shape illumination and reflectance from shading a fundamental problem in computer vision is that of inferring the intrinsic 3d structure of the world from flat 2d images of that world traditional methods for recovering scene properties such as shape reflectance or illumination rely on multiple observations of the same scene to overconstrain the problem recovering these same properties from a single image seems almost impossible in comparison there are an infinite number of shapes paint and lights that exactly reproduce a single image however certain explanations are more likely than others surfaces tend to be smooth paint tends to be uniform and illumination tends to be natural we therefore pose this problem as one of statistical inference and define an optimization problem that searches for the most likely explanation of a single image our technique can be viewed as a superset of several classic computer vision problems shape from shading intrinsic images color constancy illumination estimation etc and outperforms all previous solutions to those constituent problems

the aα spectral radius of trees and unicyclic graphs with given degree sequence abstract for any real 0 1 a g d g 1 a g is the a matrix of a graph g where a g is the adjacency matrix of g and d g is the diagonal matrix of the degrees of g this paper presents some extremal results about the spectral radius 1 a g of a g that generalize previous results about 1 a0 g and 1 a 1 2 g in this paper we study the behavior of the a spectral radius under some graph transformations for 0 1 as applications we show that the greedy tree has the maximum a spectral radius in g d when d is a tree degree sequence firstly furthermore we determine that the greedy unicyclic graph has the largest a spectral radius in g d when d is a unicyclic graphic sequence where g d g g is connected with d as its degree sequence

modulated phase of a potts model with competing binary interactions on a cayley tree we study the phase diagram for potts model on a cayley tree with competing nearest neighbor interactions j1 prolonged next nearest neighbor interactions jp and one level next nearest neighbor interactions jo vannimenus proved that the phase diagram of ising model with jo 0 contains a modulated phase as found for similar models on periodic lattices but the multicritical lifshitz point is at zero temperature later mariz et al generalized this result for ising model with jo 0 and recently ganikhodjaev et al proved similar result for the three state potts model with jo 0 we consider potts model with jo 0 and show that for some values of jo the multicritical lifshitz point be at non zero temperature we also prove that as soon as the same level interactionjo is nonzero the paramagnetic phase found at high temperatures for jo 0 disappears while ising model does not obtain such property to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular case previous work by ganikhodjaev et al for jo 0 at vanishing temperature the phase diagram is fully determined for all values and signs of j1 jp and jo at finite temperatures several interesting features are exhibited for typical values of jo j1

an exact solution on the ferromagnetic face cubic spin model on a bethe lattice the lattice spin model with q component discrete spin variables restricted to have orientations orthogonal to the faces of q dimensional hypercube is considered on the bethe lattice the recursive graph which contains no cycles the partition function of the model with dipole dipole and quadrupole quadrupole interaction for arbitrary planar graph is presented in terms of double graph expansions the latter is calculated exactly in case of trees the system of two recurrent relations which allows to calculate all thermodynamic characteristics of the model is obtained the correspondence between thermodynamic phases and different types of fixed points of the rr is established using the technique of simple iterations the plots of the zero field magnetization and quadrupolar moment are obtained analyzing the regions of stability of different types of fixed points of the system of recurrent relations the phase diagrams of the model are plotted for q 2 the phase diagram of the model is found to have three tricritical points whereas for q 2 there are one triple and one tricritical points pacs number s 05 50 q 05 70 fh

recovering intrinsic images from a single image interpreting real world images requires the ability distinguish the different characteristics of the scene that lead to its final appearance two of the most important of these characteristics are the shading and reflectance of each point in the scene we present an algorithm that uses multiple cues to recover shading and reflectance intrinsic images from a single image using both color information and a classifier trained to recognize gray scale patterns given the lighting direction each image derivative is classified as being caused by shading or a change in the surface s reflectance the classifiers gather local evidence about the surface s form and color which is then propagated using the generalized belief propagation algorithm the propagation step disambiguates areas of the image where the correct classification is not clear from local evidence we use real world images to demonstrate results and show how each component of the system affects the results

field behaviour of the xy chiral model on a cayley tree the authors present a study of the xy chiral model on a cayley tree in the presence of an external uniform magnetic field they calculate the phase diagram at zero temperature from a corresponding 1d mapping ferromagnetic commensurate and incommensurate modulated phases as well as chaotic structures are present in the zero temperature limit in this case they also obtain a devil s staircase and determine the route to chaos which is in agreement with feigenbaum s scenario for finite temperatures the phase diagrams are obtained from a 2d mapping the chaotic behaviour is present only at low temperatures

