computationally efficient nmrf model based texture synthesis

OutlinePh.D. Research Work

Conclusion and Possible Future Directions

Fast NMRF based texture synthesis algorithms

Arnab [email protected]

April 16, 2009

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

Arnab Sinha [email protected] Fast NMRF based texture synthesis algorithms



Earlier MethodsResearch problems in NMRF-tex-syn algorithms

1 OutlineEarlier MethodsResearch problems in NMRF-tex-syn algorithms

2 Ph.D. Research WorkOrder Estimation from Fourier DomainReduction of Computational complexityOrder Estimation : RevisitedInverse Texture Synthesis

3 Conclusion and Possible Future Directions





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:

Given a small texture exempler, synthesizean arbitrary sample of texture, so that thesynthesized texture is visually similar to theoriginal sample.

Original D104 Texture

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





Texture synthesis: Difficulty

Figure: Spectrum of Natural Textures





Brief History of Models

Linear

Image Domain Model Mixed Domain Model Transformed Domain Model

Texture Synthesis Algorithm

Non−Linear Models

Hard−limitedProcessJacovitti et al. (1998)

Non−Linear

Circular Harmonic Func+ Hard−limited GaussianCampasi and Scarano (2002)

Heeger and Bergen (1995)

Portilla and Simoncelli (2000)3.

2. Zhu et al. (2000)1. Zhu et al. (1997)

Chellappa and Kashyap (1985)2D−NCAR,

2D−Wold Francos et al. (1993)

2D− MAEom (1998)

Efros and Leung (1999)

Ashikhmin (2001)Wei and Levoy (2000)

Tonietto et al. (2005)

We are workingwithin thisFramework

Sampling Process

Hidden Markov TreeFan and Xia (2003)

Zhang et al. (1998)Wavelet + AR

Paget and LongstaffNNMRF

(1998) Charalampidis (2006)

Gaussian

Mathematical Models

Intuitive Models

Non−Linearity Introducedby Histogram Equalization

Pixel−based

Patch−based

Popular Methods

Kwatra et al. (2003)

Wu et al. (2004)Patch−based sampling with wavelet transformation as a feature set for graph−cut algorithm





Description of N-MRF model

• S is the lattice

• Ys is the random variable at site s ∈ S

• Concept of Neighborhood system:

• s < ℵs

• r ∈ ℵs ⇔ s ∈ ℵr

• Circular neighborhood: ℵs = {r ; s.t., |r − s|2 ≤ o2}• say, Xs = {Yr ; r ∈ ℵs }

s = (i,j), site

1st order neighbors

2nd order neighbors{ }

• 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)





Description of N-MRF model: Kernel Density Estimation

Definition of KDE, [Scott(1992)]

• single dimensional: P(x) = 1N∑N

i=1 Kh(x − xi)





Description of N-MRF model: Kernel Density Estimation

Definition of KDE, [Scott(1992)]

• single dimensional: P(x) = 1N∑N

i=1 Kh(x − xi)

• multi-dimensional: P(X) = 1N∑N

i=1∏d

j=1 Khj (X(j) − Xi(j))

• where, in case of Gaussian kernel, Khj (X(j) − Xi(j)) = 1N√

2πhjexp{− (X(j)−Xi (j))

2

2h2j

}

• and, hj = σjN−1/(d+4)






Some definitions• Input texture field: {Ys }, where, s ∈ Sin

• Output texture field: {Yq}, where, q ∈ Sout

• Definition of LCPDF: P(Yq |Xq) =

∑s∈Sin

KhY(Ys−Yq)KhX

(Xs−Xq)∑s∈Sin

KhX(Xs−Xq)

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 = ΦqHX ≈ hΦq

• Updation rule for the confidence field• Cq = min{1, 1

|r∈ℵq |∑

r∈ℵq Cr + u × e}• where, u is a random number and e is a constant scale factor






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.

