University of Groningen Institute of Mathematics and Computing Science

Universitá degli Studi di Roma TreDipartimento di Elettronica Applicata

Well Posed non-IterativeWell Posed non-Iterative

Edge and Corner Preserving SmoothingEdge and Corner Preserving Smoothing

For Artistic ImagingFor Artistic Imaging

Giuseppe Pápari, Nicolai Petkov, Patrizio Campisi

Photographical image

Output of the proposed operator

Smoothing out texture while preserving edges

Input image Gaussian smoothing Proposed operator

CoContentsntents

•Kuwahara Filter and Generalizations

•Limitations

•Proposed Operator

•Results and Comparison

•Discussion

Kuwahara Filter and GeneralizationsKuwahara Filter and Generalizations

•Four local averages: ,

1, , d d

i

i

Q x y

m x y IQ

22

,

1, , , d d

i

i i

Q x y

s x y I m x yQ

•Four local standard deviations:

Kuwahara output

For each pixel, value of mi that corresponds to the minimum standard deviation

Q1 Q2

Q3 Q4

2a

x

y

Generic pixel of the input image

Kuwahara Filter and GeneralizationsKuwahara Filter and Generalizations

Edge

Only the most homogeneous region is taken into account.

No smoothing across the edge

(x,y) = 1

Central pixel on the white side of the edge

(x,y) = 0

Central pixel on the black side of the edge

Kuwahara Filter and GeneralizationsKuwahara Filter and Generalizations

• Local averaging Smoothing

• Flipping due to Minimum Variance Criterion Edge Preserving

Kuwahara Filter and GeneralizationsKuwahara Filter and Generalizations

An example

Input image

Kuwahara output

Artifacts

on texture

Kuwahara Filter and GeneralizationsKuwahara Filter and Generalizations

Generalizations

•Number and shape of the sub-regions

»Pentagons, hexagons, circles

»Overlapping

•Weighted local averages (reducing the Gibbs phenomenon)

»Gaussian-Kuwahara

• New class of filters (Value and criterion filter structure)

»N local averages and local standard deviations (computed as convolutions)

2 2 2,i i i i im I w s I m w

»Criterion: minimum standard deviation

•Connections with the PDEs theory and morphological analysis

•Kuwahara Filter and Generalizations

•LimitationsLimitations

•Proposed Operator

•Results and Comparison

•Discussion

LimitationsLimitations

• Artifacts

(partially eliminable with weighted averages)

• Not mathematically well defined

Equal standard deviations si • Devastating instability in presence of noise

LimitationsLimitations

Simple one-dimensional example

Input signal I(t)•I(t) =

kt

Local averagesI

ttT t+T

2

1d

t

t T

k ktT

kT

Negative offset

• 1D Kuwahara filtering Two sub-windows w1 and w2 t

w2

t*

I(t)

w1

LimitationsLimitations

Simple one-dimensional example

Input signal I(t)•I(t) =

kt

Local averagesI

ttT t+T

2

1d

t

t T

k ktT

kT

Negative offset

2

1d

t T

t

kt

Tk k

T

Positive offset

• 1D Kuwahara filtering Two sub-windows w1 and w2 t

w2

t*

I(t)

w1

•I(t) = kt

Local standard deviations

2 2

1 2 4

k Ts t s t

•Equal standard deviations

I

ttT t+T

LimitationsLimitations

Simple one-dimensional example

Local averages m1(t), m2(t)

Input signal I(t) Local std. dev. s1(t), s2(t)

• 1D Kuwahara filtering Two sub-windows w1 and w2 t

w2

t*

I(t)

w1

LimitationsLimitations

Input image Kuwahara filtering Proposed approach

Synthetic two-dimensional example

Kuwahara

LimitationsLimitations

Natural image example

Input image Gauss-Kuwahara

Shadowed

area

Depleted

edge

Our approach

LimitationsLimitations

Ill-posedness of the minimum variance criterion.

Devastating effects in presence of noisy shadowed areas.