on the aα spectral radius of a graph abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 3 defined the matrix a g as a g d g 1 a g the largest eigenvalue of a g is called the a spectral radius of g in this paper we give three edge graft transformations on a spectral radius as applications we determine the unique graph with maximum a spectral radius among all connected graphs with diameter d and determine the unique graph with minimum a spectral radius among all connected graphs with given clique number in addition some bounds on the a spectral radius are obtained

commutativity and spectra of hermitian matrices abstract if two hermitian matrices commute then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order examples show that the converse is not true in general in this paper partial converses are obtained the technique involves a characterization of the equality cases for weyl s inequalities moreover a new proof on the commutativity of two hermitian matrices with property l and analogous results for the product of two positive definite hermitian matrices are included

phase diagrams of competing quadruple and binary interactions on cayley tree like lattice triangular chandelier abstract we study the phase diagrams for the ising model on a cayley tree like lattice called triangular chandelier with competing nearest neighbour interactions j 1 prolonged next nearest neighbour interactions j p and one level next nearest neighbour quadruple interactions j l 1 the phase diagrams display the multicritical points the lifshitz points that are at nonzero temperature and many modulated phases to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular case previous work of vannimenus extension result given by ganikhodjaev and uguz for k 3 at vanishing temperature the phase diagram is fully determined for all values and signs of j 1 j p and j l 1 at finite temperatures several interesting features are exhibited for typical values of j l 1 j 1 and j p j 1

star a structure and texture aware retinex model retinex theory is developed mainly to decompose an image into the illumination and reflectance components by analyzing local image derivatives in this theory larger derivatives are attributed to the changes in reflectance while smaller derivatives are emerged in the smooth illumination in this paper we utilize exponentiated local derivatives with an exponent gamma of an observed image to generate its structure map and texture map the structure map is produced by been amplified with gamma 1 while the texture map is generated by been shrank with gamma to this end we design exponential filters for the local derivatives and present their capability on extracting accurate structure and texture maps influenced by the choices of exponents gamma the extracted structure and texture maps are employed to regularize the illumination and reflectance components in retinex decomposition a novel structure and texture aware retinex star model is further proposed for illumination and reflectance decomposition of a single image we solve the star model by an alternating optimization algorithm each sub problem is transformed into a vectorized least squares regression with closed form solutions comprehensive experiments on commonly tested datasets demonstrate that the proposed star model produce better quantitative and qualitative performance than previous competing methods on illumination and reflectance decomposition low light image enhancement and color correction the code is publicly available at https github com csjunxu star

phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field we study the phase diagram for the ising model on a cayley tree with competing nearest neighbor interactionsj1 and next nearest neighbor interactionsj2 andj3 in the presence of an external magnetic field to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular cases previous works by vannimenus and by inawashiroet al at vanishing temperature the phase diagram is fully determined for all values and signs ofj2 j1 andj3 j2 in particular we verify that values ofj3 j2 high enough favor the paramagnetic phase at finite temperatures several interesting features evolution of reentrances separation of the modulated region into two disconnected pieces etc are exhibited for typical values ofj2 j1 andj3 j2

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

one dimensional mapping modulated phases and lyapunov exponent for the antiferromagnetic q state potts and multi site exchange interaction ising models abstract we describe the bifurcation structure period doubling and chaos for the antiferromagnetic q state potts model on the bethe lattice and three site interaction ising model on husimi one in a magnetic field by using the recursion relation technique a chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures the resulting one dimensional rational mapping has a positive lyapunov exponent in the region of the chaotic regime for the antiferromagnetic q state potts q 2 and three site interaction ising models we discuss modulated phases for the antiferromagnetic q state potts q 2 and q 2 and three site interaction ising model at low temperatures the q state potts model q 2 has only one modulated phase with 1 2 pinching corresponding to the 2 cycle the q state potts q 2 and three site interaction ising models have an infinite number of modulated phases with different pinching numbers we construct the first modulated phase after the first bifurcation point

on the multiplicity of an arbitrary aα eigenvalue of a connected graph abstract the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the degree diagonal matrix of g the eigenvalues of a g are called the a eigenvalues of g in this paper we prove that for a connected graph g of order n and with maximum degree 2 and for any 0 1 the multiplicity of an arbitrary a eigenvalue of g which is denoted as m g is bounded by m g 2 n 2 1 the equality holds if and only if g and satisfy one of the following conditions i g k n with n 3 and n 1 ii g c n with even order and 2 2 1 cos 2 j n j 1 2 n 2 2 iii g c n with odd order and 2 2 1 cos 2 j n j 1 2 n 1 2 iv g k n 2 n 2 with n 4 and n 2 applying this result we solve an open conjecture posed by zhou et al in 32 one year ago

