Computationally Efficient NMRF model based Texture Synthesis

Outline
Ph.D. Research Work
Conclusion and Possible Future Directions

Fast NMRF based texture synthesis algorithms

Arnab Sinha
arnab@iitk.ac.in

April 16, 2009

Thesis Supervisor: Dr. Sumana Gupta
ACES-205, Dept. of EE, Indian Institute of Technology Kanpur, India

Arnab Sinha arnab@iitk.ac.in Fast NMRF based texture synthesis algorithms

Outline
Earlier Methods
Ph.D. Research Work
Research problems in NMRF-tex-syn algorithms

1 Outline
Earlier Methods

2 Ph.D. Research Work
Order Estimation from Fourier Domain
Reduction of Computational complexity
Order Estimation : Revisited
Inverse Texture Synthesis

3 Conclusion and Possible Future Directions


Outline
Earlier Methods
Ph.D. Research Work

The significance of texture synthesis
• What defines texture ?
• Locally varying intensities and/or color values
• The local variations can be found perceptually similar within the total region
• Texture Synthesis:

Original D104 Texture
Given a small texture exempler, synthesize
an arbitrary sample of texture, so that the
synthesized texture is visually similar to the
original sample.

Synthesized texture should look alike the original texture
• Application of texture synthesis in -
• Image segmentation, classification, synthesis, etc.
• Content-based image retrieval
• Development of high-level computer vision algorithms
• Animation of real scenes
• Perceptual analysis
• Computationally fast and efficient handling of objects


Outline
Earlier Methods
Ph.D. Research Work

Texture synthesis: Difﬁculty

Figure: Spectrum of Natural Textures


Outline
Earlier Methods
Ph.D. Research Work

Brief History of Models
Texture Synthesis Algorithm

Image Domain Model Mixed Domain Model Transformed Domain Model

Non−Linear Models
Linear Non−Linear

Hidden Markov Tree Non−Linearity Introduced
by Histogram Equalization
Fan and Xia (2003)
2D−NCAR, Hard−limited Gaussian
Process
Chellappa and Kashyap (1985)
Jacovitti et al. (1998)
2D−Wold
Francos et al. (1993) Zhang et al. (1998)
Wavelet + AR
2D− MA
1. Zhu et al. (1997)
Eom (1998) Circular Harmonic Func 2. Zhu et al. (2000)
+ Hard−limited Gaussian
Campasi and Scarano (2002)
3. Portilla and Simoncelli (2000)
NNMRF
Charalampidis (2006)
Paget and Longstaff (1998)
Mathematical Models

Intuitive Models
Heeger and Bergen (1995)
Efros and Leung (1999)
Wei and Levoy (2000)
Pixel−based Ashikhmin (2001) We are working
within this
Framework
Sampling Process
Tonietto et al. (2005)
Kwatra et al. (2003)
Patch−based
Patch−based sampling with wavelet transformation
Wu et al. (2004) as a feature set for graph−cut algorithm
Popular Methods


Outline
Earlier Methods
Ph.D. Research Work

Description of N-MRF model

• S is the lattice
• Ys is the random variable at site s ∈ S
• Concept of Neighborhood system:

ℵs
•s
s = (i,j), site
• r ∈ ℵs ⇔ s ∈ ℵr
1st order neighbors
2 2
• Circular neighborhood: ℵs = {r; s.t., |r − s| ≤ o } { } 2nd order neighbors

• say, Xs = {Yr ; r ∈ ℵs }

• Say, Y(s) = {Yr ; r s}, r, s ∈ S
• Deﬁnition of MRF: P(Ys |Y(s) ) = P(Ys |{Yr ; r ∈ ℵs })
• parameteric model for P(Ys |Xs )
• semi-parameteric model for P(Ys |Xs )
• non-parameteric model for P(Ys |Xs )


