Variational Texture Synthesis with Sparsity and Sp

Authors: Guillaume Tartavel Yann Gousseau Gabriel Peyré

Publish Date: 2014/11/18

Volume: 52, Issue: 1, Pages: 124-144

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Abstract

This paper introduces a new approach for texture synthesis We propose a unified framework that both imposes first order statistical constraints on the use of atoms from an adaptive dictionary as well as second order constraints on pixel values This is achieved thanks to a variational approach the minimization of which yields local extrema each one being a possible texture synthesis On the one hand the adaptive dictionary is created using a sparse image representation rationale and a global constraint is imposed on the maximal number of use of each atom from this dictionary On the other hand a constraint on second order pixel statistics is achieved through the power spectrum of images An advantage of the proposed method is its ability to truly synthesize textures without verbatim copy of small pieces from the exemplar In an extensive experimental section we show that the resulting synthesis achieves state of the art results both for structured and small scale texturesWe would like to thank the reviewers for their valuable suggestions to improve and complete this paper We thank the authors of the VisTex database 31 for the set of textures they publicly provide Gabriel Peyré acknowledges support from the European Research Council ERC project SIGMAVisionThe case xcdot y=0 leads to A=0 and f being constant In other cases A0 the minimums psi mathrm min of the functions are strict and satisfy emathrm ipsi mathrm min=emathrm itheta =fracxcdot yleft xcdot yright hence the expression of hatu sm given in 8Updating W with the optimal weight w=varPhi nk is only one operation Updating R is in fancyscriptOL operations since only the L coefficients of the kth column is affected Similarly updating varPhi is in fancyscriptON log KN operations since the N updated coefficients must be relocated in the heap and we recall that D 0T D 0 is precomputedConclusion Since we assume that Sll Lle Nll K previous bounds can be simplified and in particular log KNin fancyscriptOlog K The initializations and precomputations are in fancyscriptOKNL Updating the heap is in fancyscriptOlambda SKN log K building and searching the heap are cheaper Hence the complexity 35

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