WqX q

{X }

X q X s( − ) Xq X s( − )t

X q X s( − ) Xq X s( − )t

Input texture

Output Texture

Input Neighborhood Vectors

Output Neighborhood

Confidence Field

Output Confidence Vector Vector

s

Matrix

Similarity Measure

N−MRF :

SinSout

q q

C

WL alg :





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

Order 4 Order 8 Order 14

Original Texture

computational complexity of texture synthesis algorithmis proportional to ’d’

Nei

ghbo

rhoo

d ve

ctor

dim

ensi

on ’d

’

Model order ’o’

0

1000

2000

3000

4000

5000

6000

7000

8000

0 10 20 30 40 50

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




Order Estimation from Fourier DomainReduction of Computational complexityOrder Estimation : RevisitedInverse Texture Synthesis

Order estimation from two fundamental frequencies

Xi

Yi

Yj

Xj

X

Y

o

c

b

a

d

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





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





D52D35

D20 D21

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

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

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





A new neighborhood system

Xi

Yi

Yj

Xj

X

Y

neighborhood

neighborhoodproposed Non−causal

circular Non−causal





Results: Proposed neighborhood system

Circular Neighborhood Proposed Neighborhood

D65

Circular Neighborhood Proposed Neighborhood

D104

D64 D95

D3 D67





Two approaches

Computational complexity affected by

1 the neighbourhood dimension, d, and

2 the number of input pixels, N

Reduction methodologies

1 Dimensionality reduction methodologies, e.g., Principal Component Analysis(PCA) – to reduce the effect of d

2 A data structure for fast search





With Dimensionality reduction methods

How the distance metric Ds,q looks after projection

(Xq − Xs)T Φq(Xq − Xs) ≈ (X̂q − X̂s)

T Φq(X̂q − X̂s)

= [PTr Pr (Xq − Xs)]

T Φq[PTr Pr (Xq − Xs)]

= [Pr (Xq − Xs)]T PrΦqPT

r [Pr (Xq − Xs)]

= (Zq − Zs)T Ψq(Zq − Zs)

• What is Ψq = PrΦqPTr ?

• Is it reducing the computational complexity ?





Simulated Annealing for Principal components

Original Ds,q

Ds,q = (Xq − Xs)T Φq(Xq − Xs)

where, Φq = DIAG{W1,W2, . . . ,Wd }

Proposed D̂s,q

D̂s,q = (Zq − Zs)T Φ̂q(Zq − Zs)

where, Φ̂q = 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”





Results

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

Texture Type Texture order d k η

NR D20 20 1516 60 21.9405NR D3 30 2820 580 3.7926NR D21 25 1960 56 29.2642NR D22 20 1256 177 6.3042NR D35 28 2452 287 6.6612NR D36 22 1516 258 5.1024

ST D7 27 2288 511 3.6438ST D13 24 1792 131 11.6006

NR+ST D18 32 3208 95 25.5666NR+ST D4 29 2628 465 4.4755NR+ST D5 29 2628 179 11.6262

IN/STR D15 23 1652 167 8.4897IN/STR D42 26 2120 293 5.9699





Results continued ...

D20

d = 1516 k=60

D7

d = 2288 k = 511

D21

d=1960 k=56

D22

d=1256 k = 177

D13

d=1792 k=131

D42

d=2120 k = 293

NNMRF Proposed Algorithm NNMRF Proposed Algorithm





With Fast Kernel Density Estimation

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

Source data vector

Target data vector

X

Y

R=100

To calculate KDEat this target pointwe only needthese two points

• Let Rn = {ts : ||ts − tn || ≤ R}• P(tn) = 1

N∑N

s=1 KH(ts − tn)

• P(tn) = 1N∑

s∈Rn KH(ts − tn)

• Let Nn = |{s < Rn}|

Err(tn ,R) =1N

∑

s<Rn

KH(ts − tn)

≤ Nn

NKH(R)

≤ KH(R)

• Rel err(R) =max(Err)

max(Probability)





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 ith cluster

• P(tn) = 1N∑M

i=1 KH(tn − Clusti)ni

• KDE of tn given the source data points at cluster centroids with a weight factorni/N

• flexibility ?





Improved Fast Gaussian Transform

• P(tn) =∑||tn−Clusti ||≤RIFGT

KH(tn − Clusti)f(tn ,Clusti)

• P(tn) = 1N∑||tn−Clusti ||≤RIFGT

∑s∈Clusti KH(tn − ts)

RIFGT

this overlapDue to

we need to consider thissource cluster

RIFGTthe radius threshold has to be To consider the source cluster

RIFGTTheand cluster shape

can vary with the overlap size

R

Source data clusters

Target data vectors

In effect it can include some source clusterwhich was not needed at all





kd-tree-based FKDE

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

e

e

e

e

e

e

ee e

e

e

ee

e e ee

ee

e

ee

e e e

e

e

e

e e

e

e

� -

6

?

