learning discriminative subspaces on random contrasts for image saliency analysis in visual saliency estimation one of the most challenging tasks is to distinguish targets and distractors that share certain visual attributes with the observation that such targets and distractors can sometimes be easily separated when projected to specific subspaces we propose to estimate image saliency by learning a set of discriminative subspaces that perform the best in popping out targets and suppressing distractors toward this end we first conduct principal component analysis on massive randomly selected image patches the principal components which correspond to the largest eigenvalues are selected to construct candidate subspaces since they often demonstrate impressive abilities to separate targets and distractors by projecting images onto various subspaces we further characterize each image patch by its contrasts against randomly selected neighboring and peripheral regions in this manner the probable targets often have the highest responses while the responses at background regions become very low based on such random contrasts an optimization framework with pairwise binary terms is adopted to learn the saliency model that best separates salient targets and distractors by optimally integrating the cues from various subspaces experimental results on two public benchmarks show that the proposed approach outperforms 16 state of the art methods in human fixation prediction

spatiotemporal saliency detection for video sequences based on random walk with restart a novel saliency detection algorithm for video sequences based on the random walk with restart rwr is proposed in this paper we adopt rwr to detect spatially and temporally salient regions more specifically we first find a temporal saliency distribution using the features of motion distinctiveness temporal consistency and abrupt change among them the motion distinctiveness is derived by comparing the motion profiles of image patches then we employ the temporal saliency distribution as a restarting distribution of the random walker in addition we design the transition probability matrix for the walker using the spatial features of intensity color and compactness finally we estimate the spatiotemporal saliency distribution by finding the steady state distribution of the walker the proposed algorithm detects foreground salient objects faithfully while suppressing cluttered backgrounds effectively by incorporating the spatial transition matrix and the temporal restarting distribution systematically experimental results on various video sequences demonstrate that the proposed algorithm outperforms conventional saliency detection algorithms qualitatively and quantitatively

stabilization of a plane periodic channel flow by noise wall normal controllers the steady state flows in a 2 d channel with periodic conditions along the axis are stabilized by a boundary feedback stochastic controller with vertical velocity observation which acts on the normal component of velocity only

pointwise control of the boussinesq system abstract we study a pointwise control problem for the boussinesq system in two dimensions the control is a source term in the heat equation and the cost functional takes into account the distance between the solution to the boussinesq system and a given profile we have recently studied similar problems for the linearized boussinesq system nguyen and raymond j optim theory appl 141 2009 147 165 studying these problems in the nonlinear case is more difficult there is some literature on the pointwise control of linear and nonlinear partial differential equations however to the authors knowledge nothing has been done for the boussinesq system in this study we prove the existence of optimal solutions and derive optimality conditions

stabilization of navier stokes equations by oblique boundary feedback controllers one designs a linear stabilizing boundary feedback controller u for the equilibrium solutions to navier stokes system on a bounded and open domain mathcal o subset mathbb r d d 2 u eta mbox 1 mathrml gamma displaystyle sum nolimits n j 1 mu j left phi j x alpha x vec n x where gamma is a connected component of the boundary partial mathcal o and mbox 1 mathrml gamma is the characteristic function of gamma here phi j are related to the eigenfunction system varphi j for the adjoint stokes oseen system vec n is the normal to partial mathcal o left is the scalar product in l 2 mathcal o d and alpha is an arbitrary c 2 function with circulation zero on partial mathcal o for d 2 3 this controller stabilizes the corresponding linearized oseen stokes system

salient region detection via integrating diffusion based compactness and local contrast salient region detection is a challenging problem and an important topic in computer vision it has a wide range of applications such as object recognition and segmentation many approaches have been proposed to detect salient regions using different visual cues such as compactness uniqueness and objectness however each visual cue based method has its own limitations after analyzing the advantages and limitations of different visual cues we found that compactness and local contrast are complementary to each other in addition local contrast can very effectively recover incorrectly suppressed salient regions using compactness cues motivated by this we propose a bottom up salient region detection method that integrates compactness and local contrast cues furthermore to produce a pixel accurate saliency map that more uniformly covers the salient objects we propagate the saliency information using a diffusion process our experimental results on four benchmark data sets demonstrate the effectiveness of the proposed method our method produces more accurate saliency maps with better precision recall curve and higher f measure than other 19 state of the arts approaches on asd cssd and ecssd data sets