Outline
Earlier Methods
Ph.D. Research Work

Description of N-MRF model: Kernel Density Estimation

Deﬁnition of KDE, [Scott(1992)]
N
1
• single dimensional: P(x) = N Kh (x − xi )
i=1


Outline
Earlier Methods
Ph.D. Research Work

Description of N-MRF model: Kernel Density Estimation

Deﬁnition of KDE, [Scott(1992)]
N
1
• single dimensional: P(x) = N Kh (x − xi )
i=1
N d
1
• multi-dimensional: P(X) = Khj (X (j) − Xi (j))
i=1 j=1
N
(X (j)−Xi (j))2
• where, in case of Gaussian kernel, Khj (X (j) − Xi (j)) = √1 }
exp{−
2hj2
N 2πhj

• and, hj = σj N −1/(d+4)


Outline
Earlier Methods
Ph.D. Research Work


Some definitions
• Input texture field: {Ys }, where, s ∈ Sin
• Output texture field: {Yq }, where, q ∈ Sout
Kh (Ys −Yq )Kh (Xs −Xq )
s∈Sin Y X
• Definition of LCPDF: P(Yq |Xq ) =
Kh (Xs −Xq )
s∈Sin X

Iterative Conditional Mode (ICM) algorithm
• Evaluate P(Yq = y|Xq ), for y = 0, 1, . . . , 255 gray values.
• Assign Yq = y, for which the above conditional probability is maximum

Local Simulated Annealing
• Define a Confidence field, Cq ; q ∈ Sout , and a matrix Φq = DIAG{Cr ; r ∈ ℵq }
• KhX (Xs − Xq ; Φq ) = exp{−(Xs − Xq )T Φh,q (Xs − Xq )}, and Φh,q = Φq HX ≈ hΦq
• Updation rule for the confidence field
1
• Cq = min{1, |r∈ℵ | r∈ℵ Cr + u × e}
q
q
• where, u is a random number and e is a constant scale factor

Outline
Earlier Methods
Ph.D. Research Work


Approximate Independent Conditional Mode (ICM) algorithm
• Ds,q = (Xs − Xq )T Φq (Xs − Xq )
• Deﬁne Sq = {r ∈ Sin } ⊂ Sin , s.t., ∀r ∈ Sq , Dr,q = constant.
• Assign Yq = yr , where r is sampled from the set Sq randomly.

S in
S out C

q q

Input texture

Output Texture Confidence Field

Output Neighborhood Output Confidence
Vector X q Vector Wq
Matrix

Input Neighborhood Vectors Similarity Measure
t
{X s} N−MRF : ( X q − X s) ( X q − X s)
t
WL alg : ( X q − X s) ( X q − X s)


Outline
Earlier Methods
Ph.D. Research Work

Research Problems

• Order estimation
• Large Computational Complexity
• Computational complexity ∝ d, the dimension of the neighborhood vector.
• Computational complexity ∝ (M × N), the input image size
• Computational complexity ∝ I, the number of iterations required to attain global
convergence

Original Texture

Neighborhood vector dimension ’d’
8000

7000
computational complexity of
6000
texture synthesis algorithm
is proportional to ’d’
5000

4000

3000

2000

1000

0
0 10 20 30 40 50
Model order ’o’
Order 4 Order 8 Order 14

Figure: Effect of order on the synthesis results and computational complexity


Outline
Ph.D. Research Work

Order estimation from two fundamental frequencies

Y

c
d
o
b
a X
Yj
Xj
Yi

Xi

Figure: Points a, b, c, d are the four corners of texton deﬁned by the fundamental spatial period
vectors [Xi Yi ] and [Xj Yj ]. The major diagonal o gives the order of causal circular neighborhood
and o/2 gives the order of non-causal circular neighborhood.