on the multiplicity of α as an eigenvalue of the aα matrix of a graph in terms of the number of pendant vertices abstract let g v g e g be a simple undirected graph with vertex set v g and edge set e g the cyclomatic number of a connected graph g is defined as g e g v g 1 the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the diagonal matrix of the vertex degrees of g let m g be the multiplicity of as an eigenvalue of a g a cluster of g is an independent set of one or more vertices of g each of which has the same set of neighbors the closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster the degree of a cluster is the cardinality of its shared set of neighbors a k cluster is a cluster of degree k the number of vertices in a k cluster is its order a collection of two or more k clusters is independent if the k clusters are pairwise disjoint and is isolated if the closures of the k clusters are pairwise disjoint cardoso et al 2 obtained a lower bound for m g in terms of the number of pendant vertices they proved that for a simple connected graph g with p g 0 pendant vertices attached at q g quasi pendant vertices m g p g q g with equality if each internal vertex is a quasi pendant vertex in this article for a graph g of order n and with t independent k clusters of orders r 1 r t we prove that m g k i 1 t r i t and if the k clusters are isolated then m g k n 2 t which extends cardoso s result to a more general case also we give an upper bound for m g in terms of p g it is proved that m g p g 2 g with g the cyclomatic number of g the graphs g whose m g attain the upper bound are characterized explicitly

on contour arguments for the three state potts model with competing interactions on a semi infinite cayley tree we consider the potts model with competing two step interactions and spin values 1 2 3 on a semi infinite cayley tree of order 2 we describe ground states and verify the peierls condition for the model using a contour argument we show the existence of three different gibbs measures associated with translation invariant ground states

on the second largest aα eigenvalues of graphs abstract let g be a graph with adjacency matrix a g and the degree diagonal matrix d g for any real 0 1 nikiforov 2017 10 defined the matrix a g as a g d g 1 a g denote its eigenvalues by 1 a g 2 a g n a g in this paper we give a lower bound of 2 a g for 0 1 as an application we show that the star is the unique graph which minimizes the second largest a eigenvalues among connected graphs for 1 2 1 in particular the graphs with k a g n 2 for k 2 are determined furthermore we determine the connected graphs with 2 a g 1 are firefly graphs moreover the firefly graphs are shown to be determined by their a spectrum for 1 2 1

diluted ising model with competing interactions abstract we introduce aperiodic but deterministic dilution of bonds in the ising model with competing ferro and antiferromagnetic interactions between first and second neighbors along the branches of a cayley tree the problem is formulated as a non linear dissipative map whose attractors correspond to solutions deep inside the tree we use a scheme of successive periodic approximations to obtain the main modulated structures of the phase diagrams the paramagnetic lines as well as some other features of the phase diagrams can be obtained from closed expressions

phase diagrams of ising models on husimi trees i pure multisite interaction systems lattice spin systems with multisite interactions have rich and interesting phase diagrams we present some results for such systems involving ising spins 1 using a generalization of the bethe lattice approximation first we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems second a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made we exhibit a full bifurcation diagram chaos period 3 windows etc for the magnetization of the base site of this system

phase diagram of the one state potts model on the cayley tree the phase diagram of the one state potts model on the closed asymmetric cayley tree with branching ratior 2 is obtained from the bethe peierls map the route to chaos via the period doubling cascade is obtained by considering the antiferromagnetic coupling limit the connection of the potts model with the percolation problem is shown by calculating the order parameter its susceptibility the internal energy and the specific heat as well as their asymptotic behavior at the paramagnetic ferromagnetic critical point due to the type of the lattice and to the polynomial character of the map this is the simplest known example of a mckay berker kirkpatrick spin glass

ising model with third neighbour interactions on the cayley tree the authors formulate the ising model with competing first second and third neighbour interactions along the branches of a cayley tree in the infinite coordination limit as a three dimensional non linear mapping problem the phase diagrams display a lifshitz point and many modulated phases the introduction of third neighbour interactions affects mainly the low temperature region of the phase diagrams

a closed form solution to retinex with nonlocal texture constraints we propose a method for intrinsic image decomposition based on retinex theory and texture analysis while most previous methods approach this problem by analyzing local gradient properties our technique additionally identifies distant pixels with the same reflectance through texture analysis and uses these nonlocal reflectance constraints to significantly reduce ambiguity in decomposition we formulate the decomposition problem as the minimization of a quadratic function which incorporates both the retinex constraint and our nonlocal texture constraint this optimization can be solved in closed form with the standard conjugate gradient algorithm extensive experimentation with comparisons to previous techniques validate our method in terms of both decomposition accuracy and runtime efficiency

