2D Fourier Theory for Image Analysis

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# 2D Fourier Theory for Image Analysis - PowerPoint PPT Presentation

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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## 2D Fourier Theory for Image Analysis

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### 2D Fourier Theory for Image Analysis

Mani Thomas

CISC 489/689

• 2D image basis
• Fourier basis
• Scale-space representation
• Gaussian pyramid
• Laplacian pyramid
• Image mosaicing
• Gabor filters
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:
• 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
• 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
• 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

v

Fourier Basis

Real

(cos) part

(u, v)

(1, 0)

(1, 1)

(0, 5)

Imaginary

(sin) part

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
• 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
• 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
• 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
• 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

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
• If L0, L1 LN is the sequence of laplacians

Li = Gi – EXPAND[Gi+1], 0<i<N

LN = GN

• 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

Seamless joining of images to get a larger view

Laplacian

level

4

Laplacian

level

2

Laplacian

level

0

left pyramid

right pyramid

blended pyramid

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

Jw

refine

warp

+

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

Odd

(sin)

Even

(cos)

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

£

=

Frequency

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

Input

image

Filter

bank

from Forsyth & Ponce

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
Conclusions
• 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