-

6

6

?

--

--

X

Y

Span

inY

dir

ecti

on

Span in X direction

σX > σY

Partition

Sub-spaces

tn

Rmax_rec_err

Error Rrec_err,nR> R+ max_rec_err

ReconstructionR

Centroid

Hyper−Sphere

Hyper−Rectangle

Eigen vectors

Y

X





Why do we need another algorithm for FKDE

Table: Why do we need another algorithm for FKDE ?

C-FKDE KD-FKDEAdvantage Clustering algorithm provides more Due to the hyper-plane boundary,

compact representation of the data one can use original radius Rspace for strict error bound

Disadvantage optimal RIFGT has to estimated kd-tree is not a good clusteringfor every tn, maximum RIFGT algorithm, therefore it does notcan increase computational provide compact representation

complexity of the data space





Principal Directive Divisive Partitioning (PDDP)[Boley(1998)]

• project each source data point within the present space onto the first principaldirection (eigen vector corresponding to the largest eigen value).

• partition the present space into two sub-spaces with respect to the mean (ormedian) of the projected values.

IV V VI VII

II III

I

2nd

1st

3rd

4th 5th

6th

7th

8th

Hieararchical BoundariesSource data

III III

VI

VII

1st 2nd 3rd 4th 5th 6th 7th 8thV

IV

Tree structure of the nodes

Leaf nodes





FKDE based on PDDP

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

If D > R => return 0lbElse if D > R => return 0Else process

rec_err Else process both childrenif D > R => process left child

vnmr

vnmr

Target data vector

D

D

2nd Direction

1st Direction

Projection of terget point

Projection of source data

Source data vectors

rec_err

Dlb

(Process child) (Which child to process)

n(t )

Projected target data vectorProjected mean vector

(a) Target point is outside

Drec_errT is within left childDrec_err

Return 0Else

Process its children

If R <

If D > RReturn 0

For the right child

lb

If R < Drec_errElse Return 0

Else Process its children

D

T

1st Direction

2nd Direction(Boundary for Partition)

Dlb

If D < R process both children

(b) Target point is inside





FKDE based on PDDP


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



vnmr

vnmr

Target data vector

D

D

2nd Direction

1st Direction



Source data vectors

rec_err

Dlb


n(t )


(c) Target point is outside


Return 0Else


If R <

If D > RReturn 0

For the right child

lb



D

T

1st Direction


Dlb


(d) Target point is inside





FKDE based on PDDP


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

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



vnmr

vnmr

Target data vector

D

D

2nd Direction

1st Direction



Source data vectors

rec_err

Dlb


n(t )


(e) Target point is outside


Return 0Else


If R <

If D > RReturn 0

For the right child

lb



D

T

1st Direction


Dlb


(f) Target point is inside





Comparison between the FKDE algorithms

(g) Image consideredfor creating the data set

1

2

3

4

(i,j)

(i−1,j)

(i,j+1)

(i,j+1)

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

{1,2} X RGB ==> 6 dimensions{1} X RGB ==> 3 dimensions

(h) Creation of the dataspace

Figure: Data set creation for FKDE analysis

Table: Time comparison

Dimension 3 6 9 12KDE: Time (sec) 981.78 1204.77 2529.22 2668.58

PDDP-FKDE: Time (sec) 11.3 23.39 52.55 65.79KD-FKDE: Time (sec) 22.55 33.88 110.62 228.44

IFGT-FKDE: Time (sec) 399.79 425.45 209.33 384.08





Comparison between the FKDE algorithms

Table: Comparative analysis of FKDE algorithms

FKDE Dimension Maximum Maximum Relative Radiusalgorithms probability Error Error threshold

PDDP-FKDE 3 4.33531e − 05 0.0003e − 04 0.0007 14.54456 1.32977e − 10 0.0000e − 10 0.0000 28.16229 9.64769e − 18 0.0014e − 18 0.0001 47.569312 1.71367e − 23 0.0000e − 25 0.0000 59.9263

