recap from monday
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Recap from Monday. DCT and JPEG Point Processing Histogram Normalization Questions: JPEG compression levels Gamma correction. JPEG Clarifications. Compression level is indeed adjusted by changing the magnitude of the quantization table. - PowerPoint PPT PresentationTRANSCRIPT
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.
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
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 = 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 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