karhunen loeve expansion of a set of rotated templates we propose a novel method for efficiently calculating the eigenvectors of uniformly rotated images of a set of templates as we show the images can be optimally approximated by a linear series of eigenvectors which can be calculated without actually decomposing the sample covariance matrix

the algebraic approach to the discrete cosine and sine transforms and their fast algorithms it is known that the discrete fourier transform dft used in digital signal processing can be characterized in the framework of the representation theory of algebras namely as the decomposition matrix for the regular module mathbb c z n mathbb c x x n 1 this characterization provides deep insight into the dft and can be used to derive and understand the structure of its fast algorithms in this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of chebyshev polynomials then we derive most of their known algorithms by pure algebraic means we identify the mathematical principle behind each algorithm and give insight into its structure our results show that the connection between algebra and digital signal processing is stronger than previously understood

spatially balanced sampling of natural resources the spatial distribution of a natural resource is an important consideration in designing an efficient survey or monitoring program for the resource generally sample sites that are spatially balanced that is more or less evenly dispersed over the extent of the resource are more efficient than simple random sampling we review a unified strategy for selecting spatially balanced probability samples of natural resources the technique is based on creating a function that maps two dimensional space into one dimensional space thereby defining an ordered spatial address we use a restricted randomization to randomly order the addresses so that systematic sampling along the randomly ordered linear structure results in a spatially well balanced random sample variable inclusion probability proportional to an arbitrary positive ancillary variable is easily accommodated the basic technique selects points in a two dimensional continuum but is also applicable to sampling finite populations or one dimension

design based maps for continuous spatial populations summarywe analyse the estimation of values of a survey variable throughout a continuum of points in a study area when a sample of points is selected by a probabilistic sampling scheme at each point the value is estimated using an inverse distance weighting interpolator and maps of the survey variable can be obtained we investigate the design based asymptotic properties of the interpolator when the study area remains fixed and the number of sampled points approaches infinity and we derive conditions ensuring design based asymptotic unbiasedness and consistency the conditions essentially require the existence of a pointwise or uniformly continuous function describing the behaviour of the survey variable and the use of spatially balanced designs to select points finally we propose a computationally simple mean squared error estimator

identifiability in bilinear inverse problems with applications to subspace or sparsity constrained blind gain and phase calibration bilinear inverse problems bips the resolution of two vectors given their image under a bilinear mapping arise in many applications without further constraints bips are usually ill posed in practice the properties of natural signals are exploited to solve bips for example subspace constraints or sparsity constraints are imposed to reduce the search space these approaches have shown some success in practice however there are few results on uniqueness in bips for most bips the fundamental question of under what condition the problem admits a unique solution is yet to be answered for example blind gain and phase calibration bgpc is a structured bip which arises in many applications including inverse rendering in computational relighting albedo estimation with unknown lighting blind phase and gain calibration in sensor array processing and multichannel blind deconvolution mbd it is interesting to study the uniqueness of such problems in this paper we define identifiability of a bip up to a group of transformations we derive necessary and sufficient conditions for such identifiability i e the conditions under which the solutions can be uniquely determined up to the transformation group these conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable although verifying these individual conditions requires problem specific procedures this framework is universally applicable to all bips applying these results to bgpc we derive sufficient conditions for unique recovery under several scenarios including subspace joint sparsity and sparsity models for bgpc with joint sparsity or sparsity constraints we develop a procedure to compute the transformation groups corresponding to inherent ambiguities we also give necessary conditions in the form of tight lower bounds on sample complexities and demonstrate the tightness of these bounds by numerical experiments the results for bgpc not only demonstrate the application of the proposed general framework for identifiability analysis but are also of interest in their own right

design based maps for finite populations of spatial units abstractthe estimation of the values of a survey variable in finite populations of spatial units is considered for making maps when samples of spatial units are selected by probabilistic sampling schemes the single values are estimated by means of an inverse distance weighting predictor the design based asymptotic properties of the resulting maps referred to as the design based maps are considered when the study area remains fixed and the sizes of the spatial units tend to zero conditions ensuring design based asymptotic unbiasedness and consistency are derived they essentially require the existence of a pointwise or uniformly continuous density function of the survey variable onto the study area some regularities in the size and shape of the units and the use of spatially balanced designs to select units the continuity assumption can be relaxed into a riemann integrability assumption when estimation is performed at a sufficiently small spatial grain and the estimates are subsequently aggregated