on the aα spectra of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 8 defined the matrix a g as a g d g 1 a g in this paper we give some results on the eigenvalues of a g for 1 2 in particular we characterize the graphs with k a g n 1 for 2 k n moreover we show that n a g 2 1 if g contains no isolated vertices

entropy minimization for shadow removal recently a method for removing shadows from colour images was developed finlayson et al in ieee trans pattern anal mach intell 28 59 68 2006 that relies upon finding a special direction in a 2d chromaticity feature space this invariant direction is that for which particular colour features when projected into 1d produce a greyscale image which is approximately invariant to intensity and colour of scene illumination thus shadows which are in essence a particular type of lighting are greatly attenuated the main approach to finding this special angle is a camera calibration a colour target is imaged under many different lights and the direction that best makes colour patch images equal across illuminants is the invariant direction here we take a different approach in this work instead of a camera calibration we aim at finding the invariant direction from evidence in the colour image itself specifically we recognize that producing a 1d projection in the correct invariant direction will result in a 1d distribution of pixel values that have smaller entropy than projecting in the wrong direction the reason is that the correct projection results in a probability distribution spike for pixels all the same except differing by the lighting that produced their observed rgb values and therefore lying along a line with orientation equal to the invariant direction hence we seek that projection which produces a type of intrinsic independent of lighting reflectance information only image by minimizing entropy and from there go on to remove shadows as previously to be able to develop an effective description of the entropy minimization task we go over to the quadratic entropy rather than shannon s definition replacing the observed pixels with a kernel density probability distribution the quadratic entropy can be written as a very simple formulation and can be evaluated using the efficient fast gauss transform the entropy written in this embodiment has the advantage that it is more insensitive to quantization than is the usual definition the resulting algorithm is quite reliable and the shadow removal step produces good shadow free colour image results whenever strong shadow edges are present in the image in most cases studied entropy has a strong minimum for the invariant direction revealing a new property of image formation

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

strange attractor in the ising model with competing interactions on the cayley tree on formule le modele d ising avec interactions de competition sur un arbre de cayley comme une application non lineaire a 2 dimensions

a note on cactus trees variational vs recursive approach in this article we deal with the variational approach to cactus trees husimi trees and the more common recursive approach that are in principle equivalent for finite systems we discuss in detail the conditions under which the two methods are equivalent also in the analysis of infinite self similar cactus trees usually investigated to the purpose of approximating ordinary lattice systems such issue is hardly ever considered in the literature we show on significant test models that the phase diagram and the thermodynamic quantities computed by the variational method when they deviates from the exact bulk properties of the cactus system generally provide a better approximation to the behavior of a corresponding ordinary system generalizing a property proved by kikuchi we also show that the numerical algorithm usually employed to perform the free energy minimization in the variational approach is always convergent

nordhaus gaddum type inequalities of the second aα eigenvalue of a graph abstract let g be a graph with adjacency matrix a g and the degree diagonal matrix d g for any real 0 1 the matrix a g of a graph is defined as a g d g 1 a g denote its eigenvalues by 1 a g 2 a g n a g in this paper for 1 2 1 the nordhaus gaddum type bound for the second largest a eigenvalue of a graph is considered we show that 2 a g 2 a g n 1 and the extremal graphs for which the equality holds are k n and s n if g k n s n k n e then 2 a g 2 a g n 1 moreover we give two upper nordhaus gaddum type bounds for the second largest a eigenvalue of a graph

shadow removal using bilateral filtering in this paper we propose a simple but effective shadow removal method using a single input image we first derive a 2 d intrinsic image from a single rgb camera image based solely on colors particularly chromaticity we next present a method to recover a 3 d intrinsic image based on bilateral filtering and the 2 d intrinsic image the luminance contrast in regions with similar surface reflectance due to geometry and illumination variances is effectively reduced in the derived 3 d intrinsic image while the contrast in regions with different surface reflectance is preserved however the intrinsic image contains incorrect luminance values to obtain the correct luminance we decompose the input rgb image and the intrinsic image each image is decomposed into a base layer and a detail layer we obtain a shadow free image by combining the base layer from the input rgb image and the detail layer from the intrinsic image such that the details of the intrinsic image are transferred to the input rgb image from which the correct luminance values can be obtained unlike previous methods the presented technique is fully automatic and does not require shadow detection