double low rank matrix recovery for saliency fusion in this paper we address the problem of fusing various saliency detection methods such that the fusion result outperforms each of the individual methods we observe that the saliency regions shown in different saliency maps are with high probability covering parts of the salient object with image regions being represented by the saliency values of multiple saliency maps the object regions have strong correlation and thus lie in a low dimensional subspace meanwhile most of the background regions tend to have lower saliency values in various saliency maps they are also strongly correlated and lie in a low dimensional subspace that is independent of the object subspace therefore an image can be represented as the combination of two low rank matrices to obtain a unified low rank matrix that represents the salient object this paper presents a double low rank matrix recovery model for saliency fusion the inference process is formulated as a constrained nuclear norm minimization problem which is convex and can be solved efficiently with the alternating direction method of multipliers furthermore to reduce the computational complexity of the proposed saliency fusion method a saliency model selection strategy based on the sparse representation is proposed experiments on five datasets show that our method consistently outperforms each individual saliency detection approach and other state of the art saliency fusion methods

structure sensitive saliency detection via multilevel rank analysis in intrinsic feature space this paper advocates a novel multiscale structure sensitive saliency detection method which can distinguish multilevel reliable saliency from various natural pictures in a robust and versatile way one key challenge for saliency detection is to guarantee the entire salient object being characterized differently from nonsalient background to tackle this our strategy is to design a structure aware descriptor based on the intrinsic biharmonic distance metric one benefit of introducing this descriptor is its ability to simultaneously integrate local and global structure information which is extremely valuable for separating the salient object from nonsalient background in a multiscale sense upon devising such powerful shape descriptor the remaining challenge is to capture the saliency to make sure that salient subparts actually stand out among all possible candidates toward this goal we conduct multilevel low rank and sparse analysis in the intrinsic feature space spanned by the shape descriptors defined on over segmented super pixels since the low rank property emphasizes much more on stronger similarities among super pixels we naturally obtain a scale space along the rank dimension in this way multiscale saliency can be obtained by simply computing differences among the low rank components across the rank scale we conduct extensive experiments on some public benchmarks and make comprehensive quantitative evaluation between our method and existing state of the art techniques all the results demonstrate the superiority of our method in accuracy reliability robustness and versatility

brief control of mixing by boundary feedback in 2d channel flow we address the problem of enhancing mixing by means of boundary feedback control in 2d channel flow this is done by first designing feedback control strategies for the stabilization of the parabolic equilibrium flow then applying this feedback with the sign of the input reversed the result is enhanced instability of the parabolic equilibrium flow which leads rapidly to highly complex flow patterns simulations of the deformation of dye blobs positioned in the flow indicate qualitatively that effective mixing is obtained for small control effort as compared with the nominal uncontrolled flow a mixedness measure p e is constructed to quantify the mixing observed and is shown to be significantly enhanced by the application of the destabilizing control feedback