algebraic signal processing theory 1 d space in our paper titled ldquoalgebraic signal processing theory foundation and 1 d timerdquo appearing in this issue of the ieee transactions on signal processing we presented the algebraic signal processing theory an axiomatic and general framework for linear signal processing the basic concept in this theory is the signal model defined as the triple a m phi where a is a chosen algebra of filters m an associated a module of signals and phi is a generalization of the z transform each signal model has its own associated set of basic sp concepts including filtering spectrum and fourier transform examples include infinite and finite discrete time where these notions take their well known forms in this paper we use the algebraic theory to develop infinite and finite space signal models these models are based on a symmetric space shift operator which is distinct from the standard time shift we present the space signal processing concepts of filtering or convolution ldquoz transform rdquo spectrum and fourier transform for finite length space signals we obtain 16 variants of space models which have the 16 discrete cosine and sine transforms dcts dsts as fourier transforms using this novel derivation we provide missing signal processing concepts associated with the dcts dsts establish them as precise analogs to the dft get deep insight into their origin and enable the easy derivation of many of their properties including their fast algorithms

multiplication free approximate algorithms for compressed domain linear operations on images we propose a method for devising approximate multiplication free algorithms for compressed domain linear operations on images e g downsampling translation filtering etc we demonstrate that the approximate algorithms give output images that are perceptually nearly equivalent to those of the exact processing while the computational complexity is significantly reduced

the discretew transform four different versions of the discretew transform dwt are introduced the dwt may be decomposed into the discrete cosine transform dct and the discrete sine transform dst eight versions of both dct and dst are introduced for the decomposition of the dwt the relationship among different versions of dwt and their relation with the discrete fourier transform dft are given convolution theorems represented by different versions of the dwt are derived

a fast recursive algorithm for computing the discrete cosine transform the discrete cosine transform dct is widely applied in various fields including image data compression because it operates like the karhunen loeve transform for stationary random data this paper presents a recursive algorithm for dct with a structure that allows the generation of the next higher order dct from two identical lower order dct s as a result the method for implementing this recursive dct requires fewer multipliers and adders than other dct algorithms

pairwise compatibility graphs a survey a graph g v e is a pairwise compatibility graph pcg if there exists an edge weighted tree t and two nonnegative real numbers d min and d max such that each leaf u of t is a node of v and there is an edge u v in e if and only if d min leq d t u v leq d max where d t u v is the sum of weights of the edges on the unique path from u to v in t in this article we survey the state of the art concerning this class of graphs and some of its subclasses

spatially balanced sampling through the pivotal method a simple method to select a spatially balanced sample using equal or unequal inclusion probabilities is presented for populations with spatial trends in the variables of interest the estimation c

corrigendum to on pairwise compatibility graphs having dilworth number two theoret comput sci 524 2014 34 40 in 4 we put in relation graphs with dilworth number at most two and the two classes lpg and mlpg in order to prove this relation we have heavily exploited a result we deduced from 1 we have now realized that this result is not always true so in this corrigendum we correctly restate the result concerning the relation between graphs with dilworth number at most two and the two classes of lpg and mlpg

continental scale relationship between bankfull width and drainage area for single thread alluvial channels we explore the bankfull width wbf versus drainage area ada relationship across a range of climatic and geologic environments and ask 1 is the relationship between ln wbf and ln ada best described by a linear function and 2 can a reliable relationship be developed for predicting wbf with ada as the only independent variable the principal data set for this study was compiled from regional curve studies and other reports that represent 1018 sites 1 m wbf 110 m and 0 50 km2 ada 22 000 km2 in the continental united states two additional data sets were used for validation after dividing the data into small medium and large size basins which respectfully correspond to ada 4 95 km2 4 95 km2 ada 337 km2 and ada 337 km2 regression lines from each data set were compared using one way analysis of covariance ancova a second ancova was performed to determine if mean annual precipitation p is an extraneous factor in the wbf versus ada relationship the ancova results reveal that using ada alone does not yield a reliable wbf versus ada relationship that is applicable across a wide range of environments and that p is a significant extraneous factor in the relationship considering data for very small basins ada 0 49 km2 and very large basins ada 1 0 105 km2 we conclude that a two segment linear model is the most probable form of the ln wbf versus ln ada relationship this study provides useful information for building complex multivariate models for predicting wbf

