understanding optimising and extending data compression with anisotropic diffusion galic et al journal of mathematical imaging and vision 31 255 269 2008 have shown that compression based on edge enhancing anisotropic diffusion eed can outperform the quality of jpeg for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation however the reasons for the good performance of eed remained unclear and they could not outperform the more advanced jpeg 2000 the goals of the present paper are threefold firstly we investigate the compression qualities of various partial differential equations this sheds light on the favourable properties of eed in the context of image compression secondly we demonstrate that it is even possible to beat the quality of jpeg 2000 with eed if one uses specific subdivisions on rectangles and several important optimisations these amendments include improved entropy coding brightness and diffusivity optimisation and interpolation swapping thirdly we demonstrate how to extend our approach to 3 d and shape data experiments on classical test images and 3 d medical data illustrate the high potential of our approach

every state on semisimple mv algebra is integral integral representation theorem will be established for finitely additive probability measures states on semisimple mv algebras this result generalizes the well known theorem of butnariu and klement in case of order continuous states on tribes of fuzzy sets precisely it will be demonstrated that every state on a separating clan of continuous fuzzy sets arises as an integral with respect to a unique borel probability measure the key technique leading to this result exploits the geometrical topological properties of the state space the set of all states on every mv algebra forms a bauer simplex

image coding based on a fractal theory of iterated contractive image transformations the author proposes an independent and novel approach to image coding based on a fractal theory of iterated transformations the main characteristics of this approach are that i it relies on the assumption that image redundancy can be efficiently exploited through self transformability on a block wise basis and ii it approximates an original image by a fractal image the author refers to the approach as fractal block coding the coding decoding system is based on the construction for an original image to encode of a specific image transformation a fractal code which when iterated on any initial image produces a sequence of images that converges to a fractal approximation of the original it is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0 5 1 0 b pixel the fractal block coder has performance comparable to state of the art vector quantizers

dynamic finite state vector quantization of digital images a vector quantization vq scheme with finite memory called dynamic finite state vector quantization dfsvq is presented the encoder consists of a large codebook so called super codebook where for each input vector a fixed number of its codevectors are chosen to generate a much smaller codebook sub codebook this sub codebook represents the best matching codevectors that could be found in the super codebook for encoding the current input vector the choice for the codevectors in the sub codebook is based on the information obtained from the previously encoded blocks where directional conditional block probability histogram matrices are used in the selection of the codevectors the index of the best matching codevector in the sub codebook is transmitted to the receiver an adaptive dfsvq scheme is also proposed in which when encoding an input vector first the sub codebook is searched for a matching codevector to satisfy a pre specified waveform distortion if such a codevector is not found in tile current sub codebook then the whole super codebook is checked for a better match if a better match is found then a signaling flag along with the corresponding index of the codevector is transmitted to the receiver both the dfsvq encoder and its adaptive version are implemented experimental results for several monochrome images with a super codebook size of 256 or 512 and different sub codebook sizes are presented

fast surface interpolation using hierarchical basis functions an alternative to multigrid relaxation that is much easier to implement and more generally applicable is presented conjugate gradient descent is used in conjunction with a hierarchical multiresolution set of basis functions the resultant algorithm uses a pyramid to smooth the residual vector before the direction is computed simulation results showing the speed of convergence and its dependence on the choice of interpolator the number of smoothing levels and other factors are presented the relationship of this approach to other multiresolution relaxation and representation schemes is also discussed

on non associative generalizations of mv algebras and lattice ordered commutative loops the aim of the paper is to describe the connections between lattice ordered commutative loops and certain basic algebras which are a non associative generalization of mv algebras and are related to commutative semicopulas this extends the well known equivalence between lattice ordered abelian groups with strong order unit and mv algebras

efficient region segmentation on compressed gray images using quadtree and shading representation abstract image segmentation which partitions the image into homogeneous regions is a fundamental operation in image processing suppose the input gray image with size n n has been compressed into the compressed image via quadtree and shading representation assume that the number of blocks in the representation is b commonly b n 2 due to the compression effect this paper first derives some closed forms for computing the mean variance of any block and for calculating the two statistical measures of any merged region in o 1 time it then presents an efficient o b b time algorithm for performing region segmentation on the compressed image directly where b is the inverse of the ackerman s function and is a very slowly growing function with the same time complexity our results extend the pioneering results by dillencourt and samet from the map image to the gray image in addition with four real images experimental results show that our proposed algorithm has about 55 4 execution time improvement ratio when compared to the previous fastest region segmentation algorithm by fiorio and gustedt whose o n 2 time algorithm is run on the original n n gray image

states on commutative basic algebras the paper deals with states on commutative basic algebras that are a non associative generalization of mv algebras or in other words the algebraic semantics for a fuzzy logic which generalizes the lukasiewicz logic in that the conjunction is not associative states are defined in the same way as mundici s states on mv algebras as normalized finitely additive 0 1 valued functions and some results analogous to the results that are known from mv algebras are proved

an efficient algorithm for computing moments on a block representation of a grey scale image computing lower order moments is important in image processing suppose the input grey image with size nxn has been compressed into the block representation where the number of blocks is k commonly k

