University of Groningen Institute of Mathematics and Computing Science Universitá degli Studi di Roma Tre Dipartimento di Elettronica Applicata Well Posed non-Iterative Edge and Corner Preserving Smoothing For Artistic Imaging Giuseppe Pápari, Nicolai Petkov, Patrizio Campisi

Photographical image

Output of the proposed operator

Smoothing out texture while preserving edges Input imageGaussian smoothingProposed operator Co ntents Kuwahara Filter and Generalizations Limitations Proposed Operator Results and Comparison Discussion

Kuwahara Filter and Generalizations Four local averages: Four local standard deviations: Kuwahara output For each pixel, value of m i that corresponds to the minimum standard deviation Generic pixel of the input image

Kuwahara 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 Generalizations Local averaging  Smoothing Flipping due to Minimum Variance Criterion  Edge Preserving

Kuwahara Filter and Generalizations An example Input image Kuwahara output Artifacts on texture

Kuwahara 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) »Criterion: minimum standard deviation Connections with the PDEs theory and morphological analysis

Kuwahara Filter and Generalizations LimitationsLimitations Proposed Operator Results and Comparison Discussion

Limitations Artifacts (partially eliminable with weighted averages) Not mathematically well defined Equal standard deviations s i   Devastating instability in presence of noise

Limitations Simple one-dimensional example Input signal I(t) I(t) = kt Local averages I  t tTtTt+T Negative offset 1D Kuwahara filtering  Two sub-windows w 1 and w 2 t w2w2 t*t* I(t)I(t) w1w1

Limitations Simple one-dimensional example Input signal I(t) I(t) = kt Local averages I  t tTtTt+T Negative offset Positive offset 1D Kuwahara filtering  Two sub-windows w 1 and w 2 t w2w2 t*t* I(t)I(t) w1w1

I(t) = kt Local standard deviations Equal standard deviations I  t tTtTt+TLimitations Simple one-dimensional example Local averages m 1 (t), m 2 (t) Input signal I(t) Local std. dev. s 1 (t), s 2 (t) 1D Kuwahara filtering  Two sub-windows w 1 and w 2 t w2w2 t*t* I(t)I(t) w1w1

Limitations Input imageKuwahara filtering Proposed approach Synthetic two-dimensional example

KuwaharaLimitations Natural image example Input image Gauss-Kuwahara Shadowed area Depleted edge Our approach

Limitations Ill-posedness of the minimum variance criterion. Devastating effects in presence of noisy shadowed areas. We propose Different weighting windows w i A different selection criterion instead of the minimum standard deviation

Kuwahara Filter and Generalizations Limitations Proposed OperatorProposed Operator Results and Comparison Discussion

Proposed Operator Gaussian mask divided in N sectors  N weighting windows N local averages and local standard deviations computed as convolutions Weighting windows

Proposed Operator Selection criterion q  Only the minimum s i survives  Criterion and value Output: » Weighted average of m i » Weights equal to proportional to (s i )  q (q is a parameter) Normalization High variance  small coefficient (s i )  q  No undetermination in case of equal standard deviations!

Proposed Operator Particular cases Equal standard deviations: s 1 = s 2 = … = s N Gaussian smoothing One standard deviation is equal to zero: s k = 0 Several values of s i are equal to zero  = Arithmetic mean of the corresponding values of m i.

Proposed Operator Edge Half of the sectors have s i = 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 Operator Color images 3 sets of local averages and local standard deviations, one for each color component with Same combination rule Not equivalent to apply the operator to each color component separately

Proposed Operator Independence on the color space Input imageRGBYC r C b L*a*b*

Proposed Operator Why independence? Linear transform.  independent Nonlinear transf.  almost independent for homogeneous regions Local averages

Proposed Operator Why independence? Linear transform.  independent Nonlinear transf.  almost independent for homogeneous regions 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 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

5  5 median filter

Results and comparison Larger set of results and Matlab implementation available at Graphical interface

Kuwahara Filter and Generalizations Limitations Proposed Operator Results and Comparison DiscussionDiscussion

Discussion 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

Discussion Proposed approach » Different windows » Different criterion  Mathematically well defined operator  Adaptive choice of the most appropriate sub-regions. Our approachGauss-KuwaharaKuwahara

Discussion Limitations » Lines are thinned » Small objects are not preserved

References 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