graph classes with structured neighborhoods and algorithmic applications given a graph in any of the following graph classes trapezoid graphs circular permutation graphs convex graphs dilworth k graphs k polygon graphs circular arc graphs and complements of k degenerate graphs we show how to compute decompositions with the number of d neighborhoods bounded by a polynomial of the input size combined with results of bui xuan telle and vatshelle 2013 1 this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes the boolean width of a graph is related to the number of 1 neighborhoods and our results imply that any of these graph classes have boolean width o logn

fast dct domain filtering using the dct and the dst a method for efficient spatial domain filtering directly in the discrete cosine transform dct domain is developed and proposed it consists of using the discrete sine transform dst and the dct for transform domain processing on the in jpeg basis of the previously derived convolution multiplication properties of discrete trigonometric transforms the proposed scheme requires neither zero padding of the input data nor kernel symmetry it is demonstrated that in typical applications the proposed algorithm is significantly more efficient than the conventional filtered spatial domain and earlier proposed dct domain methods the proposed method is applicable to any dct based image compression standard such as jpeg mpeg and h 261

gaussian predictive process models for large spatial data sets with scientific data available at geocoded locations investigators are increasingly turning to spatial process models for carrying out statistical inference over the last decade hierarchical models implemented through markov chain monte carlo methods have become especially popular for spatial modelling given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics however fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations rendering such models infeasible for large spatial data sets this computational burden is exacerbated in multivariate settings with several spatially dependent response variables it is also aggravated when data are collected at frequent time points and spatiotemporal process models are used with regard to this challenge our contribution is to work with what we call predictive process models for spatial and spatiotemporal data every spatial or spatiotemporal process induces a predictive process model in fact arbitrarily many of them the latter models project process realizations of the former to a lower dimensional subspace thereby reducing the computational burden hence we achieve the flexibility to accommodate non stationary non gaussian possibly multivariate possibly spatiotemporal processes in the context of large data sets we discuss attractive theoretical properties of these predictive processes we also provide a computational template encompassing these diverse settings finally we illustrate the approach with simulated and real data sets

on graphs that are not pcgs let t be an edge weighted tree and let d min d max be two nonnegative real numbers the pairwise compatibility graph pcg of t is a graph g such that each vertex of g corresponds to a distinct leaf of t and two vertices are adjacent in g if and only if the weighted distance between their corresponding leaves in t is in the interval d min d max similarly a given graph g is a pcg if there exist suitable t d min d max such that g is a pcg of t yanhaona bayzid and rahman proved that there exists a graph with 15 vertices that is not a pcg on the other hand calamoneri frascaria and sinaimeri proved that every graph with at most seven vertices is a pcg in this paper we construct a graph of eight vertices that is not a pcg which strengthens the result of yanhaona bayzid and rahman and implies optimality of the result of calamoneri frascaria and sinaimeri we then construct a planar graph with sixteen vertices that is not a pcg finally we prove a variant of the pcg recognition problem to be np complete

data compression properties of the hartley transform the data compression performance of the hartley transform on a markov 1 signal is theoretically compared to that of the fourier transform covariance distribution and residue correlation measurements have been computed for the hartley fourier and cosine transforms the hartley transform achieves better coding performance than the fourier transform but is inferior to the cosine transform

diagonalizing properties of the discrete cosine transforms since its introduction in 1974 by ahmed et al the discrete cosine transform dct has become a significant tool in many areas of digital signal processing especially in signal compression there exist eight types of discrete cosine transforms dcts we obtain the eight types of dcts as the complete orthonormal set of eigenvectors generated by a general form of matrices in the same way as the discrete fourier transform dft can be obtained as the eigenvectors of an arbitrary circulant matrix these matrices can be decomposed as the sum of a symmetric toeplitz matrix plus a hankel or close to hankel matrix scaled by some constant factors we also show that all the previously proposed generating matrices for the dcts are simply particular cases of these general matrix forms using these matrices we obtain for each dct a class of stationary processes verifying certain conditions with respect to which the corresponding dct has a good asymptotic behavior in the sense that it approaches karhunen loeve transform performance as the block size n tends to infinity as a particular result we prove that the eight types of dcts are asymptotically optimal for all finite order markov processes we finally study the decorrelating power of the dcts obtaining expressions that show the decorrelating behavior of each dct with respect to any stationary processes

