cv_lec10_geometric_transform.ppt

Introduction to Computer VisionLecture 10

Chapter 6: Geometric Transformations

Speaker: Yuheng WangEmail: [email protected]

Date: 01/26 /2012

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mailto:[email protected]

Outline

• Geometric Transformation Theory– Linear transformations– Affine transformations– Projective transformations

• Matlab Implementation

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Geometric Transformations

• Original Image

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• Scaling

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• Rotation

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• Affine Transformation

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• Projective Transformation

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Coordinate Mapping

• Transformation: (x, y) = T{(w, z)}

f(w, z) g(x, y )

T(w, z)

x

y

w

z

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Coordinate Mapping

• Transformation: (w, z) = T-1{(x, y)}

f(w, z) g(x, y )w x

T-1 (x, y)

z y

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Transformations: (x, y) = T{(w, z)} = (w/2, z/2)(w, z) = T-1{(x, y)} = (2x, 2y)

(x, y) = (w/2, z/2)

(w, z) = (2x, 2y)Textbook Page 280Figure 6.2

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Linear Transformations

• An operation than change the figure in some way, can be either or a combination of:– Translation: “slide”–Rotation: “turn”–Reflection: “flip”

Reference:http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry//geo-tran.html

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Translation

• Move the object without changing the image size and orientation

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Rotation

• Turn the object about around certain fixed point (the rotation center)

Rotation

Center13

Reflection

• Flip the object over a line (the line of reflection)

Line of Reflection

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Reflection

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Method to implement transformations

Find coordinates (x, y) of the original image

Use the transformation rule to change original coordinates (x, y) to (x’, y’)

Show the new image using (x’, y’)

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Translation

x’ 1 1 2

y’ 2 0 0

x -3 -3 -2

y 3 1 1

x

y

0

x’= x + 4y’= y - 1

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Rotation

x’ -5 -3 -3

y’ 1 1 2

x -3 -3 -2

y 3 1 1

x

y

0

theta = 90o

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Reflection

x’ -3 -3 -2

y’ -3 -1 -1

x -3 -3 -2

y 3 1 1

x

y

0

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Scaling

x’ -3 -3 -2

y’ -3 -1 -1

x -3 -3 -2

y 3 1 1

x

y

0

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Scaling

x’ -3 -3 2

y’ 4 1 1

x -3 -3 -2

y 3 1 1x

y

0

• multiplying each of its coordinates by a scalar

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Overall workflow

Find coordinates (x, y) of the original image

Use the transformation rule to change original coordinates (x, y) to (x’, y’)

Show the new image using (x’, y’)

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Mathematical Definition

• A geometric transformation is a matrix(vector) T that maps the pixel (x, y) to a new position (x’, y’)

x’ = Tx(x, y)

y’=Ty(x, y)

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Linear Transformations Recall:

x’ = Tx(x, y) y’=Ty(x, y)

yaxax 10' ybxby 10'

yxba

bayx *''

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00

Matrix FormatPolynomial Format

Transformation Matrix T

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Affine Transformations

210' ayaxax

210' bybxby 1*

1

0

0

1''

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11

00

yx

ba

ba

ba

yx

Matrix FormatPolynomial Format

Transformation Matrix T

Recall:

x’ = Tx(x, y) y’=Ty(x, y)

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• Perceptually, Affine Transformations are linear combinations of – Translation – Rotation– Scaling

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y

x

y

x*

cossin

sincos

yx, yx ,

x

y

0

• Rotation

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ys

xs

y

x

s

s

y

x

y

x

y

x*

0

0

x

y

0

• Scaling

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• Reflectionalong X-axis

y

x

y

x

y

x*

10

01

x

y

0

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• Reflection along Y-axis

y

x

y

x

y

x*

10

01

x

y

0

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• Shearing along X-axis

y

dyx

y

xd

y

x

10

1

x

y

0

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• Shearing along Y-axis

dxy

x

y

x

dy

x

1

01

x

y

0

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Affine Transformations• Rotation

1

*

100

0cossin

0sincos

1

y

x

y

x

yx, yx ,

x

y

0

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• Scaling

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*

100

00

00

1

ys

xs

y

x

s

s

y

x

y

x

y

x

x

y

0

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Affine Transformations• Shearing

11

*

100

010

01

1

y

dyx

y

xd

y

x

x

y

0

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• Reflection

11

*

100

010

001

1

y

x

y

x

y

x

x

y

0

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• Translation

11

*

100

10

01

1y

x

y

x

ty

tx

y

x

t

t

y

x

x

y

0

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Summary

1

*

100

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00

1

y

x

s

s

y

x

y

x

1

*

100

0cossin

0sincos

1

y

x

y

x

1

*

100

01

01

1

y

x

sh

sh

y

x

y

x

1

*

100

10

01

1

y

x

t

t

y

x

y

x

Translation Scaling

ShearingRotation

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Textbook Page 286Table 6.1

Summary

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Projective Transformations

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Vanishing Point

Vanishing Point Horizon

Line

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Matlab Implementation

• Transformation Techniques– Affine Transformations– Projective Transformations

• Matlab Implementation– function maketform– function imtransform

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help maketform

• maketform Create spatial transformation structure (TFORM).