Outline
Ph.D. Research Work

Extraction of the parameters

• Dimitri’s algorithm
• estimate the two fundamental frequecies from the two-dimensional DFT of the texture
sample.
• computational complexity is of the order of image size.
• Hays’s algorithm
• it estimates the two fundamental spatial vectors from the correlation function
• the algorithm is iterative
• computationally expensive with respect to Dimitri’s algorithm


Outline
Ph.D. Research Work

D21
D20

order = 9
order = 23 order = 4
order = 18

D52
D35

order = 48 order = 8 order = 22
order = 21

Figure: Comparison of estimated order through Dimitrios and Hays’s methods with (NR) texture
synthesis results


Outline
Ph.D. Research Work

A new neighborhood system

Y

proposed Non−causal
neighborhood
X
Yj
Xj
Yi
circular Non−causal
neighborhood
Xi


Outline
Ph.D. Research Work

Results: Proposed neighborhood system
Circular Neighborhood Proposed Neighborhood Proposed Neighborhood
Circular Neighborhood

D104
D65

D95
D64

D67
D3


Outline
Ph.D. Research Work

Two approaches

Computational complexity affected by
the neighbourhood dimension, d, and
1

the number of input pixels, N
2

Reduction methodologies
Dimensionality reduction methodologies, e.g., Principal Component Analysis
1
(PCA) – to reduce the effect of d
A data structure for fast search
2


Outline
Ph.D. Research Work

With Dimensionality reduction methods

How the distance metric Ds,q looks after projection

(Xq − Xs )T Φq (Xq − Xs ) (X q − X s )T Φ q (X q − X s )
≈
[PrT Pr (Xq − Xs )]T Φq [PrT Pr (Xq − Xs )]
=
[Pr (Xq − Xs )]T Pr Φq PrT [Pr (Xq − Xs )]
=
(Zq − Zs )T Ψq (Zq − Zs )
=

T
• What is Ψq = Pr Φq Pr ?
• Is it reducing the computational complexity ?


Outline
Ph.D. Research Work

Simulated Annealing for Principal components

Original Ds,q

(Xq − Xs )T Φq (Xq − Xs )
=
Ds,q
where, Φq = DIAG{W1 , W2 , . . . , Wd }

ˆ
Proposed Ds,q

ˆ ˆ
(Zq − Zs )T Φq (Zq − Zs )
=
Ds,q
ˆ
where, Φq = DIAG{W1 , W2 , . . . , Wk }

WHY ? Because we need only
• a steady increase in the value of conﬁdence, and
• the starting value has to be ”0” and ending value has to be ”1”


Outline
Ph.D. Research Work

Results

Table: Comparison of dimensionality; Original dimension, |ℵs | = d; Reduced dimension, k (<< d), η
is the ratio of computational complexities between earlier and proposed one

Texture Type Texture order d k η
NR D20 20 1516 60 21.9405
NR D3 30 2820 580 3.7926
NR D21 25 1960 56 29.2642
NR D22 20 1256 177 6.3042
NR D35 28 2452 287 6.6612
NR D36 22 1516 258 5.1024
ST D7 27 2288 511 3.6438
ST D13 24 1792 131 11.6006
NR+ST D18 32 3208 95 25.5666
NR+ST D4 29 2628 465 4.4755
NR+ST D5 29 2628 179 11.6262
IN/STR D15 23 1652 167 8.4897
IN/STR D42 26 2120 293 5.9699


Outline
Ph.D. Research Work

Results continued ...
NNMRF Proposed Algorithm Proposed Algorithm
NNMRF

D20 D7

k = 511
d = 2288
d = 1516 k=60

D13
D21

k=131
d=1792
d=1960 k=56

D22 D42

k = 293
d=2120
k = 177
d=1256

Outline
Ph.D. Research Work

With Fast Kernel Density Estimation

Assumption: The h parameters along all the directions are equal.

