Affine Invariant Total Variation Models

Transcription

1 Affine Invariant Total Variation Models Helen Balinsky, Alexander Balinsky Media Technologies aboratory HP aboratories Bristol HP-7-94 Jne 6, 7* Total Variation, affine restoration, Sobolev ineqality, image processing This report relates to the field of image restorations and featres extracting from noisy and blrred images. Since their introdction in a classical paper by Rdin, Osher and Fatemi, Total Variation (TV) minimising models have become one of the most poplar and sccessfl tools for image restorations. Whilst invariance nder affine transformations is very important for many image processing tasks, the total variation fnctional is not invariant nder general affine transformation. In the crrent report we introdce for the first time a new affine invariant reglarization fnctional which has many properties similar to total variation and can be sed for affine invariant denoising and restoration tasks. The explicit formla for calclation of this reglarization fnctional is given. * Internal Accession Date Only Cardiff School of Mathematics, Cardiff University, Cardiff, Wales Copyright 7 Hewlett-Packard Development Company,.P. Approved for External Pblication

2 Affine Invariant Total Variation Models Alexander Balinsky Cardiff School of Mathematics Cardiff University Cardiff Helen Balinsky Media Technologies aboratory HP aboratories Bristol Keywords: Total Variation, affine restoration, Sobolev ineqality, image processing Abstract This report relates to the field of image restorations and featres extracting from noisy and blrred images. Since their introdction in a classical paper by Rdin, Osher and Fatemi [], Total Variation (TV) minimising models have become one of the most poplar and sccessfl tools for image restorations. Whilst invariance nder affine transformations is very important for many image processing tasks, the total variation fnctional is not invariant nder general affine transformation. In the crrent report we introdce for the first time a new affine invariant reglarization fnctional which has many properties similar to total variation and can be sed for affine invariant denoising and restoration tasks. The explicit formla for calclation of this reglarization fnctional is given. Introdction Variational models have been extremely sccessfl in a wide variety of restoration problems (denoising, deblrring, blind deconvoltion, and impainting), and remain one of the most active areas of research in image processing and compter vision. Variational models exhibit the soltion of these problems as minimizers of appropriately chosen energy fnctionals. Assme that a given image is noisy and blrred: = K+ n. Then the Bayesian restoration energy proposed in [] is () E [ ] = TV ( ) +λe [ ]

3 for a certain tning parameter λ >, and TV( ): = denotes the total variation of. Used as a reglarization term, the TV fnctional is particlarly relevant in recovering piecewise smooth fnctions withot smoothing the sharp discontinities, in contrast with other reglarization fnctionals generally based on a qadratic norm. The revoltionary aspect of the model () is its reglarization term TV( ) that allows for discontinities, bt at the same time disfavors oscillations. However, this reglarization term is not invariant nder general linear transformation. This limits application of this method for image registration problems and for analysis of images obtained nder different angles. Affine invariant total variation energy Since we are going to introdce an affine invariant energy we will think abot R as an abstract two dimensional real vector space V withot any preselected basis. et : V R be any smooth enogh fnction on V with compact spport. V is commtative locally compact grop, so it has a well defined invariant Haar measre µ ( dx) defined p to constant. This Haar measre is a mltiple the standard ebesge measre, bt we want to constrct everything coordinate free. Now, sing the fnction and the measre µ ( dx), we define a norm on the same vector space V. et v V be any vector from V. The derivative v of the fnction in direction of the vector v is defined as sal by d( x+ tv) ( v )( x) =. dt t= v does not involve any inner prodct or norm on V. We defined v as () v = ( v)( x) µ ( dx). V Now ( V, ) is the two-dimensional Banach space that we shall associate with. Its nit ball B( ) = { v V : v } is a symmetric convex body in V and or new affine invariant total variation energy of is defined as ATV ( ) =, Vol( B ( )) Vol B is jst µ measre of B ( ). This ATV energy is obviosly where ( ( )) invariant nder linear measre preserving transformation. Moreover, as was shown in [4] the following Sobolev-type ineqality holds ATV ( ) const, which opens the way for sing fnctional analysis techniqes to analyse these ATVtype models. For practical calclation another expression for Vol( B ( )) is more sefl. If we introdce any Eclidean strctre on V then

4 where Vol( B ( )) = v dv, S is the nit circle with canonical measre dv and v = v + v orthogonal coordinate system ( x= ( x, x ), v= ( v, v ) ). S x x in the Remark. ) ATV ( ) TV ( ). To see this, note that from the Hölder ineqality 3/ and Fbini s theorem we have v v dv dv = x ( ), v dxdv R = x ( ), v dvdx = v v dv = x ( ) dx, S S S R S S R where v is any fixed nit vector. From this we have TV ( ) which Vol( B ( )) implies ATV ( ) TV ( ). 3/ ) If data has additional smoothness then we can define p-version of ATV by sing p norm in (). Conclsions In this report we have introdced new affine invariant total variation energy. Using this energy as a reglarization term reslts in affine invariant total variation models. This ATV energy has the following Advantages: ) ATV energy is bonded from above by total variation energy ATV() p -3/ TV() and ths is capable of handling edges. This is becase edges are precisely the case of finite TV. ) Stability and efficiency of TV models heavily depend on Sobolev ineqality (Sobolev ineqality allows to prove existence of decompositions and to control error in nmerical schemes). New proposed ATV energy satisfies the same ineqality. 3) Since ATV is affine invariant by constrction, the reslts of restoration will also be affine invariant. 4) Despite of being affine invariant ATV energy can be effectively calclated in any orthogonal coordinate system. 3

5 References []. Rdin and S. Osher, Total variation based image restoration with free local constrains, In Proc. st IEEE ICIP, Volme, pages 3-35, 994. []. Rdin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D., 6:59-68, 99. [3] T. F. Chan and J. Shen, Image Processing and Analysis: PDE, wavelets, and stochastic methods, SIAM, 5. [4] G. Zhang, The affine Sobolev ineqality, J. Differential Geom. 53 (999), no.,