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2D Fourier Theory for Image Analysis. Mani Thomas CISC 489/689. Roadmap. 2D image basis Fourier basis Scale-space representation Gaussian pyramid Laplacian pyramid Image mosaicing Gabor filters. Different basis representation.

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Presentation Transcript
  • 2D image basis
  • Fourier basis
  • Scale-space representation
    • Gaussian pyramid
    • Laplacian pyramid
  • Image mosaicing
  • Gabor filters
different basis representation
Different basis representation
  • Recall our discussion of basis vectors for coordinate systems:
    • Describe point as linear combination of ortho-gonal basis vectors: x=a1v1+ ... +anvn
  • The standard basis for images is the set of unit vectors corresponding to each pixel. A toy example:
hadamard basis
Hadamard basis
  • The standard basis is not the only one we can use to describe an image
  • E.g., the Hadamard basis (basis images shown

here for 2 x 2 images, where black = +1,

white = -1)

    • For the previous example, we can express the image

with these new (normalized) basis vectors as:

    • Coefficients of sum = projection of I onto new basis (dot product)
    • These are the coordinates of the image in “Hadamard space”
    • We can also say that I has undergone a Hadamard transform H:
sinusoidal basis
Sinusoidal Basis
  • Binary-valued, rectangular wave pattern of Hadamard basis doesn’t capture real image gradients well
    • Idea: Use smoothly-varying sinusoidal patterns at different frequencies, angles for basis images
fourier basis
Fourier Basis
  • The Fourier basis uses the family of complex sinusoidal functions
2d dft
  • Forward 2D DFT
  • Inverse 2D DFT
    • (u, v) are the frequency coordinates while (x, y) are the spatial coordinates
    • M, N are the number of spatial pixels along the x, y coordinates
fourier basis8


Fourier Basis


(cos) part

(u, v)

(1, 0)

(1, 1)

(0, 5)


(sin) part

fourier transform in matlab
Fourier transform in Matlab
  • Discrete, 2-D Fourier & inverse Fourier transforms are computed by fft2 and ifft2, respectively
  • fftshift: Move origin (DC component) to image center for display
  • Example:

>> I = imread(‘test.png’); % Load grayscale image

>> F = fftshift(fft2(I)); % Shifted transform

>> imshow(log(abs(F)),[]); % Show log magnitude

>> imshow(angle(F),[]); % Show phase angle

phase and magnitude
Phase and Magnitude
  • Output of the Fourier transform is a complex number
    • Decompose the complex number as the magnitude and phase components
  • In Matlab: u = real(z), v = imag(z), r = abs(z), and theta = angle(z)
image pyramid representation
Image pyramid representation
  • Smoothing means removing high frequencies
    • Smoothing required to avoid aliasing
  • Fourier transform of a Gaussian is a Gaussian
    • Convolution is a multiplication Gaussian suppresses high frequencies
gaussian pyramid
Gaussian Pyramid
  • Downsampling: Cut width, height in half at each iteration:
  • Upsampling S"(I): Double size of image, interpolate missing pixels
  • Let the base (the finest resolution) of an n-level Gaussian pyramid be defined as P0=I. Then the ith level is reduced from the level below it by:

from Forsyth & Ponce

Gaussian pyramid

laplacian pyramid
Laplacian pyramid
  • The tip (the coarsest resolution) of an n-level Laplacian pyramid is the same as the Gaussian pyramid at that level: Ln(I) =Pn(I)
  • The ith level is expanded from the level above according to Li(I) =Pi(I) ¡S"(Pi+1(I))
  • Synthesizing the original image: Get I back by summing upsampled Laplacian pyramid levels
gaussian and laplacian
Gaussian and Laplacian

courtesy of Wolfram

  • Gaussian – Smoothing pyramid
    • Each level is a smoothed and decimated signal of the previous
  • Laplacian – Band pass filter of the images
    • Each level is the difference of a more smoothed and less smoothed image
summation property
Summation Property
  • If L0, L1 LN is the sequence of laplacians

Li = Gi – EXPAND[Gi+1], 0<i<N


  • The steps used to construct the Laplacian can be reversed to get the original
    • Expand Li and add it to Li-1 to Gi-1

G0 = i=0N Li

applications image mosaicing
Applications – Image Mosaicing

Seamless joining of images to get a larger view











left pyramid

right pyramid

blended pyramid

image mosaicing
Image mosaicing
  • Automatic mosaicing
    • Cross correlation to compute translation between images
  • Matlab demo – Burt and Adelson’s paper
    • http://www.cs.huji.ac.il/course/2003/impr/spline83.pdf





u=1.25 pixels

u=2.5 pixels

u=5 pixels

u=10 pixels

image J

image J

image I

image I

Pyramid of image J

Pyramid of image I

Application - Coarse-to-Fine Estimation

Slide from CS 223-B L9 by Richard Szeliski

gabor filters





Gabor filters
  • Gaussian windowed Fourier Transform
    • Make convolution kernels from product of Fourier basis images and Gaussians




texture representation filter responses
Texture Representation: Filter Responses
  • Choose a group of filters
    • Edge/Bar filters: Something like Gabor filters at different orientations, scales
    • Spot filters: Center-surround filters like a Gaussian/difference of Gaussians at multiple scales
  • Run filters over image to get a set of response images
    • Each contains specific texture information
example filter responses
Example: Filter Responses





from Forsyth & Ponce

texture similarity based on response statistics
Texture Similarity based on Response Statistics
  • Collect statistics of responses over an image or subimage
    • Mean of squared response
    • Mean and variance of squared response
  • Euclidean distance between vectors of response statistics for two images is measure of texture similarity
  • 2D Fourier Theory
  • Image pyramid representation
    • Gaussian pyramid
    • Laplacian pyramid
  • Applications of Image Pyramids
    • Image Mosaicing
      • Gaussian + Laplacian pyramids (Burt and Adelson)
  • Texture statistics
    • Gabor filters