• Let Rn = {ts : ||ts − tn || ≤ R}
N
1
• P(tn ) = N s=1 KH (ts − tn )
Y
1
• P(tn ) = N s∈Rn KH (ts − tn )
• Let Nn = |{s Rn }|
R=100
1
= KH (ts − tn )
Err(tn , R)
N
s Rn
Nn
K (R)
≤
To calculate KDE
NH
at this target point
X we only need
KH (R)
≤
Source data vector these two points
Target data vector
max(Err)
• Rel err(R) =
max(Probability)


Outline
Ph.D. Research Work

Earlier FKDE algorithms

• Improved Fast Gaussian Transform (IFGT)
[Yang et al.(2003)Yang, Duraiswami, Gumerov, and Davis]
• kd-tree based FKDE [Gray and Moore(2003)]
• Reconstructionhistogram [Zhang et al.(2005)Zhang, Tang, and Kwok]

Reconstructionhistogram
• Clustering: {Clusti ; i = 1 . . . M}
• Let ni as the number of source data vectors within i th cluster
M
1
• P(tn ) = N i=1 KH (tn − Clusti )ni
• KDE of tn given the source data points at cluster centroids with a weight factor
ni /N
• ﬂexibility ?


Outline
Ph.D. Research Work

Improved Fast Gaussian Transform
• P(tn ) = KH (tn − Clusti )f (tn , Clusti )
||tn −Clusti ||≤RIFGT
1
• P(tn ) = N KH (tn − ts )
||tn −Clusti ||≤RIFGT s∈Clusti

Source data clusters

Target data
vectors
Due to
this overlap
we need to
consider this
source cluster
R IFGT R

To consider the source cluster
In effect it can include some source cluster
R
the radius threshold has to be IFGT
which was not needed at all

The R IFGT can vary with the overlap size
and cluster shape


Outline
Ph.D. Research Work

kd-tree-based FKDE

Build up the kd-tree and Search according to the radius R.

Y
6

σ X > σY
Hyper−Sphere
Y
-6 -
Partition
Span in Y direction

e
6 e
ee
R max_rec_err
Hyper−Rectangle
e ee e
e
e e
e
e
e
Eigen vectors
e
e
e e e
e
e
e
Centroid
e
e
-
e ee
Sub-spaces
e
e
-
e
?
e
X
e
Reconstruction
X Error R rec_err,n
-
R
tn
> R+ R max_rec_err
-
?
Span in X direction


Outline
Ph.D. Research Work

Why do we need another algorithm for FKDE

Table: Why do we need another algorithm for FKDE ?

C-FKDE KD-FKDE
Advantage Clustering algorithm provides more Due to the hyper-plane boundary,
one can use original radius R
compact representation of the data
space for strict error bound
optimal RIFGT has to estimated
Disadvantage kd-tree is not a good clustering
for every tn , maximum RIFGT algorithm, therefore it does not
can increase computational provide compact representation
complexity of the data space


Outline
Ph.D. Research Work

Principal Directive Divisive Partitioning (PDDP)[Boley(1998)]
• project each source data point within the present space onto the ﬁrst principal
direction (eigen vector corresponding to the largest eigen value).
• partition the present space into two sub-spaces with respect to the mean (or
median) of the projected values.

Hieararchical Boundaries
Source data

Tree structure of the nodes
6th
IV I
1st
VI 7th
2nd II III
I
III
II VII
5th V VI VII
4th IV

8th
V 8th
6th 7th
2nd 3rd 4th 5th
1st
3rd
Leaf nodes


Outline
Ph.D. Research Work

FKDE based on PDDP

If the present node is a leaf then evaluate KDE.
1

Source data vectors
Projection of source data
Target data vector
(t n) Projection of terget point
D rec_err
2nd Direction T is within left child
D If R D rec_err
T
Return 0
Drec_err Else
If D R process both children
Process its children D
2nd Direction
Dlb
(Boundary for Partition)
1st Direction
mr
vn D
For the right child lb
If Dlb R

(Which child to process)
(Process child) Return 0
If Dlb R = return 0 if D R = process left child Else If R D rec_err
Else if Drec_err R = return 0 Else process both children Return 0
Else process Else
Process its children
vn Projected target data vector
mr Projected mean vector 1st Direction