a note on the aα spectral radius of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 2017 7 defined the matrix a g as a g d g 1 a g let u and v be two vertices of a connected graph g suppose that u and v are connected by a path w 0 v w 1 w s 1 w s u where d w i 2 for 1 i s 1 let g p s q u v be the graph obtained by attaching the paths p p to u and p q to v let s 0 1 nikiforov and rojo 2018 9 conjectured that g p s q u v g p 1 s q 1 u v if p q 2 in this paper we confirm the conjecture as applications firstly the extremal graph with maximal a spectral radius with fixed order and cut vertices is characterized secondly we characterize the extremal tree which attains the maximal a spectral radius with fixed order and matching number these results generalize some known results

an ising model with three competing interactions on a cayley tree in this paper we consider an ising model with three competing restricted interactions on the cayley tree j2 j3 the translation invariant and periodic gibbs measures for these models are investigated and the problem of the phase transition in these classes is solved

potts model on a cayley tree and logistic equation the q state potts model on a cayley tree can be solved by a recursion formula depending on the properties at the surface the model on a bethe lattice is obtained by extrapolating the interior of a cayley tree sufficiently far from the surface in order to have a stable fixed point of the recursion for q 2 we find a second order transition of percolation type at the bethe peierls temperature and a first order transition at a higher temperature for coordination number z 3 the recursion extrapolated to q 1 is identical to the logistic equation the feigenbaum route to chaos appears for antiferromagnetic coupling of the potts model the first period doubling corresponds to a multicritical point in the phase diagram of the potts model

complete devil s staircase in an ising model with competing interactions an ising model on a regular cayley tree with competing ferro and antiferromagnetic nearest neighbour interactions can be formulated as a discrete two dimensional mapping the authors use this mapping to obtain sequences of modulated phases at low temperatures associated with complete devil s staircases the fractal dimensionality of the staircases increase with temperature at high temperatures the incommensurate phases may occupy regions of finite measure of the phase diagram

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

a note on the spectral characterization of strongly connected bicyclic digraphs let d be a digraph with vertex set v d and a be the adjacency matrix of d in this paper we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs furthermore we show that any strongly connected bicyclic digraph is determined by the spectrum

automatic estimation and removal of noise from a single image image denoising algorithms often assume an additive white gaussian noise awgn process that is independent of the actual rgb values such approaches cannot effectively remove color noise produced by today s ccd digital camera in this paper we propose a unified framework for two tasks automatic estimation and removal of color noise from a single image using piecewise smooth image models we introduce the noise level function nlf which is a continuous function describing the noise level as a function of image brightness we then estimate an upper bound of the real nlf by fitting a lower envelope to the standard deviations of per segment image variances for denoising the chrominance of color noise is significantly removed by projecting pixel values onto a line fit to the rgb values in each segment then a gaussian conditional random field gcrf is constructed to obtain the underlying clean image from the noisy input extensive experiments are conducted to test the proposed algorithm which is shown to outperform state of the art denoising algorithms

linear shape from shading in many situations the reflectance function of a surface is approximately linear and there is an effielent closed form solution to the shape from shading problem when boundary conditions e g edges singular points are not available good estimates of shape may still be extracted by using the assumption of general viewing position an improved method for estimating the illuminant direction is also presented

on the second largest eigenvalue of the signless laplacian abstract let g be a graph of order n and let q 1 g q n g be the eigenvalues of the q matrix of g also known as the signless laplacian of g we give a necessary and sufficient condition for the equality q k g n 2 where 1 k n in particular this result solves an open problem raised by wang belardo huang and borovicanin we also show that q 2 g g and determine all graphs for which equality holds

modulated structures of an ising model with competing nearest neighbour interactions the authors use a two dimensional mapping to analyse an ising model with competing ferromagnetic and antiferromagnetic nearest neighbour interactions on a cayley tree the phase diagram which had some resemblance to the case of an ising spin glass displays some tricritical points and many modulated phases

intrinsic image decomposition using a sparse representation of reflectance intrinsic image decomposition is an important problem that targets the recovery of shading and reflectance components from a single image while this is an ill posed problem on its own we propose a novel approach for intrinsic image decomposition using reflectance sparsity priors that we have developed our sparse representation of reflectance is based on a simple observation neighboring pixels with similar chromaticities usually have the same reflectance we formalize and apply this sparsity constraint on local reflectance to construct a data driven second generation wavelet representation we show that the reflectance component of natural images is sparse in this representation we further propose and formulate a global sparse constraint on reflectance colors using the assumption that each natural image uses a small set of material colors using this sparse reflectance representation and the global constraint on a sparse set of reflectance colors we formulate a constrained l 1 norm minimization problem for intrinsic image decomposition that can be solved efficiently our algorithm can successfully extract intrinsic images from a single image without using color models or any user interaction experimental results on a variety of images demonstrate the effectiveness of the proposed technique