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- 1. Frequency Domain : 1 Frequency DomainFrequency Domain
- 2. Frequency Domain : 2 Fourier Series and TransformFourier Series and Transform
- 3. Frequency Domain : 3 Fourier Transform of ContinuousFourier Transform of Continuous VariableVariable 2 ( ) ( ) j t F f t e dtπµ µ ∞ − −∞ = ∫ { }1 2 ( ) ( ) ( ) j t F f t F e dπµ µ µ µ ∞ − −∞ ℑ = = ∫ 2 ( ) ( ) j t f t e dtπµ µ ∞ − −∞ ℑ = ∫ ( ) ( )[cos(2 ) sin(2 )]F f t t j t dtµ πµ πµ ∞ −∞ = −∫
- 4. Frequency Domain : 4 Discrete Fourier Transform (DFT)Discrete Fourier Transform (DFT) 1 2 / 0 ( ) ( ) 1,2,3,..., 1 M j ux M x F u f x e u Mπ − − = = = −∑ 1 2 / 0 1 ( ) ( ) 1,2,3,..., 1 M j ux M u f t F u e u M M π − = = = −∑
- 5. Frequency Domain : 5 Fourier Transform: VisualizationFourier Transform: Visualization
- 6. Frequency Domain : 6 2-D Discrete Fourier Transform2-D Discrete Fourier Transform 1 1 2 ( / / ) 0 0 ( , ) ( , ) M N j ux M vy N x y F u v f x y e π − − − + = = = ∑ ∑ 1 1 2 ( / / ) 0 0 1 ( , ) ( , ) M N j ux M vy N u v f x y F u v e MN π − − + = = = ∑ ∑
- 7. Frequency Domain : 7 2-D Fourier Transform: Visualization2-D Fourier Transform: Visualization
- 8. Frequency Domain : 8 2-D Fourier Transform:2-D Fourier Transform: ImplementationImplementation
- 9. Frequency Domain : 9 2-D Fourier Transform:2-D Fourier Transform: ImplementationImplementation
- 10. Frequency Domain : 10 Basic Steps of Filtering in FrequencyBasic Steps of Filtering in Frequency DomainDomain 1. Multiply input f(x,y) by (-1)x+y to center transform 2. Compute DFT of image, F(u,v) 3. Multiply F(u,v) by filter function H(u,v) to get G(u,v) 4. Compute inverse DFT of G(u,v) to get g(x,y) 5. Multiply g(x,y) by (-1)x+y to get filtered image
- 11. Frequency Domain : 11 Image Characteristics in FrequencyImage Characteristics in Frequency DomainDomain Low frequencies responsible for general appearance of image over smooth areas High frequencies responsible for detail (e.g., edges and noise) Intuitively, modifying different frequency coefficients affects different characteristics of an image
- 12. Frequency Domain : 12 Example: DC component removalExample: DC component removal Suppose we remove the DC component from the Fourier transform of an image
- 13. Frequency Domain : 13 Why does it look like that?Why does it look like that? DC component characterizes the mean of the image intensities
- 14. Frequency Domain : 14 Examples of Frequency DomainExamples of Frequency Domain FilteringFiltering
- 15. Frequency Domain : 15 Correspondence between Filtering inCorrespondence between Filtering in Spatial and Frequency DomainsSpatial and Frequency Domains Basic spatial filtering is essentially 2D discrete convolution between an image f and filter function h Convolution in spatial domain becomes multiplication in frequency domain ( , ) ( , ) ( , )g x y f x y h x y= ∗ ( , ) ( , ) ( , )G u v F v v H u v=
- 16. Frequency Domain : 16 Correspondence between Filtering inCorrespondence between Filtering in Spatial and Frequency DomainsSpatial and Frequency Domains What does this mean? Given a filter in frequency domain Corresponding filter in spatial domain can be obtained by taking inverse Fourier transform Given a filter in spatial domain, Corresponding filter in frequency domain can be obtained by taking Fourier transform
- 17. Frequency Domain : 17 Correspondence between Filtering inCorrespondence between Filtering in Spatial and Frequency DomainsSpatial and Frequency Domains

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