Digital Image Processing Image Filtering - · PDF fileDigital Image Processing Image Filtering ... rFom R.C. Gonzalez, R.E. Woods, Digital image ropcessing. 19. ... Digital Image Processing

Table of Content

Digital Image ProcessingImage Filtering

O. Le [email protected]

Univ. of Rennes 1http://www.irisa.fr/temics/staff/lemeur/

January 2, 2011

1

mailto:[email protected]://www.irisa.fr/temics/staff/lemeur/

Table of Content

1 Introduction

2 Point-to-point transformation

3 Linear ltering (neighborhoodoperator)

4 Non Linear ltering

5 Conclusion

2

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Introduction

1 Introduction

2 Point-to-point transformation

3 Linear ltering (neighborhoodoperator)

4 Non Linear ltering

5 Conclusion

3

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Introduction: image transformation

There exist 3 types of transformation:

Point to point transformation:The output value at a specic coordinateis dependent only on one input value butnot necessarily at the same coordinate;

Local to point transformation:The output value at a specic coordinateis dependent on the input values in theneighborhood of that same coordinate;

Global to point transformation:The output value at a specic coordinateis dependent on all the values in the inputimage.

Note that the complexity increases with the size of the considered neighborhood...

4

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Point-to-point transformation

1 Introduction

2 Point-to-point transformationSpatial coordinates-basedtransformationsPixel values-based transformations

3 Linear ltering (neighborhoodoperator)

4 Non Linear ltering

5 Conclusion

5

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Spatial coordinates-based transformations

Remark:

This section is composed of several pictures extracted fromhttp://eeweb.poly.edu/~onur/lectures/lectures.html.

Let im[x, y ] be an input image of size N N.

A spatial coordinates-based transformation, also called warping, aims at providing animage IM[k, l ] from the input image im[x, y ]:

IM[k, l ] = im[x(k, l), y(k, l)]

x(k, l) and y(k, l) are the transformations or the pixel warping functions. These functionsjust modify the spatial coordinates of a pixel not its value;

Special cases to take into consideration:

the new coordinates [x(k, l), y(k, l)] is out of the image IM. In this case,IM[k, l ] = 0;the new coordinates must be integers (Rounding operation, nearest integers...).

6

http://eeweb.poly.edu/~onur/lectures/lectures.html

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Transpose

The transpose tansformation is given by

x(k, l) = l

y(k, l) = k

7

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Vertical Flip

The vertical ip tansformation is given by

x(k, l) = N ky(k, l) = l

The horizontal ip tansformation is given by

x(k, l) = k

y(k, l) = N l

8

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Translation

The translation tansformation is given by

x(k, l) = k + Tk

y(k, l) = l + Tl

where Tk and Tl are the transalation values for the x-axis and the y-axis respectively.

In the example below, we have (Tl = 50)

IM[k, l ] = im[x(k, l), y(k, l)]

IM[k, l ] = im[k, l + Tl ]

Dierent transformations can be obtained depending on Tk and Tl values.

9

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Rotation

The rotation tansformation is given by

x(k, l) = (k x0)cos (l y0)sin + x0y(k, l) = (k x0)sin + (l y0)cos + y0

where [x0, y0] is the spatial coordinates of the center of the rotation and the angle.

Extracted from http://eeweb.poly.edu/ onur/lectures/lectures.html.

10

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Wave

x(k, l) = k

y(k, l) = l + sin( l)x(k, l) = k + sin( k)y(k, l) = l

and can be used to strengthen the eect.

11

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Warp and swirl

x(k, l) = k

y(k, l) =sign(l y0)

y0 (l y0)2 + y0

The swirl eect is a rotation but theangle of the rotation varies with thepixel distance from the center of theimage [x0, y0]:

d =

(k x0)2 + (l y0)2

=

512r

If r 0, is small...

12

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Glass eect

A glass eect is obtained by adding a small and random displacement to each pixel:

x(k, l) = k + (RAND(1, 1) 1/2) 10y(k, l) = l + (RAND(1, 1) 1/2) 10

Extracted from http://eeweb.poly.edu/ onur/lectures/lectures.html.

13

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Summary

All 2D linear transformations: [x

y

]=

[a b

c d

] [k

l

]Scale, rotation, mirror...

Properties:

Origin maps to origin;

Lines map to lines;

Ratios are preserved...

Ane transformations (linear transf. + translation):xyw

=a b cd e f0 0 1

klw

Properties:

Origin does not necessarily map to origin;

Lines map to lines;

Ratios are preserved...

14

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Summary

Ane transformations (linear transf. + translation):

Translation: xy1

=1 0 tx0 1 ty0 0 1

kl1

Scale: xy

1

=sx 0 00 sy 00 0 1

kl1

2D in-plane rotation: xy

1

=cos sin 0sin cos 0

0 0 1

kl1

15

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Pixel values-based transformations

Let im[x , y ] be an input image of size N N.

A pixel values-based transformation aims at providing an image IM[k, l ] from theinput image im[x , y ]:

IM[k, l ] = f (im[k, l ])

Noticed that the spatial coordinates of pixel are not modied. The function f isused to modify the pixel values.

The simplest one is the identity function: f (p) = p:

IM[k, l ] = f (im[k, l ])

IM[k, l ] = im[k, l ]

16

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Histogram

An image histogram is a graphical representation of the tonal distribution in a digitalimage. It plots the number of pixels for each tonal value.

Histogram gives information about the global distribution of an image.

Histogram plots the number of pixels in the image (vertical axis) with a particularbrightness value (horizontal axis).

(a) Original (b) Histogram (intensity)

17

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Histogram

(a) Original (b) Histogram (intensity)

(c) Original (d) Histogram (intensity)

18

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Histogram

High dynamic range in the last caseproviding the best quality.

From R.C. Gonzalez, R.E. Woods, Digital image processing.

19

IntroductionPoint-to-point transformation

Linear ltering (neighborhood operator)Non Linear ltering

Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Histogram equalization

The goal is to increase the global contrast of images, especially when the usable dataof the image is repr