KD-FKDE 3 4.33531e − 05 0.0005e − 04 0.0011 14.54456 1.32977e − 10 0.0001e − 10 0.0000 28.16229 9.64769e − 18 0.0014e − 18 0.0001 47.569312 1.71367e − 23 0.0000e − 25 0.0000 59.9263

IFGT-FKDE 3 4.33531e − 05 0.2618e − 04 0.6040 18.18076 1.32977e − 10 0.5389e − 10 0.4053 35.20289 9.64769e − 18 0.4183e − 18 0.0434 59.461612 1.71367e − 23 0.132e − 25 0.0007703 74.9079





Computationally efficient Texture synthesis algorithm with FKDEalgorithms

Problems

1 how to include the effect of Wq (the temperature field) within the PDDP-based treestructure for the implementation of FKDE, and

2 there are two joint densities corresponding to {Yq ,Xq} and Xq; therefore, itrequires 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






Problems



Inclusion of Wq


• Ending State:{Wq,i = 1} ⇒ P(Xq; Wq) = P(Xq)⇒ Rnew = R






Problems



Inclusion of Wq


• Ending State:{Wq,i = 1} ⇒ P(Xq; Wq) = P(Xq)⇒ Rnew = R

Rnew =Rcq

abs(vn −mr ) ≤ Rnew

⇒ abs(vn −mr ) ≤ Rcq

⇒ abs(vn −mr )cq ≤ R





Results with comparisons

Original NNMRF IFGT kd−tree Proposed

D102

D49

D20






Original NNMRF IFGT Proposedkd−tree

D53

D104

D4

D82






Original NNMRF IFGT kd−tree Proposed

D110

D60

D93

D97






Table: Time taken in texture synthesis: input texture size 128 × 128 and output texture size256 × 256

NNMRF C-FKDE KD-FKDE PDDP-FKDEhours 8 5 8 6

minutes 7 55 34 0seconds 39 12 56 41





Maximum Log-Pseudo-likelihood

LPL =∑

s∈Sin

log[P(Ys |Xs)]

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





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

LPL is not saturating rather it is increasing

LPL

Order

−80000

−70000

−60000

−50000

−40000

−30000

−20000

−10000

0 5 10 15 20 25 30 35 40

D102: Near regularD104: Near regular

D110: StochasticD60: StochasticD93: Stochastic

LPL is getting saturated before 2 ordernd

Order

LPL

−450

−400

−350

−300

−250

−200

−150

−100

−50

0

0 1 2 3 4





Why does the original LPL measure, not saturate ?

Why ?

• original LCPDF: p(Yq |Xq) =∑

s∈S Kh (Ys−Yq)Kh (Xs−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.





A new definition for LCPDF

p(Ys |Xs) =

∑q∈S δ(Ys − Yq)Kh(Xs − Xq)∑

q∈S Kh(Xs − Xq)

Two reasons in the support for this new definition

• From the texture synthesis algorithm point of view

• From a numerical point of view

0 50 100 150 200 250 3000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Gray Levels

Pro

babi

lity

calc

ulat

ed w

ith G

auss

ian

kern

el

D104 Near Regular Texture

D110 Stochastic Texture

Probability = 0.0002963

probaility = 0.003544





Results: D104

D104

2 4 6 8

10 14 16

18 20 22 24

12





Results: D9

D9

31 5 7

11 13 15

17 19 21 23

9





Results for Near-regular textures

Original NNMRF Our Synthesis Algorithm

D104 D20

D22 D34

o = 12 o = 18

0 = 13 o = 17






Results for Stochastic textures


D4 D9

D93 D97

O = 10 O = 9

O = 16O = 9






Results for Some other textures


D53 D55

D80 D82

O = 17 O = 14

O = 12


O = 14





Problem Definition





Problem Definition

Texture synthesis

HOW ?





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.





Definition of Objective Functions

According to N-MRF model

• Distance between two LCPDF’s evaluated w.r.t. both input and output texturepatches.








• What distance function ?









• Computationally expensive









• Computationally expensive

According to N-MRF model: Intuitively

• Size of the output patch

• Do the input neighborhood vectorsexist within output patch ?

