Recap from Monday

DCT and JPEGPoint ProcessingHistogram Normalization

Questions:JPEG compression levelsGamma correction

JPEG ClarificationsCompression level is indeed adjusted by changing the

magnitude of the quantization table.

JPEG often preferentially compresses color information.

Gamma CorrectionIs an instance of these

power law intensity transformations.

Typically, gamma = 2.2 for a display device, and 1/2.2 for encoding.

The result is a more perceptually uniform encoding.

Gamma CorrectionImage encoded for

various display gammas.

Source: Wikipedia

Image Warping

cs195g: Computational PhotographyJames Hays, Brown, Spring 2010

Slides from Alexei Efros and Steve Seitz

http://www.jeffrey-martin.com

Image Transformations

image filtering: change range of imageg(x) = T(f(x))

f

x

Tf

x

f

x

Tf

x

image warping: change domain of imageg(x) = f(T(x))

Image Transformations

T

T

f

f g

g

image filtering: change range of image

g(x) = T(f(x))

image warping: change domain of imageg(x) = f(T(x))

Parametric (global) warpingExamples of parametric warps:

translation rotation aspect

affineperspective

cylindrical

Parametric (global) warping

Transformation T is a coordinate-changing machine:

p’ = T(p)

What does it mean that T is global?• Is the same for any point p• can be described by just a few numbers (parameters)

Let’s represent T as a matrix:

p’ = Mp

T

p = (x,y) p’ = (x’,y’)

y

x

y

xM

'

'

ScalingScaling a coordinate means multiplying each of its components by

a scalarUniform scaling means this scalar is the same for all components:

2

Non-uniform scaling: different scalars per component:

Scaling

X 2,Y 0.5

Scaling

Scaling operation:

Or, in matrix form:

byy

axx

'

'

y

x

b

a

y

x

0

0

'

'

scaling matrix S

What’s inverse of S?

2-D Rotation

(x, y)

(x’, y’)

x’ = x cos() - y sin()y’ = x sin() + y cos()

2-D Rotation

x = r cos ()y = r sin ()x’ = r cos ( + )y’ = r sin ( + )

Trig Identity…x’ = r cos() cos() – r sin() sin()y’ = r sin() cos() + r cos() sin()

Substitute…x’ = x cos() - y sin()y’ = x sin() + y cos()

(x, y)

(x’, y’)

2-D RotationThis is easy to capture in matrix form:

Even though sin() and cos() are nonlinear functions of ,• x’ is a linear combination of x and y

• y’ is a linear combination of x and y

What is the inverse transformation?• Rotation by –• For rotation matrices

y

x

y

x

cossin

sincos

'

'

TRR 1

R

2x2 MatricesWhat types of transformations can be

represented with a 2x2 matrix?

2D Identity?

yyxx

''

yx

yx

1001

''

2D Scale around (0,0)?

ysy

xsx

y

x

*'

*'

y

x

s

s

y

x

y

x

0

0

'

'

2x2 MatricesWhat types of transformations can be

represented with a 2x2 matrix?

2D Rotate around (0,0)?

yxyyxx

*cos*sin'*sin*cos'

y

x

y

x

cossin

sincos

'

'

2D Shear?

yxshy

yshxx

y

x

*'

*'

y

x

sh

sh

y

x

y

x

1

1

'

'

2x2 MatricesWhat types of transformations can be

represented with a 2x2 matrix?

2D Mirror about Y axis?

yyxx

''

yx

yx

1001

''

2D Mirror over (0,0)?

yyxx

''

yx

yx

1001

''

2x2 MatricesWhat types of transformations can be

represented with a 2x2 matrix?

2D Translation?

y

x

tyy

txx

'

'

Only linear 2D transformations can be represented with a 2x2 matrix

NO!

All 2D Linear Transformations

Linear transformations are combinations of …• Scale,• Rotation,• Shear, and• Mirror

Properties of linear transformations:• Origin maps to origin• Lines map to lines• Parallel lines remain parallel• Ratios are preserved• Closed under composition

y

x

dc

ba

y

x

'

'

yx

lkji

hgfe

dcba

yx''

Homogeneous CoordinatesQ: How can we represent translation as a 3x3

matrix?

y

x

tyy

txx

'

'

Homogeneous Coordinates

Homogeneous coordinates• represent coordinates in 2

dimensions with a 3-vector

1

coords shomogeneou y

x

y

x homogenouscoords

Homogeneous Coordinates

Add a 3rd coordinate to every 2D point• (x, y, w) represents a point at location (x/w, y/w)• (x, y, 0) represents a point at infinity• (0, 0, 0) is not allowed

Convenient coordinate system to

represent many useful transformations

1 2

1

2(2,1,1) or (4,2,2) or (6,3,3)

x

y

Homogeneous CoordinatesQ: How can we represent translation as a 3x3

matrix?

A: Using the rightmost column:

100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx

'

'

TranslationExample of translation

11100

10

01

1

'

'

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2ty = 1

Homogeneous Coordinates

Basic 2D TransformationsBasic 2D transformations as 3x3 matrices

1100

0cossin

0sincos

1

'

'

y

x

y

x

1100

10

01

1

'

'

y

x

t

t

y

x

y

x

1100

01

01

1

'

'

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

'

'

y

x

s

s

y

x

y

x

Scale

Matrix CompositionTransformations can be combined by

matrix multiplication

wyx

sysx

tytx

wyx

1000000

1000cossin0sincos

1001001

'''

p’ = T(tx,ty) R() S(sx,sy) p

Affine Transformations

Affine transformations are combinations of …• Linear transformations, and

• Translations

Properties of affine transformations:• Origin does not necessarily map to origin

• Lines map to lines

• Parallel lines remain parallel

• Ratios are preserved

• Closed under composition

• Models change of basis

Will the last coordinate w always be 1?

wyx

fedcba

wyx

100''

Projective Transformations

Projective transformations …• Affine transformations, and

• Projective warps

Properties of projective transformations:• Origin does not necessarily map to origin

• Lines map to lines

• Parallel lines do not necessarily remain parallel

• Ratios are not preserved

• Closed under composition

• Models change of basis

wyx

ihgfedcba

wyx

'''

2D image transformations

These transformations are a nested set of groups• Closed under composition and inverse is a member

Matlab Demo

“help imtransform”