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

image compression by b tree triangular coding this paper describes an algorithm for still image compression called b tree triangular coding bttc the coding scheme is based on the recursive decomposition of the image domain into right angled triangles arranged in a binary tree the method is attractive because of its fast encoding o n log n and decoding spl theta n where n is the number of pixels and because it is easy to implement and to parallelize experimental studies indicate that bttc produces images of satisfactory quality from a subjective and objective point of view one advantage of bttc over jpeg is its shorter execution time

states on semi divisible generalized residuated lattices reduce to states on mv algebras a semi divisible residuated lattice is a residuated lattice l satisfying an additional condition weaker than that of divisibility such structures are related to mathematical fuzzy logic as well as to extended probability theory by the fact that the subset of complemented elements induces an mv algebra we define generalized residuated lattices by omitting commutativity of the corresponding monoidal operation and study semi divisibility in such structures we show that given a good generalized residuated lattice l the set of complemented elements of l denoted by mv l forms a pseudo mv algebra if and only if l is semi divisible maximal filters on a semi divisible generalized residuated lattice l are in one to one correspondence with maximal filters on mv l we study states on semi divisible generalized residuated lattices riecan states on a semi divisible generalized residuated lattice l are determined by riecan states on mv l the same holds true for bosbach states whenever l is a good divisible generalized residuated lattice extremal riecan states on a semi divisible generalized residuated lattice l are in one to one correspondence with maximal and semi normal filters on l

measures states and de finetti maps on pseudo bck algebras in this paper we extend the notions of states and measures presented in dvurecenskij and pulmannova 2000 12 to the case of pseudo bck algebras and study similar properties we prove that under some conditions the notion of a state in the sense of dvurecenskij and pulmannova 2000 12 coincides with the bosbach state and we extend to the case of pseudo bck algebras some results proved by kuhr only for pseudo bck semilattices we characterize extremal states and show that the quotient pseudo bck algebra over the kernel of a measure can be embedded into the negative cone of an archimedean group additionally we introduce a borel state and using results by kuhr and mundici 2007 28 we prove a relationship between de finetti maps bosbach states and borel states

how to choose interpolation data in images we introduce and discuss shape based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the laplace equation the shape analysis is done in the framework of gamma convergence from two different points of view first we propose a continuous pde model and get pointwise information on the importance of each pixel by a topological asymptotic method second we introduce a finite dimensional setting into the continuous model based on fat pixels balls with positive radius and study by gamma convergence the asymptotics when the radius vanishes in this way we obtain relevant information about the optimal distribution of the best interpolation pixels we show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations numerical computations are presented that confirm the usefulness of our theoretical findings for pde based image compression

on spectral disjointness of powers for rank one transformations and mobius orthogonality abstract we study the spectral disjointness of the powers of a rank one transformation for a large class of rank one constructions including those for which the cutting and stacking parameters are bounded and other examples such as rigid generalized chacon s maps and katok s map we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum our proof involves the existence in the weak closure of u t k k z of sufficiently many analytic functions of the operator u t then we apply these disjointness results to prove sarnak s conjecture for the possibly non uniquely ergodic symbolic models associated to these rank one constructions all sequences realized in these models are orthogonal to the mobius function

mobius disjointness for homogeneous dynamics we prove sarnak s mobius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected lie groups namely we show that if g is any such group g a lattice and u g an ad unipotent element then for every x g and every function f continuous on the 1 point compactification of g the sequence f xun cannot correlate with the mobius function on average

interpolative btc image coding with vector quantization the authors suggest two interpolative block truncation coding btc image coding schemes with vector quantization and median filters as the interpolator the first scheme is based on quincunx subsampling and the second one on every other row and every other column subsampling it is shown that the schemes yield a significant reduction in bit rate at only a small performance degradation and in general better channel error resisting capabilities as compared to the absolute moment btc the methods are further demonstrated to outperform the corresponding btc schemes with pure vector quantization at the same bit rate and require minimal computations for the interpolation

mixing of all orders of lie groups actions we show that for lie groups whose adjoint representation reflects their topology mixing implies mixing of all orders in particular we prove that mixing actions of a semisimple lie group are mixing of all orders answering a conjecture of b marcus