• T = maketform(TRANSFORMTYPE,...) creates a multidimensional spatial transformation structure (a 'TFORM struct') that can be used with TFORMFWD, TFORMINV, FLIPTFORM, IMTRANSFORM, or TFORMARRAY. TRANSFORMTYPE can be 'affine', 'projective', 'custom', 'box', or 'composite'. Spatial transformations are also called geometric transformations.

• ……

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Several Forms

• T = maketform('affine',A);• T = maketform('affine',U,X);• T = maketform('projective',A);• T = maketform('projective',U,X);• T =

maketform('custom',NDIMS_IN,NDIMS_OUT, FORWARD_FCN,INVERSE_FCN, TDATA);

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T = maketform('custom',NDIMS_IN, NDIMS_OUT,FORWARD_FCN,INVERSE_FCN, TDATA

• Parameters– NDIMS_IN and NDIMS_OUT are the numbers of input

and output dimensions. – FORWARD_FCN and INVERSE_FCN are function

handles to forward and inverse functions. – TDATA can be any MATLAB array and is typically used

to store parameters of the custom transformation. It is accessible to FORWARD_FCN and INVERSE_FNC via the "tdata" field of T.

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help imtransform

• imtransform Apply 2-D spatial transformation to image.

• B = imtransform(A,TFORM) transforms the image A according to the 2-D spatial transformation defined by TFORM, which is a tform structure as returned by MAKETFORM or CP2TFORM. If ndims(A) > 2, such as for an RGB image, then the same 2-D transformation is automatically applied to all 2-D planes along the higher dimensions.

• ……

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clear all, close all, clcIm = checkerboard(50);figure, imshow(Im), axis on;title('Original Checkerboard');

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Im = imread('cameraman.tif');figure, imshow(Im), axis on;title(‘Fig.1. Original Gray Scale Image');

axis on

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Matlab for Scaling

sx=0.5; sy=1; Tscale=[sx 0 0; 0 sy 0; 0 0 1];tScale=maketform('affine',Tscale);ImScale=imtransform(Im, tScale);figure, imshow(ImScale), axis on;title('Scaled Checkerboard');

Textbook Page 289

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Matlab for Scaling

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Scaling

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Matlab for Rotationtheta = pi/6;Trot = [cos(theta),sin(theta),0; -sin(theta),cos(theta),0; 0, 0, 1];tRot = maketform('affine',Trot); ImRot = imtransform(Im, tRot);figure, imshow(ImRot), axis on;title('Rotated Checkerboard'); Textbook

Page 29052

Matlab for Rotation

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Rotation

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Matlab for Shearing

shx=2;shy=3; Tshear=[1 shx 0;

shy 1 0; 0 0 1];

tShear=maketform('affine',Tshear);ImShear=imtransform(Im,tShear);figure, imshow(ImShear),axis on;title('Sheared Checkerboard');

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Matlab for Shearing

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Matlab for Shearing

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Matlab for Projection

Tpoj=[0.4788, 0.0135, -0.0009; 0.0135, 0.4788, -0.0009; 0.5059, 0.5059, 1];

tPoj=maketform('projective',Tpoj);ImPoj=imtransform(Im,ImPoj);figure, imshow(ImPoj), axis on;title('Projected Checkerboard');

Textbook Page 291 58

Matlab for Projection

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Projection

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Affine Transformation Example

Im = imread('cameraman.tif'); T = maketform('affine',[.5 0 0; .5 2 0; 0 0 1]); tformfwd([10 20],T); ImTrans= imtransform(Im,T); figure, imshow(Im), axis on; figure, imshow(ImTrans), axis on;

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Result

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Take home assignments• Know– Concepts of Affine/ Projective transformations– Transformation Matrices

• Practice– Play with different transformation matrices T and

see how they are used in image transformations

• Master– Function maketform and function imtransform– How they are implemented in Matlab

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Take Home Message

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1

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0cossin

0sincos

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sh

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Translation Scaling

ShearingRotation

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cv_lec10_geometric_transform.ppt

Documents

pixel x

original coordinates

new position x

linear transformations

transformation rule

matrixvector t

original imageuse

new image