Lecture 4 Image Enhancement in Frequency Domain Machine Vision R. Ebrahimpour, 2008

Objective Basic understanding of the frequency domain, Fourier series, and Fourier transform. Image enhancement in the frequency domain.

Summary of FT, DTFT/DSFT, DFT and FFT Fourier Transform (FT): (continuous) (continuous) Discrete Time/Space Fourier Transform (DTFT/DSFT): (discrete) (continuous, periodic)

Summary of FT, DTFT/DSFT, DFT and FFT Discrete Fourier Transform (DFT): (discrete, finite) (discrete, finite) Fast Fourier Transform (FFT): Fast algorithm for computing DFT

Two-Dimensional Discrete Fourier Transform (2D-DFT)

2D DFT and Inverse DFT (IDFT) f(x, y) F(u, v) often used short notation: M, N: image size x, y: image pixel position u, v: spatial frequency

The Meaning of DFT and Spatial Frequencies Important Concept Any signal can be represented as a linear combination of a set of basic components Fourier components: sinusoidal patterns Fourier coefficients: weighting factors assigned to the Fourier components Spatial frequency: The frequency of Fourier component Not to confused with electromagnetic frequencies (e.g., the frequencies associated with light colors)

Real Part, Imaginary Part, Magnitude, Phase, Spectrum Magnitude-phase representation: Magnitude (spectrum): Phase (spectrum): Power Spectrum:

2D DFT Properties Mean of image/ DC component: Highest frequency component: “Half-shifted” Image: Conjugate Symmetry: Magnitude Symmetry:

2D DFT Properties Spatial domain differentiation: Frequency domain differentiation: Distribution law: Laplacian: Spatial domain Periodicity: Frequency domain periodicity:

Computation of 2D-DFT To compute the 1D-DFT of a 1D signal x (as a vector): To compute the inverse 1D-DFT: To compute the 2D-DFT of an image X (as a matrix): To compute the inverse 2D-DFT:

Computation of 2D-DFT Fourier transform matrices: remember W N - 1 · (N-1) 2 2(N-1) W N - 1 · (N-1) 2 2(N-1) relationship: relationship: In particular, for N = 4: In particular, for N = 4:

Computation of 2D-DFT: Example A 4x4 image Compute its 2D-DFT: MATLAB function: fft2

Computation of 2D-DFT: Example Real part: Imaginary part: Magnitude: Phase:

Computation of 2D-DFT: Example MATLAB function: ifft2

Spectrum of images

Centered Representation MATLAB function: fftshift Example:

Log-Magnitude Visualization 2D-DFT centered

2D-DFT  centered  log intensity transformation Apply to Images 2D-DFT  centered  log intensity transformation

2D-DFT (Frequency) Domain Filtering

Convolution Theorem f (x,y) h(x,y) g(x,y) G(u,v) = F(u,v) H(u,v) input image impulse response (filter) output image DFT IDFT DFT IDFT DFT IDFT G(u,v) = F(u,v) H(u,v)

Frequency Domain Filtering Filter design: design H(u,v)

2D-DFT Domain Filter Design Ideal lowpass, bandpass and highpass

2D-DFT Domain Filter Design- Ideal LP Project # 5 Salt and pepper denoising with Median and ILP 2D-DFT Domain Filter Design- Ideal LP Ideal lowpass, bandpass and highpass Cutoff frequency

2D-DFT Domain Filter Design- Ideal LP Ideal lowpass filtering with cutoff frequencies set at radii values of 5, 15, 30, 80, and 230, respectively Blurring &Ringing Effect

2D-DFT Domain Filter Design- Ideal LP

2D-DFT Domain Filter Design- Ideal LP

2D-DFT Domain Filter Design- Ideal LP

Cutoff frequency

No Visible Ringing Effect Blurring & No Visible Ringing Effect

2D-DFT Domain Filter Design- Butterworth LP

2D-DFT Domain Filter Design- Butterworth LP

2D-DFT Domain Filter Design- Gaussian LP Gaussian lowpass

2D-DFT Domain Filter Design- Gaussian LP Effect of Gaussian lowpass filter

2D-DFT Domain Filter Design- Gaussian LP Effect of Gaussian lowpass filter Blurring & No Ringing Effect

2D-DFT Domain Filter Design- Gaussian LP

2D-DFT Domain Filter Design- Gaussian LP Effect of Gaussian lowpass filter

2D-DFT Domain Filter Design Gaussian lowpass filtering Gaussian highpass filtering

2D-DFT Domain Filter Design Choices of highpass filters Ideal Butterworth Gaussian

2D-DFT Domain Filter Design

2D-DFT Domain Filter Design- Ideal HP

2D-DFT Domain Filter Design- Butterworth HP

2D-DFT Domain Filter Design- Gaussian HP

2D-DFT Domain Filter Design Ideal Butterworth Gaussian

2D-DFT Domain Filter Design Gaussian filter with different width

2D-DFT Domain Filter Design- Laplacian HP

2D-DFT Domain Filter Design- Laplacian HP

2D-DFT Domain Filter Design- Laplacian HP

2D-DFT Domain Filter Design- Laplacian HP

2D-DFT Domain Filter Design Orientation selective filters

2D-DFT Domain Filter Design Narrowband Filtering by combining radial and orientation selection *

Homomorphic filtering

Homomorphic filtering

Homomorphic filtering C controls the sharpness of the slope

Homomorphic filtering