Raster image morphingcgg.mff.cuni.cz/~pepca/lectures/pdf/2d-05-morphing.en.pdftransition between two raster images –continuous transformation –„in-between” frame generation

Morphing 2015 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 1 / 21

Image morphing

© 1998-2015 Josef Pelikán

CGG MFF UK Praha

[email protected]

http://cgg.mff.cuni.cz/~pepca/


Morphing – „image transition”

transition between two raster images– continuous transformation– „in-between” frame generation

geometric deformation (warping)– image space warp (domain transform)

change of attribute (image) function– color/attribute transition (range transform)


Diagram

image coordinates

[x,y] U R2 [u,v] U’ R2

g

f0

attribute space (color)

[R,G,B,..]

f1

time

source target


Interpolation of deform. function

g x y t x y U Rt , , ,0 1 2

Introducing time variable into g function:

g x y x y

g x y g x y u v0

1

, ,

, , ,

Boundary conditions:

g0g0.5

g1


Interpolation of image function

f g x y t f x y t f g x y

t x y U R

t t , , ,

, ,

1

0 1

0 1 1

2

Image function in time t:

can be precomputed

f0 f0.5 f1


defined by [triangle] meshes– individual vertices can move in time (triangle mesh,

spline net, ..)

defined by a set of features– interpolation of arrow endpoints– better: arrow centers, lengths and orientations

deformation of a planar polygon– transition function should minimize deformation

energy

Deformation


shape transition between two planar polygons– need not to have the same number of vertices– continuous transform (for „in-betweening”)

method based on physics– elastic wire model – deformation work minimization– „stretching” and „bending” work

Planar polygon blending


Linear vertex-position interpolation

P P P P P

P P P P PN

N

0 0 0 0 0

1 1 1 1 10 1 0

0 1 0

, ,

, ,

Polygons with equal number of vertices:

P t t P t Pi i i 1 0 1Intermediate position of Pi:

P(0)

P(1)

P(0.5)


Angle interpolation

i 0

Pi(0)

Pi-1(0)

Pi+1(0)Bi(0)Fi(0)

Pi(0.5)

Pi(1)Pi-1(0.5)

Pi-1(1)

Pi+1(0.5)

Pi+1(1)

i 0 5.

i 1


Inner angles

i i i i

i i

t P t P t P t

B t F t

1 1

0

, ,

, ,

tan , ,

A BA BA B

x y x yx x y y

A B B A

A B A B

0

For angle between two adjacent line segments:

tan ii i

i i

tB t F tB t F t

y t y t t y t

x t x t t x t

02

1 22

02

1 22

1 2 1

1 2 1


Inner angles

Q x y F B F B

Q x yF B F B

F B F B

Q x y F B F B

i i i i

i i i i

i i i i

i i i i

0 0 0

1 1 1

12

12

2 2 2

0 0 0 0

1 0 0 1

1 0 0 1

1 1 1 1

, ,

,,

, ,

Q t Q t Q t t Q t 02

1 221 2 1

Quadratic Bézier curve defined by control points Qi:

while this holds: i t Q t 1 0 0 0, , , ,


Evolution of an inner angle

0

90

180

270

monotone Q0

Q1

Q2

Q0

Q1Q2

one turning point

Q0

Q1Q2

zero crossing


„Stretching” work

WA EL

2

02From physics we have:

A cross section

E elastic modulus

L0 original length of the segment

(absolute) stretching

There would be infinite work to shrinking a segmentinto one point!


„Stretching” work

W

k L L

c L L c L LSS

e

S S

S

1 0

0 1 0 11 min , max ,

Modified formula:

k A ES elastic modulus times cross-section

eS elastic „exponent“ (1 .. plastic, 2 .. elastic)

cS degeneration penalty


„Bending” work

W k m p QB B Be

BB

total inner angle change

kB bending elastic modulus

mB penalty for non-monotony

„fold“ of the inner anglepB penalty for zero-crossing („turnover”),

zero if Q(t) does not cross zero


sum of all the „stretching” and „bending” components– „stretching” work for each segment– „bending” work for each vertex

polygon size normalization– „stretching” work is not scale-invariant– uniform scaling to the same size of bounding

rectangle's largest side

Total deformation work


looking for an optimal source-to-target vertex assignment– transition of vertices is given (linear)– to minimize total deformation work– not expecting equal number of vertices edge

degenerations– too large edges could be split in preprocessing

dynamic algorithm .. O(MN)– assignment of short sequences largest sequences

Global optimization


Assignment graph

1:1 assignment

P(0)

P(1)

0 1 M-1 M=0

1

2

N-1

N=0

2

collapse in P(0)

collapse in P(1)

„stretching” work(one edge)

„bending” work(two adjacent edges)

forbidden !


Algorithm

looking for a path from [0,0] to [M,N] with minimal cost– the path cannot return or turn by 90°

each edge has its „stretching“ work

each couple of adjacent edges define „bending” work

dynamic programming – each path prefix has three values in the working matrix– if it went E, SE or S


General algorithm

looking for a path from [i,0] to [i,N] with minimal cost

trying all potential starting points [i,0]

cyclic connection of left/right boundaries (toroidal topology)

one level higher complexity

O(M N M) working matrix


The End

More info:

J. Gomes et al.: Warping and Morphing of Graphical Objects, C.N., SIGGRAPH’95

T. Sederberg, E. Greenwood: A Physically Based Approach to 2D Shape Blending, SIGGRAPH’92

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Raster image morphingcgg.mff.cuni.cz/~pepca/lectures/pdf/2d-05-morphing.en.pdftransition between two raster images –continuous transformation –„in-between” frame generation