• M × N

• 1|Sin |∑

s∈Sinmin{||Xs − Xq ||2, where

q ∈ Sout }





Problem with these two objective functions

Neighborhood

A B

Scaled up versions of solutions

approximatelysame

difficult to find within "B"Deformation/variation within "A"





Three objectives

• Say Sout = {s ∈ Sin, s.t., (si − i)2 ≤ M2 and (sj − j)2 ≤ N2}• Define S {i,j,M,N}in = S − Sout

• 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

F1 =1|Sin |

∑

s∈Sin

min{||Xs − Xq ||2; q ∈ Sout }

F2 =1|Sout |

∑

q∈Sout

min{||Xq − Xs ||2; s ∈ S {i,j,M,N}in }

F3 = M × N





Multi-objective Framework

minx

f1(x)f2(x)...

fm(x)

such thatinequality constraints: gj(x) ≥ 0, j = 1, 2, ..., J

equality constraints: hk (x) = 0, k = 1,2, ...,K

solution space: xLi ≤xi ≤ xU

i , i = 1, 2, ...,N





Multi-objective Framework

minx

f1(x)f2(x)...

fm(x)

such thatinequality constraints: gj(x) ≥ 0, j = 1, 2, ..., J

equality constraints: hk (x) = 0, k = 1,2, ...,K

solution space: xLi ≤xi ≤ xU

i , i = 1, 2, ...,N

DominationA vector x ∈ RN is said to dominate y ∈ RN if both the conditions stated below holdtrue:

fi(x) ≤ fi(y), ∀i ∈ [1 . . .m]

∃ j ∈ [1 . . .m], such that, fj(x) < fj(y)





Pareto-optimal Front

F1

2F

2FF1

F1

2F

2nd

obje

ctiv

e fu

nctio

n

1st objective function

Worst in Best in

Best in Worst in

All are optimalsolutions





Genetic Algorithm

Why not classical optimization algorithms

F1 =1|Sin |

∑

s∈Sin

min{||Xs − Zt ||2; t ∈ Sout }

F2 =1|Sout |

∑

t∈Sout

min{||Zt − Xs ||2; s ∈ S {i,j,M,N}in }

F3 = M × N

Genetic Algorithm

• It is an intuitive algorithm, biologically inspired,

• Based upon the phylosophy of survival of the fittest

• Crossover, mutation operators

• There are many algorithms, we choose NSGA [Srinivas and Deb(1994)]





Redundancy of objectives

F1 F2 F3

F1 + − −F2 − + +F3 − + +

Table: Conflict matrix: In each case of Fabric.0014, D22, Fabric.0009, and Grass textures the sametrait has been observed.

600 700 800 9000

50

100

150

200

250

300

F1

F2

(a) F1 conflicts F2

0 2000 4000 6000 80000

50

100

150

200

250

300

F3

F2

(b) non-conflicting F2 and F3

0 2000 4000 6000 8000600

650

700

750

800

850

900

F3

F1

(c) F1 conflicts F3

Figure: Two-dimensional views of Pareto-optimal front for the Fabric.0014 texture: the positivity ornegativity of the cross correlation between the objective functions can be understood





Results: D22

2500 3000 3500 40000

100

200

300

400

1

2

3 4 5

F1

F2

ExtractedExemplerOriginal Textured

Region

ExtractedExempler

SynthesisResult

SynthesisResult





Results: Fabric.0009

F1

1st solution

2nd

3rd

Original Textured Region

F2

500

1000

1500

2000

2500

3000

1900 1950 2000 2050 2100 2150 2200 2250





Results: Grass

F1

Synthesized textures fromthe extracted solutions

1st solution

2nd solution

3rd solution

Original Textured Region

F2

0

20

40

60

80

100

120

140

160

180

200

260 280 300 320 340 360 380 400 420




Conclusion• The problem of Order estimation• The problem of computational complexity reduction

• With the incorporation of Dimensionality Reduction methodology, e.g., PCA• With fast estimation of Kernel Density Estimation with an improvised data structure

• An inverse application of texture synthesis with NMRF model• Objective functions• Analysis of objective functions• Multi-objective framework

Possible Future Directions• Order estimation for in-homogeneous textures or globally varying textures

• three-dimansional variation of surface• structural variation• time specific variation

• How to incorporate a control field within the texture analysis

• How to choose a particular solution from the multi-objective framework, dependingupon the application in hand




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computationally efficient nmrf model based texture synthesis

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synthesized texture

research work conclusion

denes texture

syn algorithms conclusion

total region texture

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research workresearch

s s denition of mrf