(a) Target point is outside (b) Target point is inside


Outline
Ph.D. Research Work

FKDE based on PDDP

1

Is there any need to go further for the children of the present node ?
2

Source data vectors
Target data vector
D rec_err
D If R D rec_err
T
Return 0
Drec_err Else
2nd Direction
Dlb
1st Direction
mr
vn D
If Dlb R

Else process Else

(c) Target point is outside (d) Target point is inside


Outline
Ph.D. Research Work

FKDE based on PDDP

1

Is there any need to go further for the children of the present node ?
2

Which child node (left or right or both) of the present node to process further ?
3

Source data vectors
Target data vector
D rec_err
D If R D rec_err
T
Return 0
Drec_err Else
2nd Direction
Dlb
1st Direction
mr
vn D
If Dlb R

Else process Else

(e) Target point is outside (f) Target point is inside


Outline
Ph.D. Research Work

Comparison between the FKDE algorithms
(i−1,j)

2
(i,j+1)
3
(i,j) 1
4

(i,j+1)

{1} X RGB == 3 dimensions
{1,2} X RGB == 6 dimensions
{1,2,3} X RGB == 9 dimensions
{1,2,3,4} X RGB == 12 dimensions

(g) Image considered (h) Creation of the data
for creating the data set space

Figure: Data set creation for FKDE analysis

Table: Time comparison

Dimension 3 6 9 12
KDE: Time (sec) 981.78 1204.77 2529.22 2668.58
PDDP-FKDE: Time (sec) 11.3 23.39 52.55 65.79
KD-FKDE: Time (sec) 22.55 33.88 110.62 228.44
IFGT-FKDE: Time (sec) 399.79 425.45 209.33 384.08


Outline
Ph.D. Research Work

Comparison between the FKDE algorithms

Table: Comparative analysis of FKDE algorithms

FKDE Dimension Maximum Maximum Relative Radius
algorithms probability Error Error threshold
4.33531e − 05 0.0003e − 04
PDDP-FKDE 3 0.0007 14.5445
1.32977e − 10 0.0000e − 10
6 0.0000 28.1622
9.64769e − 18 0.0014e − 18
9 0.0001 47.5693
1.71367e − 23 0.0000e − 25
12 0.0000 59.9263
4.33531e − 05 0.0005e − 04
KD-FKDE 3 0.0011 14.5445
1.32977e − 10 0.0001e − 10
6 0.0000 28.1622
9.64769e − 18 0.0014e − 18
9 0.0001 47.5693
1.71367e − 23 0.0000e − 25
12 0.0000 59.9263
4.33531e − 05 0.2618e − 04
IFGT-FKDE 3 0.6040 18.1807
1.32977e − 10 0.5389e − 10
6 0.4053 35.2028
9.64769e − 18 0.4183e − 18
9 0.0434 59.4616
1.71367e − 23 0.132e − 25
12 0.0007703 74.9079


Outline
Ph.D. Research Work

Computationally efficient Texture synthesis algorithm with FKDE
algorithms

Problems
how to include the effect of Wq (the temperature field) within the PDDP-based tree
1
structure for the implementation of FKDE, and
there are two joint densities corresponding to {Yq , Xq } and Xq ; therefore, it
2
requires two FKDE structure, which is not computationally efficient.

Inclusion of Wq
• Starting State:
{Wq,i = 0} ⇒ P(Xq ; Wq ) = constant
⇒ P(Xq ) is uniform
⇒ Each Xs has equal effect upon Xq
⇒ Every Xs should be considered in the KDE
⇒ Rnew is very large


Outline
Ph.D. Research Work

algorithms

Problems
1
2

Inclusion of Wq
• Starting State:
{Wq,i = 0} ⇒ P(Xq ; Wq ) = constant
⇒ P(Xq ) is uniform
⇒ Rnew is very large
• Ending State:
{Wq,i = 1} ⇒ P(Xq ; Wq ) = P(Xq )
⇒ Rnew = R

