# Introduction to Computer Vision - PowerPoint PPT Presentation

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Lecture 05 Roger S. Gaborski. Introduction to Computer Vision. Quiz Review In Class Exercise Correlation – Convolution Filtering. Example. Histogram PDF  CDF Equalized Image Filtering using Correlation continued. Histogram Equalization.

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Introduction to Computer Vision

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Lecture 05

Roger S. Gaborski

### Introduction to Computer Vision

Roger S. Gaborski

• Quiz Review

• In Class Exercise

• Correlation – Convolution

• Filtering

Roger S. Gaborski

### Example

• Histogram PDF  CDF Equalized Image

• Filtering using Correlation continued

### Histogram Equalization

• Consider an image with the following gray level values:

• Construct the pdf

• Construct the cdf

• Equalize the image using the cdf (not histeq)

### Histogram Equalization

• Consider an image with the following gray level values:

• Construct the pdf

1/9 2/9 3/9 4/9 5/9

.1 .2 .3 .4 .5

pdf

cdf

1/9 2/9 3/9 4/9 5/9

.1 .2 .3 .4 .5

### Histogram Equalization

Look Up Table

1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 1

cdf

probability

.1 .2 .3 .4 .5

Gray level value

Gray level value

### Histogram Equalization

• Consider an image with the following gray level values:

• Construct the pdf

• Construct the cdf

• Equalize the image using the cdf (not histeq)

### Spatial Filtering

• Neighborhood processing

• Define center point (x, y)

• Perform operations involving only pixels in the neighborhood

• Result of operation is response to process at that point

• Moving the pixel results in a new neighborhood

• Repeat process for every point in the image

Roger S. Gaborski

### Linear and Nonlinear Spatial Filtering

• Linear operation

• Multiply each pixel in the neighborhood by the corresponding coefficient and sum the results to get the response for each point (x, y)

• If neighborhood is m x n , then mn coefficients are required

• Coefficients are arranged in a matrix, called

• filter / filter mask / kernel / template

• Mask sizes are typically odd numbers (3x3, 5x5, etc.)

Roger S. Gaborski

Image origin

y

Kernel coefficients

x

Roger S. Gaborski

### Correlation and Convolution

• Correlation

• Place mask w on the image array f as previously described

• Convolution

• First rotate mask w by 180 degrees

• Place rotated mask on image as described previously

• Convolution = 180 degree rotation + correlation

Roger S. Gaborski

### Example: 1D Correlation

• Assume w and f are one dimensional

• Origin of f is its left most point

• Place w so that its right most point coincides with the origin of f

• Pad f with 0s so that there are corresponding f points for each w point (also pad end with 0s)

• Multiply corresponding points and sum

• In this case (example on next page) result is 0

• Move w to the right one value, repeat process

• Continue process for whole length of f

Roger S. Gaborski

### Reminder

• ‘full’ is the result we obtain from the operations on the previous slide. If instead of aligning the left most element of f with the right most element of w we aligned the center element of w with the left most value of f we would obtain the ‘same’ result, same indicating the result is the same length of the original w

Roger S. Gaborski

Chapter 3

www.prenhall.com/gonzalezwoodseddins

Roger S. Gaborski

‘Full’

correlation

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‘Same’

correlation

etc.

Roger S. Gaborski

### Example - Convolution

• Convolution is the same procedure, but the filter is first rotated 180 degrees.

• Convolution = 180 degree rotation + correlation

• If the filter is symmetric, correlation and convolution results are the same

Roger S. Gaborski

Chapter 3

www.prenhall.com/gonzalezwoodseddins

This can be simply extend to images

Roger S. Gaborski

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Roger S. Gaborski

### Linear Filtering in MATLAB

g = imfilter(f, w, filtering mode, boundary, size)

• filters the imput image f with the filter mask w.

• f is input image. It can be of any class (logical/numeric) and dimension.

• g is output image

• filter mode:

- 'corr' : correlation, and default mode

- 'conv' : convolution

Roger S. Gaborski

### Parameters

• g = imfilter(f, w, filtering mode, boundary, size)

Boundary options

- X pad boundary with value X. Default X = 0.

Size options

- 'same' g is the same size of f (default mode)

- 'full' g is full filtered by w, so size of g is increased

Roger S. Gaborski

### MATLAB function for filtering: imfilter

• g = imfilter(f, w, ‘replicate’)

• Correlation is the default filtering mode.

• If filters are pre-rotated 180 degrees, can simply use default(corr) for convolution

• If filter is symmetric, doesn’t matter

Roger S. Gaborski

Chapter 3

www.prenhall.com/gonzalezwoodseddins

Roger S. Gaborski

### Example:Smoothing

• w = ones(31); (31x31 filter)

• % Normally the coefficients (w) are scaled to sum to one

• % In this example only coefficients are not scaled by 312

• % Convolution should result in a blurred result

• gd = imfilter(f, w);

• % Default mode: correlation filtering

• imshow(gd, [ ]);

Roger S. Gaborski

Chapter 3

www.prenhall.com/gonzalezwoodseddins

‘symmetric’ ‘circular’ ‘replicate’, uint8

Roger S. Gaborski