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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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lecture 05 roger s gaborski
Lecture 05

Roger S. Gaborski

Introduction to Computer Vision

Roger S. Gaborski

slide2

Quiz Review

  • In Class Exercise
  • Correlation – Convolution
    • Filtering

Roger S. Gaborski

example
Example
  • Histogram PDF  CDF Equalized Image
  • Filtering using Correlation continued
histogram equalization
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 equalization1
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

histogram equalization2
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 equalization3
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
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 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

slide10

Image origin

y

Kernel coefficients

mask

x

Image region under mask

Roger S. Gaborski

correlation and convolution
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
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
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

slide14

Chapter 3

www.prenhall.com/gonzalezwoodseddins

Roger S. Gaborski

slide15

‘Full’

correlation

Roger S. Gaborski

slide18

‘Same’

correlation

etc.

Roger S. Gaborski

example convolution
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

slide20

Chapter 3

www.prenhall.com/gonzalezwoodseddins

This can be simply extend to images

Roger S. Gaborski

linear filtering in matlab
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
Parameters
  • g = imfilter(f, w, filtering mode, boundary, size)

Boundary options

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

- \'symmetric\' symmetric padding

- \'replicate\' replicate padding

- \'circular\' circular padding

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
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

slide29

Chapter 3

www.prenhall.com/gonzalezwoodseddins

Roger S. Gaborski

example smoothing
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

slide31

Chapter 3

www.prenhall.com/gonzalezwoodseddins

Input Default padding ‘replicate’

‘symmetric’ ‘circular’ ‘replicate’, uint8

Roger S. Gaborski

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