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University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Image processing
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Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

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Page 1: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Image processing

Page 2: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 2

Reading

Jain, Kasturi, Schunck, Machine Vision.McGraw-Hill, 1995. Sections 4.2-4.4,4.5(intro), 4.5.5, 4.5.6, 5.1-5.4.

Page 3: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 3

Image processingAn image processing operation typically defines a newimage g in terms of an existing image f.The simplest operations are those that transform each pixelin isolation. These pixel-to-pixel operations can bewritten:

Examples: threshold, RGB grayscaleNote: a typical choice for mapping to grayscale is to applythe YIQ television matrix and keep the Y.!

g(x,y) = t( f (x,y))

!

Y

I

Q

"

#

$ $ $

%

&

' ' '

=

0.299 0.587 0.114

0.596 (0.275 (0.321

0.212 (0.523 0.311

"

#

$ $ $

%

&

' ' '

R

G

B

"

#

$ $ $

%

&

' ' '

Page 4: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 4

Pixel movement

Some operations preserve intensities, but movepixels around in the image

Examples: many amusing warps of images

[Show image sequence.]

% %( , ) ( ( , ), ( , ))g x y f x x y y x y=

Page 5: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 5

Noise Image processing is also useful for noise reduction and edge

enhancement. We will focus on these applications for the remainder ofthe lecture…

Common types of noise: Salt and pepper noise:

contains randomoccurrences of black andwhite pixels

Impulse noise: containsrandom occurrences ofwhite pixels

Gaussian noise:variations in intensity drawnfrom a Gaussian normaldistribution

Page 6: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 6

Ideal noise reduction

Page 7: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 7

Ideal noise reduction

Page 8: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 8

Practical noise reduction

How can we “smooth” away noise in a single image?

Is there a more abstract way to represent this sort ofoperation? Of course there is!

Page 9: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 9

Discrete convolution

For a digital signal, we define discrete convolution as:

where

!

g[i] = f [i]" h[i]

= f [ # i ]h[i $ # i ]# i

%

= f [ # i ]) h [ # i $ i]

# i

%

!

) h [i] = h["i]

Page 10: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 10

Discrete convolution in 2D

Similarly, discrete convolution in 2D becomes:

where

!

g[i, j] = f [i, j]" h[i, j]

= f [ # i , # j ]h[i $ # i , j $ # j ]# j

%# i

%

= f [ # i , # j ]) h [ # i $ i, # j $ j]

# j

%# i

%

!

) h [i, j] = h["i," j]

Page 11: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 11

Convolution representation

Since f and h are defined over finite regions, we can writethem out in two-dimensional arrays:

Note: This is not matrix multiplication!Q: What happens at the edges?

Page 12: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 12

Mean filters

How can we represent our noise-reducingaveraging filter as a convolution diagram(know as a mean filter)?

Page 13: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 13

Effect of mean filters

Page 14: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 14

Gaussian filters

Gaussian filters weigh pixels based on their distance fromthe center of the convolution filter. In particular:

This does a decent job of blurring noise while preservingfeatures of the image.What parameter controls the width of the Gaussian?What happens to the image as the Gaussian filter kernelgets wider?What is the constant C? What should we set it to?

2 2 2( ) /(2 )

[ , ]i j

eh i j

C

!" +

=

Page 15: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 15

Effect of Gaussian filters

Page 16: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 16

Median filters

A median filter operates over an mxmregion by selecting the median intensity inthe region.What advantage does a median filter haveover a mean filter?Is a median filter a kind of convolution?

Page 17: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 17

Effect of median filters

Page 18: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 18

Comparison: Gaussian noise

Page 19: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 19

Comparison: salt and pepper noise

Page 20: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 20

Edge detection

One of the most important uses of imageprocessing is edge detection:

Really easy for humansReally difficult for computers

Fundamental in computer visionImportant in many graphics applications

Page 21: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 21

What is an edge?

Q: How might you detect an edge in 1D?

Page 22: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 22

Gradients

The gradient is the 2D equivalent of the derivative:

Properties of the gradientIt’s a vectorPoints in the direction of maximum increase of fMagnitude is rate of increase

How can we approximate the gradient in a discrete image?

( , ) ,f f

f x yx y

! "# #$ = % &' # # (

Page 23: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 23

Less than ideal edges

Page 24: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 24

Steps in edge detection

Edge detection algorithms typically proceedin three or four steps:

Filtering: cut down on noiseEnhancement: amplify the difference betweenedges and non-edgesDetection: use a threshold operationLocalization (optional): estimate geometry ofedges beyond pixels

Page 25: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 25

Edge enhancementA popular gradient magnitude computation is the Sobeloperator:

We can then compute the magnitude of the vector (sx, sy).

1 0 1

2 0 2

1 0 1

1 2 1

0 0 0

1 2 1

x

y

s

s

!" #$ %

= !$ %$ %!& '

" #$ %

= $ %$ %! ! !& '

Page 26: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 26

Results of Sobel edge detection

Page 27: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 27

Second derivative operators

The Sobel operator can produce thick edges. Ideally, we’re looking forinfinitely thin boundaries.

An alternative approach is to look for local extrema in the firstderivative: places where the change in the gradient is highest.

Q: A peak in the first derivative corresponds to what in the secondderivative?

Q: How might we write this as a convolution filter?

Page 28: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 28

Localization with the Laplacian

An equivalent measure of the second derivative in 2D isthe Laplacian:

Using the same arguments we used to compute thegradient filters, we can derive a Laplacian filter to be:

Zero crossings of this filter correspond to positions ofmaximum gradient. These zero crossings can be used tolocalize edges.

2 22

2 2( , )

f ff x y

x y

! !" = +

! !

2

0 1 0

1 4 1

0 1 0

! "# $% = &# $# $' (

Page 29: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 29

Localization with the Laplacian

Original Smoothed

Laplacian (+128)

Page 30: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 30

Marching squares

We can convert these signed values intoedge contours using a “marching squares”technique:

Page 31: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 31

Sharpening with the Laplacian

Original Laplacian (+128)

Original + Laplacian Original - LaplacianWhy does the sign make a difference?How can you write each filter that makes each bottom image?

Page 32: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 32

Spectral impact of sharpeningWe can look at the impact of sharpening on the Fourier spectrum:

2

0 1 0

1 5 1

0 1 0

!

"# $% &" ' = " "% &% &"( )

Spatial domain Frequency domain

Page 33: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 33

Summary

What you should take away from this lecture:The meanings of all the boldfaced terms.How noise reduction is doneHow discrete convolution filtering worksThe effect of mean, Gaussian, and median filtersWhat an image gradient is and how it can be computedHow edge detection is doneWhat the Laplacian image is and how it is used in either edgedetection or image sharpening

Page 34: Image processing - cs.utexas.edu...Image processing An image processing operation typically defines a new image g in terms of an existing image f. The simplest operations are those

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 34

Next time: Affine Transformations

Topic:How do we represent the rotations, translations, etc.

needed to build a complex scene from simpler objects?Read:

• Watt, Section 1.1.

Optional:• Foley, et al, Chapter 5.1-5.5.• David F. Rogers and J. Alan Adams,

Mathematical Elements for Computer Graphics, 2ndEd., McGraw-Hill, New York, 1990, Chapter 2.