We propose

• Different weighting windows wi

• A different selection criterion instead of the minimum standard deviation

•Kuwahara Filter and Generalizations

•Limitations

•Proposed OperatorProposed Operator

•Results and Comparison

•Discussion

Proposed OperatorProposed Operator

• Gaussian mask divided in N sectors N weighting windows

• N local averages and local standard deviations computed as convolutions

2 2 2,i i i i im I w s I w m

Weighting windows

2 2

222

1, ,

2

x y

ii

w x y g x y e

Proposed OperatorProposed Operator

Selection criterion

• q Only the minimum si survives Criterion and value

• Output:

ii

qi

qi

i

s

ms

» Weighted average of mi

» Weights equal to proportional to (si)q (q is a parameter)

Normalization

• High variance small coefficient (si)q

No undetermination in case of equal standard deviations!

Proposed OperatorProposed Operator

Particular cases

• Equal standard deviations: s1 = s2 = … = sN

1i

i

mN

Gaussian smoothing

I g

• One standard deviation is equal to zero: sk = 0

km

• Several values of si are equal to zero

= Arithmetic mean of the corresponding values of mi.

Proposed OperatorProposed Operator

• Edge

Half of the sectors have si = 0. The other ones are not considered

An example

• Edgeless areas:

All std. dev. similar Gaussian smoothing (no Gibbs phenomenon)

• Corner preservation

Automatic selection of the prominent sectors

Proposed OperatorProposed Operator

Color images

• 3 sets of local averages and local standard deviations, one for each color component

with

qi i

iqi

i

s

s

m

• Same combination rule

3 21 2 3

1, , ,

T ci i i i i i

cm m m s s

m

Not equivalent

to apply the operator

to each color component separately

Proposed OperatorProposed Operator

Independence on the color space

Input image RGB YCrCb L*a*b*

Proposed OperatorProposed Operator

Why independence?

qi i

iqi

i

s

s

m

Linear transform. independent

Nonlinear transf. almost independent for homogeneous regions

• Local averages

Proposed OperatorProposed Operator

Why independence?

Linear transform. independent

Nonlinear transf. almost independent for homogeneous regions

qi i

iqi

i

s

s

m

• Local averages

Low for homogeneous regions.

The degree of homogeneity of a region does not depend on the color space.

• Local standard deviations

•Kuwahara Filter and Generalizations

•Limitations

•Proposed Operator

•Results and ComparisonResults and Comparison

•Discussion

Results and comparisonResults and comparison

Existing algorithm for comparison

• Kuwahara filter and generalizations

• Bilateral filtering

• Morphological filters

• Median filters

Input image

Proposed approach

Gauss-Kuwahara filter

Input image (blurred)

Proposed approach (deblurred)

Bilateral filtering (not deblurred)

Input image

Proposed approach

Morphological closing (Struct. elem.: Disk of radius 5px)

Morphological area open-closing

Input image

Morphological area open-closing

Proposed approach

Input image

Proposed approach

Kuwahara Filter

Morphological area open-closing

Input image

Proposed approach

Bilateral Filtering

Input image

proposed aproach

55 median filter

Results and comparisonResults and comparison

Larger set of results and Matlab implementation available at

http://www.cs.rug.nl/~imaging/artisticsmoothing

• Graphical interface

•Kuwahara Filter and Generalizations

•Limitations

•Proposed Operator

•Results and Comparison

•DiscussionDiscussion

DiscussionDiscussion

• Edge/corner preserving smoothing

• Undetermination for equal standard deviation

» Instability in presence of noise

» Discontinuities in presence of shadowed areas

• Criterion and value filter structure

» Local averaging Smoothing

» Minimum variance criterion Edge preserving

DiscussionDiscussion

• Proposed approach

» Different windows

» Different criterion

Mathematically well defined operator

Adaptive choice of the most appropriate sub-regions.

Our approachGauss-KuwaharaKuwahara

DiscussionDiscussion

• Limitations

» Lines are thinned

» Small objects are not preserved

ReferencesReferences

•G. Papari, N. Petkov, P. Campisi

Artistic Edge and Corned Preserving Smoothing

To appear on IEEE Transactions on Image Processing, 2007

•G. Papari, N. Petkov, P. Campisi

Edge and Corned Preserving Smoothing for Artistic Imaging

Proceedings SPIE 2007 Image Processing: Algorithms and Systems, San Jose, CA