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EECE\CS 253 Image Processing

Richard Alan Peters II

Department of Electrical Engineering and

Computer Science

Fall Semester 2007

Lecture Notes

Lecture Notes: Rotating Images

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6/25/2012 26/25/2012 21999-2007 by Richard Alan Peters II

Geometric Remapping

Input Image

Warping Function

Interpolation Function

Output Image


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Image Rotation

Isize,,inin

BCR

2

in

2

inCRD

in

in1tan

C

RA

image size:

aspect angle:

length of diagonal:


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Image Rotation

Compute the dimensionsof the output image:number of rows.

if 0 < 90.

,round sinDout AR


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Image Rotation

Compute the dimensionsof the output image:number of columns.

AC cosDroundout

if 0 < 90.


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Image Rotation

The output dimension calculation depends on the value of as follows:

If 0 < 90:

,round sinDout AR

.round cosDout AC

If90 < 0: ,round sinDout AR

.round cosDout AC

If 90 < 180:

,round 90cosDout AR

.round 90sinDout AC

If180


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Image Rotation

.1,1,

out21

out21

out0out0

CRCR

0out

0out

C

R

Allocate an outputimage with dimensions(Rout,Cout), where

1

AR sinDroundout

AC cosDroundout

and center point

1Rout and Cout below are valid for 0 < 90. Otherwise see slide 7.


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Image Rotation

0in

0in

0out

0out1

:,,

C

R

Cc

Rr

c

rcr

f

f

cossin

sincos

1

Work backward. Forevery output loc. (r,c)select an input loc. (rf,cf)by rotating (r,c) aroundthe image center by -.

0out

0out

C

R


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Image Rotation

0in

0in

0out

0out1

:,,

C

R

Cc

Rr

c

rcr

f

f

After rotation by -,J(r,c) is in nearly thesame position asI(rf,cf).


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Image Rotation

0in

0in

0out

0out1

:,,

C

R

Cc

Rr

c

rcr

f

f



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Image Rotation

0in

0in

0out

0out1

:,,

C

R

Cc

Rr

c

rcr

f

f



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Image Rotation

.,I;,J ff crcr Interpolation:The output pixel value is(usually) a function of the

values on a neighborhood a set of pixels thatsurrounds (rf,cf).

Bilinear Interp. uses a 22neighborhood, bicubic

uses a 44.

Nearest neighbor is simply,J(r,c) = I(ri ,ci ) where(ri ,ci ) = round(rf,cf).


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Image Rotation with Interpolation

Original image: San Francisco financial district


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Image Rotation with Nearest Neighbor Interpolation

Bicubic Nearest Neighbor Nearest Neighbor Bilinear


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Image Rotation with Bilinear Interpolation

Nearest Neighbor Bilinear Bilinear Bicubic


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Image Rotation with Bicubic Interpolation

Bilinear Bicubic Bicubic Nearest Neighbor


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Example of Warping: Map Image to Sphere

0 out out1

min , .2

R C

out

out

0

2 2

out out

1

1

,

tan ,

sin .

r

c

r c

0 in in1

max , .2

d R C

20

in

in

sin

cos

d d

r d

c d

For each output pixel,

For the output image, For the input image,

For each output pixel,the input pixel loc is,


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Example of Warping: Map Image to Sphere

d: radial distancefrom center ofinput image.: same for

output image.

eece253 13 rotation

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