tensor tensor products with invertible linear transforms abstract research in tensor representation and analysis has been rising in popularity in direct response to a the increased ability of data collection systems to store huge volumes of multidimensional data and b the recognition of potential modeling accuracy that can be provided by leaving the data and or the operator in its natural multidimensional form in recent work 1 the authors introduced the notion of the t product a generalization of matrix multiplication for tensors of order three which can be extended to multiply tensors of arbitrary order 2 the multiplication is based on a convolution like operation which can be implemented efficiently using the fast fourier transform fft the corresponding linear algebraic framework from the original work was further developed in 3 and it allows one to elegantly generalize all classical algorithms from numerical linear algebra in this paper we extend this development so that tensor tensor products can be defined in a so called transform domain for any invertible linear transform in order to properly motivate this transform based approach we begin by defining a new tensor tensor product alternative to the t product we then show that it can be implemented efficiently using dcts and that subsequent definitions and factorizations can be formulated by appealing to the transform domain using this new product as our guide we then generalize the transform based approach to any invertible linear transform we introduce the algebraic structures induced by each new multiplication in the family which is that of c algebras and modules finally in the spirit of 4 we give a matrix algebra based interpretation of the new family of tensor tensor products and from an applied perspective we briefly discuss how to choose a transform we demonstrate the convenience of our new framework within the context of an image deblurring problem and we show the potential for using one of these new tensor tensor products and resulting tensor svd for hyperspectral image compression

dct dst and gauss markov fields conditions for equivalence the correspondence addresses the intriguing question of which random models are equivalent to the discrete cosine transform dct and discrete sine transform dst common knowledge states that these transforms are asymptotically equivalent to first order gauss causal markov random processes we establish that the dct and the dst are exactly equivalent to homogeneous one dimensional 1 d and two dimensional 2 d gauss noncausal markov random fields defined on finite lattices with appropriate boundary conditions

symmetric convolution using unitary transform matrices we demonstrate relationships between the convolutional and unitary forms of certain trigonometric transform matrices we then use the new relationships to derive forms of the symmetric convolution multiplication property based on unitary rather than convolutional forms of the transform matrices

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

on dynamic threshold graphs and related classes abstract this paper deals with the well known classes of threshold and difference graphs both characterized by separators i e node weight functions and thresholds we design an efficient algorithm to find the minimum separator and we show how to maintain minimum its value when the input threshold or difference graph is fully dynamic i e edges nodes are inserted removed moreover exploiting the data structure used for maintaining the minimality of the separator we study the disjoint union and the join of two threshold graphs showing that the resulting graphs are threshold signed graphs i e a superclass of both threshold and difference graphs finally we consider the complement operation on all the three introduced classes of graphs all these operations produce in output the modified graph in terms of their separator and require time linear w r t the number of different degrees we observe that recomputing from scratch the separator would run either in linear for threshold and difference graphs or quadratic for threshold signed graphs time w r t the number of nodes of the graph

exploring pairwise compatibility graphs a graph g v e is called a pairwise compatibility graph pcg if there exists a tree t a positive edge weight function w on t and two non negative real numbers d m i email protected d m a x such that each leaf l u of t corresponds to a vertex email protected v and there is an edge u v e if and only if d m i email protected d t w l u l v d m a x where d t w l u l v is the sum of the weights of the edges on the unique path from l u to l v in t in this paper we analyze the class of pcgs in relation to two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves concern only d m a x lpg or only d m i n mlpg in particular we show that the union of lpg and mlpg classes does not coincide with the whole class of pcgs their intersection is not empty and that neither of the classes lpg and mlpg is contained in the other finally we study the closure properties of the classes pcg mlpg and lpg under some common graph operations in particular we consider the following operations adding an isolated or universal vertex adding a pendant vertex adding a false or a true twin taking the complement of a graph and taking the disjoint union of two graphs