Outline
Ph.D. Research Work

algorithms

Problems
1
2

Inclusion of Wq
• Starting State:
{Wq,i = 0} ⇒ P(Xq ; Wq ) = constant R
=
Rnew
⇒ P(Xq ) is uniform cq
abs(vn − mr ) ≤ Rnew
R
⇒ Rnew is very large ⇒ abs(vn − mr ) ≤
cq
• Ending State:
⇒ abs(vn − mr )cq ≤ R
{Wq,i = 1} ⇒ P(Xq ; Wq ) = P(Xq )
⇒ Rnew = R

Outline
Ph.D. Research Work

Results with comparisons

Original NNMRF kd−tree
IFGT Proposed

D102

D49

D20


Outline
Ph.D. Research Work

Original kd−tree
NNMRF IFGT Proposed

D53

D104

D4

D82


Outline
Ph.D. Research Work


Original NNMRF IFGT kd−tree Proposed

D110

D60

D93

D97


Outline
Ph.D. Research Work


Table: Time taken in texture synthesis: input texture size 128 × 128 and output texture size
256 × 256

NNMRF C-FKDE KD-FKDE PDDP-FKDE
hours 8 5 8 6
minutes 7 55 34 0
seconds 39 12 56 41


Outline
Ph.D. Research Work

Maximum Log-Pseudo-likelihood

LPL = log[P(Ys |Xs )]
s∈Sin

For 1st order neighborhood system For 2nd order neighborhood system


Outline
Ph.D. Research Work

How to estimate MLPL ?

• parametric MRF model
• non-parametric MRF model: what should be the kernel ?
• Gaussian kernel: as used in [Paget and Longstaff(1998)]
• Dirac-delta kernel
• Some other solution

Effect of kernel upon the MLPL estimate
nd
LPL is getting saturated before 2 order
LPL is not saturating rather it is increasing
0
−10000
−50
−20000
−100
−30000 −150
−40000
LPL

−200
LPL
−250
−50000
D102: Near regular
−300
D104: Near regular
−60000
−350
D110: Stochastic
−70000 −400
D60: Stochastic
D93: Stochastic −450
−80000
0 1 2 3 4
0 5 10 15 20 25 30 35 40
Order
Order


Outline
Ph.D. Research Work

Why does the original LPL measure, not saturate ?

Why ?
Kh (Ys −Yq )Kh (Xs −Xq )
s∈S
• original LCPDF: p(Yq |Xq ) =
q∈S Kh (Xs −Xq )

• Changing terms with order:
• hy = σy N −1/(d+4) : changes due to change in d and N, with order
√
• In case of LCPDF the normalizing term becomes: 2πhy ;
• Moreover, hy also affect the argument within the exponential term.

• One can not neglect this term.


Outline
Ph.D. Research Work

A new deﬁnition for LCPDF

δ(Ys − Yq )Kh (Xs − Xq )
q∈S
p(Ys |Xs ) =
Kh (Xs − Xq )
q∈S

Two reasons in the support for this new deﬁnition
• From the texture synthesis algorithm point of view
• From a numerical point of view

