for x 1 MN M 1 N-1 u= 0 v= 0 F(u, v) exp(jπ (ux/m + vy/n)), = 0,1,,,M -1and y = 0,1,,, N -1. (10) 10

11 Frequency Domain Filtering Frequency is directly related to rate of change, so frequencies in the Fourier transform may be related to pa\erns of intensity varia<ons in the image. Slowest varying frequency at u = v = 0 corresponds to average gray level of the image. Low frequencies correspond to slowly varying components in the image- for example, large areas of similar gray levels. Higher frequencies correspond to faster gray level changes- such as edges, noise etc. 11

12 Procedure for Filtering in the Frequency Domain 1. Mul<ply the input image by (- 1) x+y to centre the transform. Compute the DFT F(u,v) of the resul<ng image 3. Mul<ply F(u,v) by a filter H(u,v) 4. Compute the inverse DFT transform g*(x,y) 5. Obtain the real part g(x,y) of 4 6. Mul<ply the result by (- 1) x+y 1

13 Example: Notch Filter We wish to force the average value of an image to zero. We can achieve this by sedng F(0, 0) =0, and then taking its inverse transform. So choose the filter func<on as: H (u, v) = 0 if (u, v) = (M/, N/) H (u, v) = 1otherwise. Called the notch filter- constant func<on with a hole (notch) at the origin. A filter that a\enuates high frequencies while allowing low frequencies to pass through is called a lowpass filter. A filter that a\enuates low frequencies while allowing high frequencies to pass through is called a highpass filter. 13

14 Convolu<on Theorem: correspondence between spa<al and frequency filtering Let F(u, v) and H( u, v) be the Fourier transforms of f(x, y) and h(x,y). Let * be spa<al convolu<on, and mul<plica<on be element- by- element product. Then f(x, y) * h(x, y) and F(u, v) H(u, v) cons<tute a Fourier transform pair, i.e. spa<al convolu<on (LHS) can be obtained by taking the inverse transform of RHS, and conversely, the RHS can be obtained as the forward Fourier transform of LHS. Analogously, convolu<on in the frequency domain reduces to mul<plica<on in the spa<al domain, and vice versa. Using this theorem, we can also show that filters in the spa<al and frequency domains cons<tute a Fourier transform pair. 14

15 Exploi<ng the correspondence If filters in the spa<al and frequency domains are of the same size, then filtering is more efficient computa<onally in frequency domain. Spa<al filters tend to be smaller in size. Filtering is also more intui<ve in frequency domain- so design it there. Then, take the inverse transform, and use the resul<ng filter as a guide to design smaller filters in the spa<al domain. 15

16 Example of Smoothing an Image In spa<al domain, we just convolve the image with a Gaussian kernel to smooth it In frequency domain, we can mul<ply the image by a filter achieve the same effect 16

17 Example of Smoothing an Image 1. Mul<ply the input image by (- 1) x+y to center the transform. Compute the DFT F(u,v) of the resul<ng image 3. Mul<ply F(u,v) by a filter G(u,v) 4. Computer the inverse DFT transform h*(x,y) 5. Obtain the real part h(x,y) of 4 6. Mul<ply the result by (- 1) x+y 17

18 Example of Smoothing an Image 1. Mul<ply the input image by (- 1) x+y to center the transform. Compute the DFT F(u,v) of the resul<ng image 3. Mul<ply F(u,v) by a filter G(u,v) 4. Computer the inverse DFT transform h*(x,y) 5. Obtain the real part h(x,y) of 4 6. Mul<ply the result by (- 1) x+y 18

19 Example of Smoothing an Image 1. Mul<ply the input image by (- 1) x+y to center the transform. Compute the DFT F(u,v) of the resul<ng image 3. Mul<ply F(u,v) by a filter G(u,v) 4. Computer the inverse DFT transform h*(x,y) 5. Obtain the real part h(x,y) of 4 Х 6. Mul<ply the result by (- 1) x+y 19

20 Example of Smoothing an Image 1. Mul<ply the input image by (- 1) x+y to center the transform. Compute the DFT F(u,v) of the resul<ng image 3. Mul<ply F(u,v) by a filter G(u,v) 4. Compute the inverse DFT transform h*(x,y) 5. Obtain the real part h(x,y) of 4 6. Mul<ply the result by (- 1) x+y 0

21 Example of Smoothing an Image 1. Mul<ply the input image by (- 1) x+y to center the transform. Compute the DFT F(u,v) of the resul<ng image 3. Mul<ply F(u,v) by a filter G(u,v) 4. Computer the inverse DFT transform h*(x,y) 5. Obtain the real part h(x,y) of 4 6. Mul<ply the result by (- 1) x+y f (x, y)* g(x, y) F(u, v)g(u, v) 1

22 Gaussian Filter Gaussian filters are important because their shapes are easy to specify, and both the forward and inverse Fourier transforms of a Gaussian func<on are real Gaussian func<ons. Let H(u) be a one dimensional Gaussian filter specified by: H(u) = A exp where σ is the standard devia<on of the Gaussian curve. The corresponding filter in the spa<al domain is h(x) = This is usually a lowpass filter. u σ πσ A exp π σ x

23 DoG Filter Difference of Gaussians may be used to construct highpass filters: H(u) = A exp with A B and δ 1 > δ. The corresponding filter in the spa<al domain is u σ 1 B exp u σ h(x) = πσ 1 π σ x π σ x 1 A exp πσ B exp 3

24 Image Pyramids An image pyramid is a collec<on of decreasing resolu<on images arranged in the shape of a pyramid. 4

25 Image Pyramids System block diagram for crea<ng image pyramids 1. Compute a reduced- resolu<on approxima<on of the input image by filtering and downsampling (mean, Gaussian, subsampling). Upsample the output of step 1 and filter the result (possibly with interpola<on) 3. Compute the difference between the predic<on of step and the input to step 1 Repea<ng, produce approxima<on and predic<on residual pyramids 5

26 Image Pyramids Two image pyramids and their sta<s<cs Upsample and filtering the lowest resolu<on approxima<on image Add the 1- level higher Laplacian s predic<on residual 6

27 References and Acknowledgement Gonzalez and Woods, 00, Chapter , 7.1 Szeliski Chapter Some material, including images and tables, were drawn from the textbook, Digital Image Processing by Gonzalez and Woods, and P.C. Rossin s presenta<on. 7