a novel multiresolution spatiotemporal saliency detection model and its applications in image and video compression salient areas in natural scenes are generally regarded as areas which the human eye will typically focus on and finding these areas is the key step in object detection in computer vision many models have been proposed to simulate the behavior of eyes such as saliencytoolbox stb neuromorphic vision toolkit nvt and others but they demand high computational cost and computing useful results mostly relies on their choice of parameters although some region based approaches were proposed to reduce the computational complexity of feature maps these approaches still were not able to work in real time recently a simple and fast approach called spectral residual sr was proposed which uses the sr of the amplitude spectrum to calculate the image s saliency map however in our previous work we pointed out that it is the phase spectrum not the amplitude spectrum of an image s fourier transform that is key to calculating the location of salient areas and proposed the phase spectrum of fourier transform pft model in this paper we present a quaternion representation of an image which is composed of intensity color and motion features based on the principle of pft a novel multiresolution spatiotemporal saliency detection model called phase spectrum of quaternion fourier transform pqft is proposed in this paper to calculate the spatiotemporal saliency map of an image by its quaternion representation distinct from other models the added motion dimension allows the phase spectrum to represent spatiotemporal saliency in order to perform attention selection not only for images but also for videos in addition the pqft model can compute the saliency map of an image under various resolutions from coarse to fine therefore the hierarchical selectivity hs framework based on the pqft model is introduced here to construct the tree structure representation of an image with the help of hs a model called multiresolution wavelet domain foveation mwdf is proposed in this paper to improve coding efficiency in image and video compression extensive tests of videos natural images and psychological patterns show that the proposed pqft model is more effective in saliency detection and can predict eye fixations better than other state of the art models in previous literature moreover our model requires low computational cost and therefore can work in real time additional experiments on image and video compression show that the hs mwdf model can achieve higher compression rate than the traditional model

quaternion generic fourier descriptor for color object recognition by introducing quaternions fourier transforms the fundamental tool in signal and image processing were extended to treat a color image in a holistic manner on the other hand color image description has long been a major topic of pattern recognition in this work inspired by the quaternion fourier transforms we propose a quaternion generic fourier descriptor to describe color images holistically and effectively without losing any color information the theoretical framework to construct a set of invariants with respect to geometric transformation is also provided moreover several experiments are conducted on color image databases to illustrate the effectiveness of the proposed invariants the results provide supports for that the quaternion generic fourier descriptor invariants have striking robustness to geometric transformation and noise remarkable progress in accuracy robustness and efficiency of color object recognition has been made by the proposed descriptor in addition it presents superior performances on real recognition scenarios highlightswe propose an image descriptor with the quaternion algebra to describe color images a framework to develop robust invariants to geometric transformation is provided experiments are conducted to validate the performance of the proposed descriptor the proposed descriptor performs robustly to geometric transformation and noise a progress in accuracy of recognition has been made by the proposed descriptor on real recognition scenarios the proposed descriptor also performs superiorly

a multisize superpixel approach for salient object detection based on multivariate normal distribution estimation this paper presents a new method for salient object detection based on a sophisticated appearance comparison of multisize superpixels those superpixels are modeled by multivariate normal distributions in cie lab color space which are estimated from the pixels they comprise this fitting facilitates an efficient application of the wasserstein distance on the euclidean norm formula see text to measure perceptual similarity between elements saliency is computed in two ways on the one hand we compute global saliency by probabilistically grouping visually similar superpixels into clusters and rate their compactness on the other hand we use the same distance measure to determine local center surround contrasts between superpixels then an innovative locally constrained random walk technique that considers local similarity between elements balances the saliency ratings inside probable objects and background the results of our experiments show the robustness and efficiency of our approach against 11 recently published state of the art saliency detection methods on five widely used benchmark data sets

internal stabilization by noise of the navier stokes equation we show that the navier stokes equation in mathcal o subset r d d 2 3 around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller v t xi sum i 1 n v i t psi i xi dot beta i t xi in mathcal o where beta i i 1 n are independent brownian motions and psi i i 1 n is a system of functions on mathcal o with support in an arbitrary open subset mathcal o 0 subset mathcal o the stochastic control input v i i 1 n is found in feedback form the corresponding result for the linearized navier stokes equation was established in e barbu the internal stabilization by noise of the linearized navier stokes equation esaim control optim calc var to appear

stabilization of a plane channel flow by wall normal controllers abstract the steady state flows in a two dimensional channel with periodic conditions along the axis can be stabilized by boundary controllers with vertical velocity observation which acts on the normal component of velocity only