0.018

0.016
Probability calculated with Gaussian kernel

D104 Near Regular Texture
0.014
D110 Stochastic Texture
0.012

0.01
probaility = 0.003544
0.008

0.006 Probability = 0.0002963

0.004

0.002

0
0 50 100 150 200 250 300
Gray Levels


Outline
Ph.D. Research Work

Results: D104

D104

8
2 6
4

10 14
12 16

22 24
18 20

Outline
Ph.D. Research Work

Results: D9

D9

5
1 3 7

9 11 13 15

19
17 21 23

Outline
Ph.D. Research Work

Results for Near-regular textures

Original Original NNMRF Our Synthesis Algorithm
NNMRF Our Synthesis Algorithm

D104 D20
o = 12 o = 18

D22 D34
o = 17
0 = 13


Outline
Ph.D. Research Work

Results for Stochastic textures

Our Synthesis Algorithm
Original NNMRF Original NNMRF
Our Synthesis Algorithm

D4 D9
O=9
O = 10

D93 D97
O = 16
O=9


Outline
Ph.D. Research Work

Results for Some other textures

Original NNMRF Our Synthesis Algorithm Original NNMRF Our Synthesis Algorithm

D53 D55
O = 14
O = 17

D82
D80
O = 14 O = 12


Outline
Ph.D. Research Work

Problem Deﬁnition


Outline
Ph.D. Research Work

Problem Deﬁnition

Texture synthesis

HOW ?


Outline
Ph.D. Research Work

Motivation

Applications of Inverse Texture Synthesis
• Understanding of Textures
• Content-based image/video retrieval
• Perceptual Image/Video compression
• Computer Vision Tasks
• Perceptual understanding of textures within the image
• Creation of animation – Collecting information from natural images/sequences
• Perceptual Understanding of temporal texture – such as, dance sequence, walk
sequence, music sequence etc.


Outline
Ph.D. Research Work

Deﬁnition of Objective Functions

According to N-MRF model
• Distance between two LCPDF’s evaluated w.r.t. both input and output texture
patches.


Outline
Ph.D. Research Work


patches.
• What distance function ?


Outline
Ph.D. Research Work


patches.
• Computationally expensive


Outline
Ph.D. Research Work


patches.
• Computationally expensive

According to N-MRF model: Intuitively
• M×N
• Size of the output patch
1
min{||Xs − Xq ||2 , where
• |S |
• Do the input neighborhood vectors s∈Sin
in
exist within output patch ? q ∈ Sout }


Outline
Ph.D. Research Work

Problem with these two objective functions

Scaled up versions of solutions

A B

approximately
same
Neighborhood
Deformation/variation within quot;Aquot;
difficult to find within quot;Bquot;


Outline
Ph.D. Research Work

Three objectives

• Say Sout = {s ∈ Sin , s.t., (si − i)2 ≤ M 2 and (sj − j)2 ≤ N 2 }
{i,j,M,N}
= S − Sout
• Define S
in
• First objective finds neighborhood from input texture within the output texture
• Second objective finds neighborhood from output texture within the input texture,
excluding the part of Sout

1
min{||Xs − Xq ||2 ; q ∈ Sout }
=
F1
|Sin |
s∈Sin
1 {i,j,M,N}
min{||Xq − Xs ||2 ; s ∈ Sin
= }
F2
|Sout |
q∈Sout
= M×N
F3


Outline
Ph.D. Research Work

Multi-objective Framework

 
 f1 (x) 
 

 f2 (x) 
 inequality constraints: gj (x) ≥ 0, j = 1, 2, ..., J
 
 
  such that
 
minx  . 
  equality constraints: hk (x) = 0, k = 1, 2, ..., K
.
 
 
.
 
 
solution space: xiL ≤xi ≤ xiU , i = 1, 2, ..., N
fm (x)


Outline
Ph.D. Research Work

Multi-objective Framework

 
 f1 (x) 
 

 f2 (x) 
 inequality constraints: gj (x) ≥ 0, j = 1, 2, ..., J
 
 
  such that
 
minx  . 
  equality constraints: hk (x) = 0, k = 1, 2, ..., K
.
 
 
.
 
 
solution space: xiL ≤xi ≤ xiU , i = 1, 2, ..., N
fm (x)

Domination
A vector x ∈ RN is said to dominate y ∈ RN if both the conditions stated below hold
true:

fi (x) ≤ fi (y), ∀i ∈ [1 . . . m]
∃ j ∈ [1 . . . m], such that, fj (x) fj (y)


Outline
Ph.D. Research Work

Pareto-optimal Front

Best in F 1
2nd objective function F 2
Worst in F 2

All are optimal
solutions Best in F 2
Worst in F 1

1st objective function F1


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