some classes of graphs that are not pcgs abstract a graph g v e is a pairwise compatibility graph pcg if there exists an edge weighted tree t and two non negative real numbers d m i n and d m a x d m i n d m a x such that each node u v is uniquely associated to a leaf of t and there is an edge u v e if and only if d m i n d t u v d m a x where d t u v is the sum of the weights of the edges on the unique path p t u v from u to v in t understanding which graph classes lie inside and which ones outside the pcg class is an important issue in this paper we show that some interesting classes of graphs have empty intersection with pcg they are wheels strong product of a cycle and p 2 and the square of an n node cycle with n sufficiently large as a side effect we show that the smallest planar graph not to be pcg has not 20 nodes as previously known but only 8 it is c 8 2

design based asymptotics for two phase sampling strategies in environmental surveys summary we analyse design based properties of two phase strategies for estimating totals and nonlinear functions of totals for environmental populations when the sampling schemes are uniquely determined by points placed in the study region in the first phase points are located using tessellation stratified sampling whereas in the second phase a finite population sampling scheme is adopted we give sufficient conditions on second phase designs that ensure consistency and we investigate the variance convergence rate for some familiar schemes

generating matrices for the discrete sine transforms in this correspondence we obtain a general form for the generating matrices of the eight types of discrete sine transforms these matrix forms can be used to study the performance of the different discrete sine transforms as substitutes of the karhunen loeve transform and will allow us to show in a very straightforward way that the discrete sine transforms as the discrete cosine transforms are asymptotically optimal for any finite order markov process

symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms this paper uses the fact that the discrete fourier transform diagonalizes a circulant matrix to provide an alternate derivation of the symmetric convolution multiplication property for discrete trigonometric transforms derived in this manner the symmetric convolution multiplication property extends easily to multiple dimensions using the notion of block circulant matrices and generalizes to multidimensional asymmetric sequences the symmetric convolution of multidimensional asymmetric sequences can then be accomplished by taking the product of the trigonometric transforms of the sequences and then applying an inverse trigonometric transform to the result an example is given of how this theory can be used for applying a two dimensional 2 d finite impulse response fir filter with nonlinear phase which models atmospheric turbulence

on pairwise compatibility graphs having dilworth number two a graph g v e is called a pairwise compatibility graph pcg if there exists a tree t a positive edge weight function w on t and two non negative real numbers d m i n and d m a x d m i n d m a x such that v coincides with the set of leaves of t and there is an edge u v e if and only if d m i n d t w u v d m a x where d t w u v is the sum of the weights of the edges on the unique path from u to v in t when the constraints on the distance between the pairs of leaves concern only d m a x or only d m i n the two subclasses lpgs leaf power graphs and mlpgs minimum leaf power graphs are defined the dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another it is known that lpg mlpg is not empty and that threshold graphs i e dilworth one graphs are contained in it in this paper we prove that dilworth two graphs belong to the set lpg mlpg too our proof is constructive since we show how to compute all the parameters t w d m a x and d m i n exploiting the usual representation of dilworth two graphs in terms of node weight function and thresholds for graphs with dilworth number two that are also split graphs i e split permutation graphs we provide another way to compute t w d m i n and d m a x when these graphs are given in terms of their intersection model

on the use of discrete cosine transforms for multicarrier communications in this correspondence the conditions to use any kind of discrete cosine transform dct for multicarrier data transmission are derived the symmetric convolution multiplication property of each dct implies that when symmetric convolution is performed in the time domain an element by element multiplication is performed in the corresponding discrete trigonometric domain therefore appending symmetric redundancy as prefix and suffix into each data symbol to be transmitted and also enforcing symmetry for the equivalent channel impulse response the linear convolution performed in the transmission channel becomes a symmetric convolution in those samples of interest furthermore the channel equalization can be carried out by means of a bank of scalars in the corresponding discrete cosine transform domain the expressions for obtaining the value of each scalar corresponding to these one tap per subcarrier equalizers are presented this study is completed with several computer simulations in mobile broadband wireless communication scenarios considering the presence of carrier frequency offset cfo the obtained results indicate that the proposed systems outperform the standardized ones based on the dft