brief paper exponential stabilization of the linearized navier stokes equation by pointwise feedback noise controllers we address the problem of exponential stabilization in probability of the linearized navier stokes equations in an equilibrium point this is done by designing a linear stochastic feedback controller with support in a point or on a discrete set of points of the domain this controller consists of a steady state impulse component with support in a finite set of points modulated by an unsteady feedback noise controller

salient region detection using diffusion process on a two layer sparse graph diffusion based salient region detection has recently received intense research attention in this paper we present some effective improvements concerning two important aspects of diffusion based methods the construction of the diffusion matrix and the seed vector first we construct a two layer sparse graph which is generated by connecting each node to its neighboring nodes and the most similar node that shares common boundaries with its neighboring nodes compared with the most frequently used two layer neighborhood graph our graph not only effectively uses local spatial relationships but also removes dissimilar redundant nodes second we use the spatial variance of superpixel clusters to obtain the seed vector and compared with the previously most used boundary prior our approach can better distinguish saliency seeds from the background seeds especially when salient objects appear near the image boundaries finally we calculate two preliminary saliency maps using the saliency and background seed vectors and more accurate results are obtained using the manifold ranking diffusion method integrating these two diffusion based saliency maps we obtain the final saliency map extensive experiments in which we compare our method with 20 existing state of the art methods on five benchmark data sets asd dut omron ecssd msra5k and msra10k show that the proposed method performs better in terms of various evaluation metrics

scale invariant amplitude spectrum modulation for visual saliency detection saliency detection is one of the key issues in simulating visual attention selection most attention models adopt the competitive structure to simulate the human visual system although these models provide remarkable results and convincing biological plausibility they are still confronted with many difficulties in practical applications because of their extreme time cost and parameter sensitivity recently a new saliency detection approach based on fourier transform as represented by spectral residual sr and phase fourier transform pft has been attracting much attention for its excellent accuracy and computational speed all these models can be unified into one framework called amplitude spectrum modulation asm the aim of this paper is to explore the intrinsic mechanism of asm and develop an advanced asm model after analyzing sr and pft we give a mathematical description for the fundamental idea and the inherent limitations of the existing asm models a new saliency detective model based on the scale invariant asm scene and context based modulation and competitive structure is also proposed breaking through the limitations of the traditional asm models simulation results suggest that the proposed model is more accurate in predicting human eye fixation and is more robust against different types of stimulus when compared with competing models

efficient implementation of quaternion fourier transform convolution and correlation by 2 d complex fft the concepts of quaternion fourier transform qft quaternion convolution qcv and quaternion correlation which are based on quaternion algebra have been found to be useful for color image processing however the necessary computational algorithms and their complexity still need some attention we develop efficient algorithms for qft qcv and quaternion correlation the conventional complex two dimensional 2 d fourier transform ft is used to implement these quaternion operations very efficiently with these algorithms we only need two complex 2 d fts to implement a qft six complex 2 d fts to implement a one side qcv or a quaternion correlation and 12 complex 2 d fts to implement a two side qcv and the efficiency of these quaternion operations is much improved meanwhile we also discuss two additional topics the first one is about how to use qft and qcv for quaternion linear time invariant qlti system analysis this topic is important for quaternion filter design and color image processing besides we also develop the spectrum product qcv it is an improvement of the conventional form of qcv for any arbitrary input functions it always corresponds to the product operation in the frequency domain it is very useful for quaternion filter design

the internal stabilization of the stokes oseen equation by feedback point controllers abstract one designs a stabilizable feedback controller for the stokes oseen equation in an open domain o r d d 2 with support in a finite set of points k o j 1 m more precisely m d n where n is the number of unstable eigenvalues of the stokes oseen operator the controller is expressed in terms of eigenfunctions j 1 j n of the dual stokes oseen operator

