Advanced Multimedia
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Transcript of Advanced Multimedia
Advanced Multimedia
Warping & Morphing
Tamara Berg
Image Warping
Slides from: Alexei Efros & Steve Seitz
http://www.jeffrey-martin.com
D'Arcy Thompson http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/darcy.html
http://en.wikipedia.org/wiki/D'Arcy_Thompson
Importance of shape and structure in evolution
Slide by Durand and Freeman
Image Warping in Biology
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 ( + )
(x, y)
(x’, y’)
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()
(x, y)
(x’, y’)
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:
What is the inverse transformation?
y
x
y
x
cossin
sincos
'
'
R
2-D RotationThis is easy to capture in matrix form:
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
''
2x2 MatricesWhat types of transformations can be
represented with a 2x2 matrix?
2D Identity?
yyxx
''
yx
yx
1001
''
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
*'
*'
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
''
2x2 MatricesWhat types of transformations can be
represented with a 2x2 matrix?
2D Mirror about Y axis?
yyxx
''
yx
yx
1001
''
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
''
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
'
'
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 CoordinatesHomogeneous coordinates
• represent coordinates in 2 dimensions with a 3-vector
1
coords shomogeneou y
x
y
x
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?
y
x
tyy
txx
'
'
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
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
Recovering Transformations
What if we know f and g and want to recover the transform T?• willing to let user provide correspondences
– How many do we need?
x x’
T(x,y)y y’
f(x,y) g(x’,y’)
?
Translation: # correspondences?
How many correspondences needed for translation?
x x’
T(x,y)y y’
?
Translation: # correspondences?
How many correspondences needed for translation?
How many Degrees of Freedom?
x x’
T(x,y)y y’
?
Translation: # correspondences?
How many correspondences needed for translation?
How many Degrees of Freedom?
What is the transformation matrix?
x x’
T(x,y)y y’
?
100
'10
'01
yy
xx
pp
pp
M
Euclidian: # correspondences?
How many correspondences needed for translation+rotation?
How many DOF?
x x’
T(x,y)y y’
?
Affine: # correspondences?
How many correspondences needed for affine?
How many DOF?
x x’
T(x,y)y y’
?
Projective: # correspondences?
How many correspondences needed for projective?
How many DOF?
x x’
T(x,y)y y’
?
Example: warping triangles
Given two triangles: ABC and A’B’C’ in 2D (12 numbers)
Need to find transform T to transfer all pixels from one to the other.
What kind of transformation is T?
How can we compute the transformation matrix:
T(x,y)
?
A
B
C A’C’
B’
Source Destination
11001
'
'
y
x
fed
cba
y
x
Image warping
Given a coordinate transform (x’,y’) = T(x,y) and a source image f(x,y), how do we compute a transformed image g(x’,y’) = f(T(x,y))?
x x’
T(x,y)
f(x,y) g(x’,y’)
y y’
f(x,y) g(x’,y’)
Forward warping
Send each pixel f(x,y) to its corresponding location
(x’,y’) = T(x,y) in the second image
x x’
T(x,y)
Q: what if pixel lands “between” two pixels?
y y’
f(x,y) g(x’,y’)
Forward warping
Send each pixel f(x,y) to its corresponding location
(x’,y’) = T(x,y) in the second image
x x’
T(x,y)
Q: what if pixel lands “between” two pixels?
y y’
A: distribute color among neighboring pixels (x’,y’)– Known as “splatting”– Check out griddata in Matlab
f(x,y) g(x’,y’)x
y
Inverse warping
Get each pixel g(x’,y’) from its corresponding location
(x,y) = T-1(x’,y’) in the first image
x x’
Q: what if pixel comes from “between” two pixels?
y’T-1(x,y)
f(x,y) g(x’,y’)x
y
Inverse warping
Get each pixel g(x’,y’) from its corresponding location
(x,y) = T-1(x’,y’) in the first image
x x’
T-1(x,y)
Q: what if pixel comes from “between” two pixels?
y’
A: Interpolate color value from neighbors– nearest neighbor, bilinear, Gaussian, bicubic
– Check out interp2 in Matlab
Forward vs. inverse warpingQ: which is better?
A: usually inverse—eliminates holes• however, it requires an invertible warp function—not always possible...
Homework 3A
On course webpage. Due Tues, March 31. Start now and come ask me questions asap. My solution is < 20 lines of code, but hardest part will be conceptually figuring out how to do it.
Homework 3B will build on the implementation of 3A. So, I will release a solution to 3A on Thurs, April 2 and no late homeworks will be accepted after that.
Overview: Implement image warping and composition in matlab.
Homework 3A
+
Target Source
Homework 3A
+
Target Source
User clicks on 4 points in target, 4 points in source.
Homework 3A
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Target Source
User clicks on 4 points in target, 4 points in source.Find best affine transformation between source subimg and target subimg.
Homework 3A
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Target Source
User clicks on 4 points in target, 4 points in source.Find best affine transformation between source subimg and target subimg.Transform the source subimg accordingly.
Homework 3A
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Target Source
User clicks on 4 points in target, 4 points in source.Find best affine transformation between source subimg and target subimg.Transform the source subimg accordingly.Insert the transformed source subimg into the target image.
Homework 3A
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Target Source
User clicks on 4 points in target, 4 points in source.Find best affine transformation between source subimg and target subimg.Transform the source subimg accordingly.Insert the transformed source subimg into the target image.
