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Convolution and Edge Detection. 15-463: Computational Photography Alexei Efros, CMU, Fall 2005. Some slides from Steve Seitz. Fourier spectrum . Fun and games with spectra. Gaussian filtering. A Gaussian kernel gives less weight to pixels further from the center of the window

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Convolution and edge detection
Convolution and Edge Detection

15-463: Computational Photography

Alexei Efros, CMU, Fall 2005

Some slides from Steve Seitz




Gaussian filtering
Gaussian filtering

  • A Gaussian kernel gives less weight to pixels further from the center of the window

  • This kernel is an approximation of a Gaussian function:



Convolution
Convolution

  • Remember cross-correlation:

  • A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image:

  • It is written:

  • Suppose H is a Gaussian or mean kernel. How does convolution differ from cross-correlation?


The convolution theorem

The greatest thing since sliced (banana) bread!

The Fourier transform of the convolution of two functions is the product of their Fourier transforms

The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms

Convolution in spatial domain is equivalent to multiplication in frequency domain!

The Convolution Theorem



2d convolution theorem example
2D convolution theorem example

|F(sx,sy)|

f(x,y)

*

h(x,y)

|H(sx,sy)|

g(x,y)

|G(sx,sy)|



Image gradient

Image gradient

  • The gradient of an image:

  • The gradient points in the direction of most rapid change in intensity


Effects of noise
Effects of noise

  • Consider a single row or column of the image

    • Plotting intensity as a function of position gives a signal

How to compute a derivative?

  • Where is the edge?


Solution smooth first

Solution: smooth first

  • Where is the edge?


Derivative theorem of convolution
Derivative theorem of convolution

  • This saves us one operation:


Laplacian of gaussian
Laplacian of Gaussian

  • Consider

Laplacian of Gaussian

operator

  • Where is the edge?

  • Zero-crossings of bottom graph


2d edge detection filters

Laplacian of Gaussian

  • is the Laplacian operator:

2D edge detection filters

Gaussian

derivative of Gaussian


Matlab demo
MATLAB demo

g = fspecial('gaussian',15,2);

imagesc(g)

surfl(g)

gclown = conv2(clown,g,'same');

imagesc(conv2(clown,[-1 1],'same'));

imagesc(conv2(gclown,[-1 1],'same'));

dx = conv2(g,[-1 1],'same');

imagesc(conv2(clown,dx,'same');

lg = fspecial('log',15,2);

lclown = conv2(clown,lg,'same');

imagesc(lclown)

imagesc(clown + .2*lclown)



What does blurring take away1
What does blurring take away?

smoothed (5x5 Gaussian)


Edge detection by subtraction
Edge detection by subtraction

Why does

this work?

smoothed – original


Gaussian image filter
Gaussian - image filter

FFT

Gaussian

delta function

Laplacian of Gaussian



Unsharp masking

-

=

+ a

=

Unsharp Masking


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