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

stochastic stabilization and destabilization it is shown in this paper that any nonlinear systems dot x t f x t t in rd can be stabilized by brownian motion provided f hook x t k x for some k 0 on the other hand this system can also be destabilized by brownian motion if the dimension d 2 similar results are also obtained for any given stochastic differential equation dx t f hook x t t g x t t dw t

generalization ability of optimal cluster separation networks optimal separation of two clusters of normalized vectors can be performed in a neural network with adjustable threshold and weights which is trained to maximum stability generalization from arbitrarily selected training clusters to a given bipartitioning of input space is studied the network s threshold becomes a global optimization and order parameter this causes the generalization ability to increase rapidly with the distance of the cluster separation plane from the origin separation is shown to be stochastic for small and deterministic for large training cluster sizes

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

hypercomplex fourier transforms of color images fourier transforms are a fundamental tool in signal and image processing yet until recently there was no definition of a fourier transform applicable to color images in a holistic manner in this paper hypercomplex numbers specifically quaternions are used to define a fourier transform applicable to color images the properties of the transform are developed and it is shown that the transform may be computed using two standard complex fast fourier transforms the resulting spectrum is explained in terms of familiar phase and modulus concepts and a new concept of hypercomplex axis a method for visualizing the spectrum using color graphics is also presented finally a convolution operational formula in the spectral domain is discussed

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

training of optimal cluster separation networks finding the optimal separation of two clusters of normalized vectors corresponds to training thresholds and weights in a neural network of maximum stability in order to achieve this two local iterative algorithms are presented which treat threshold and weights all in one avoiding the need to calculate any intermediate test quantities convergence is proved and the separation stability obtained is shown to match theoretical predictions and to be superior to existing algorithms

stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers application to the navier stokes system let a mathcal d a to mathcal x be the generator of an analytic semigroup and b mathcal v to mathcal d a a relatively bounded control operator in this paper we consider the stabilization of the system y ay bu where u is the linear combination of a family v 1 ldots v k our main result shows that if a b satisfies a unique continuation property and if k is greater than or equal to the maximum of the geometric multiplicities of the unstable modes of a then the system is generically stabilizable with respect to the family v 1 ldots v k with the same functional framework we also prove the stabilizability of a class of nonlinear systems when using feedback or dynamical controllers we apply these results to stabilize the navier stokes equations in two and three dimensions by using boundary controls

visual saliency based on scale space analysis in the frequency domain we address the issue of visual saliency from three perspectives first we consider saliency detection as a frequency domain analysis problem second we achieve this by employing the concept of nonsaliency third we simultaneously consider the detection of salient regions of different size the paper proposes a new bottom up paradigm for detecting visual saliency characterized by a scale space analysis of the amplitude spectrum of natural images we show that the convolution of the image amplitude spectrum with a low pass gaussian kernel of an appropriate scale is equivalent to an image saliency detector the saliency map is obtained by reconstructing the 2d signal using the original phase and the amplitude spectrum filtered at a scale selected by minimizing saliency map entropy a hypercomplex fourier transform performs the analysis in the frequency domain using available databases we demonstrate experimentally that the proposed model can predict human fixation data we also introduce a new image database and use it to show that the saliency detector can highlight both small and large salient regions as well as inhibit repeated distractors in cluttered images in addition we show that it is able to predict salient regions on which people focus their attention

generalization performance of bayes optimal classification algorithm for learning a perceptron the generalization error of the bayes optimal classification algorithm when learning a perceptron from noise free random training examples is calculated exactly using methods of statistical mechanics it is shown that if an assumption of replica symmetry is made then in the thermodynamic limit the error of the bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm by a factor approaching ensuremath surd 2 as the ratio of the number of training examples to perceptron weights grows in addition it is shown that approximations to the generalization error of the bayes optimal algorithm can be achieved by learning algorithms that use a two layer neutral net to learn a perceptron