Useful matlab functions: ginput, cp2transform, imtransform. Look these up!
Homework 3A
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Target Source
= Result
Image Morphing - Women in Art video
http://youtube.com/watch?v=nUDIoN-_Hxs
Morphing = Object Averaging
The aim is to find “an average” between two objects• Not an average of two images of objects…• …but an image of the average object!• How can we make a smooth transition in time?
– Do a “weighted average” over time t
How do we know what the average object looks like?• We haven’t a clue!• But we can often fake something reasonable
– Usually required user/artist input
P
Qv = Q - P
P + 0.5v= P + 0.5(Q – P)= 0.5P + 0.5 Q
P + 1.5v= P + 1.5(Q – P)= -0.5P + 1.5 Q(extrapolation)Linear Interpolation
(Affine Combination):New point aP + bQ,defined only when a+b = 1So aP+bQ = aP+(1-a)Q
Averaging Points
P and Q can be anything:• points on a plane (2D) or in space (3D)• Colors in RGB or HSV (3D)• Whole images (m-by-n D)… etc.
What’s the averageof P and Q?
Idea #1: Cross-Dissolve
Interpolate whole images:
Imagehalfway = (1-t)*Image1 + t*image2
This is called cross-dissolve in film industry
But what if the images are not aligned?
Idea #2: Align, then cross-disolve
Align first, then cross-dissolve• Alignment using global warp – picture still valid
Dog Averaging
What to do?• Cross-dissolve doesn’t work• Global alignment doesn’t work
– Cannot be done with a global transformation (e.g. affine)
• Any ideas?
Feature matching!• Nose to nose, tail to tail, etc.• This is a local (non-parametric) warp
Idea #3: Local warp, then cross-dissolve
Morphing procedure: for every t,1. Find the average shape (the “mean dog”)
• local warping
2. Find the average color• Cross-dissolve the warped images
Local (non-parametric) Image Warping
Need to specify a more detailed warp function• Global warps were functions of a few (2,4,8) parameters• Non-parametric warps u(x,y) and v(x,y) can be defined
independently for every single location x,y!• Once we know vector field u,v we can easily warp each pixel
(use backward warping with interpolation)
Image Warping – non-parametricMove control points to specify a spline warp
Spline produces a smooth vector field
Warp specification - denseHow can we specify the warp?
Specify corresponding spline control points• interpolate to a complete warping function
But we want to specify only a few points, not a grid
Warp specification - sparseHow can we specify the warp?
Specify corresponding points• interpolate to a complete warping function
• How do we do it?
How do we go from feature points to pixels?
Triangular Mesh
1. Input correspondences at key feature points
2. Define a triangular mesh over the points• Same mesh in both images!• Now we have triangle-to-triangle correspondences
3. Warp each triangle separately from source to destination• How do we warp a triangle?• 3 points = affine warp!• Just like texture mapping
TriangulationsA triangulation of set of points in the plane is a partition
of the convex hull to triangles whose vertices are the points, and do not contain other points.
There are an exponential number of triangulations of a point set.
An O(n3) Triangulation AlgorithmRepeat until impossible:
• Select two sites.• If the edge connecting them does not intersect previous
edges, keep it.
“Quality” Triangulations
Let (T) = (1, 2 ,.., 3t) be the vector of angles in the triangulation T in increasing order.
A triangulation T1 will be “better” than T2 if (T1) > (T2) lexicographically.
The Delaunay triangulation is the “best” • Maximizes smallest angles
good bad
Improving a TriangulationIn any convex quadrangle, an edge flip is possible. If
this flip improves the triangulation locally, it also improves the global triangulation.
If an edge flip improves the triangulation, the first edge is called illegal.
Illegal Edges
Lemma: An edge pq is illegal iff one of its opposite vertices is inside the circle defined by the other three vertices.
Proof: By Thales’ theorem.
Theorem: A Delaunay triangulation does not contain illegal edges.
Corollary: A triangle is Delaunay iff the circle through its vertices is empty of other sites.
Corollary: The Delaunay triangulation is not unique if more than three sites are co-circular.
p
q
Naïve Delaunay Algorithm
Start with an arbitrary triangulation. Flip any illegal edge until no more exist.
Could take a long time to terminate.
Delaunay Triangulation by Duality
General position assumption: There are no four co-circular points.
Draw the dual to the Voronoi diagram by connecting each two neighboring sites in the Voronoi diagram.
Corollary: The DT may be constructed in O(nlogn) time.
This is what Matlab’s delaunay function uses.
Image MorphingWe know how to warp one image into the other, but
how do we create a morphing sequence?1. Create an intermediate shape (by interpolation)
2. Warp both images towards it
3. Cross-dissolve the colors in the newly warped images
Warp interpolationHow do we create an intermediate warp at time t?
• Assume t = [0,1]• Simple linear interpolation of each feature pair• (1-t)*p1+t*p0 for corresponding features p0 and p1
Morphing & matting
Extract foreground first to avoid artifacts in the background
Slide by Durand and Freeman
Other Issues
Beware of folding• You are probably trying to do something 3D-ish
Morphing can be generalized into 3D• If you have 3D data, that is!
Extrapolation can sometimes produce interesting effects• Caricatures
Dynamic Scene