nonlocal center surround reconstruction based bottom up saliency estimation many saliency models consider the feature extraction as the algorithmic core and the performance of their methods relies on the selection of the features to a great extent however there can hardly be a set of features effective to pop out the salient regions under various visual environments moreover because saliency is not tuned to certain visual features a location winning the spatial competition in any feature space can be defined as salient instead of seeking for or learning the features to highlight the difference between the salient areas and the background we focus more on the sparsity and uniqueness carried by the original image itself the source of all the features to propose a nonlocal reconstruction based saliency model in the proposed approach the saliency is measured by the sparse reconstruction residual of representing the central patch with a linear combination of its surrounding patches sampled in a nonlocal manner in addition this is generalized to model the global aspect saliency which provides a complement to the nonlocal saliency and improves the performance further as a generalization of itti et al s classical center surround comparison scheme the proposed approach performs well on images where itti et al s method fails as well as on general natural images numerical experiments show the proposed approach produces better results compared with the state of the art algorithms on three public databases highlightsa nonlocal reconstruction based bottom up saliency estimation method is proposed nonlocal image processing is employed to provide a better treatment of textures sparsity pursuit is very effective for removing duplicated similar image regions reconstruction residual is inherently coherent with the bottom up saliency our method is stable under three public datasets and consistent with human fixations

h infty feedback boundary stabilization of the two dimensional navier stokes equations we study the robust or h infty exponential stabilization of the linearized navier stokes equations around an unstable stationary solution in a two dimensional domain omega the disturbance is an unknown perturbation in the boundary condition of the fluid flow we determine a feedback boundary control law robust with respect to boundary perturbations by solving a max min linear quadratic control problem next we show that this feedback law locally stabilizes the navier stokes system similar problems have been studied in the literature in the case of distributed controls and disturbances to the authors knowledge it is the first time that the robust stabilization of the navier stokes equations is studied for boundary controls and boundary disturbances

a novel graph based optimization framework for salient object detection in traditional graph based optimization framework for salient object detection an image is over segmented into superpixels and mapped to one single graph the saliency value of each superpixel is then computed based on the similarity between connected nodes and the saliency related queries when applying the traditional graph based optimization framework to the salient object detection problem in natural scene images we observe at least two limitations only one graph is employed to describe the information contained in an image and no cognitive property about visual saliency is explicitly modeled in the optimization framework in this work we propose a novel graph based optimization framework for salient object detection firstly we employ multiple graphs in our optimization framework a natural scene image is usually complex employing multiple graphs from different image properties can better describe the complex information contained in the image secondly we model one popular cognitive property about visual saliency visual rarity in our graph based optimization framework making this framework more suitable for saliency detection problem specifically we add a regularization term to constrain the saliency value of each superpixel according to visual rarity in our optimization framework our experimental results on four benchmark databases with comparisons to fifteen representative methods demonstrate that our graph based optimization framework is effective and computationally efficient highlightsa novel graph based optimization framework for salient object detection is proposed in the paper multiple graphs are employed in our optimization framework to better describe a natural scene image visual rarity is modeled as a regularization term in our framework to better detect saliency experimental results on four datasets with fifteen methods prove the effectiveness of our method

abstract settings for tangential boundary stabilization of navier stokes equations by high and low gain feedback controllers abstract the present paper seeks to continue the analysis in barbu et al tangential boundary stabilization of navier stokes equations memoir ams to appear on tangential boundary stabilization of navier stokes equations d 2 3 as deduced from well posedness and stability properties of the corresponding linearized equations it intends to complement v barbu i lasiecka r triggiani tangential boundary stabilization of navier stokes equations memoir ams to appear on two levels i by casting the riccati based results of barbu et al tangential boundary stabilization of navier stokes equations memoir ams to appear for d 2 3 in an abstract setting thus extracting the key relevant features so that the resulting framework may be applicable also to other stabilizing boundary feedback operators as well as to other parabolic like equations of fluid dynamics ii by including in the case d 2 this time also the low level gain counterpart of the results in barbu et al tangential boundary stabilization of navier stokes equations memoir ams to appear with both riccati based and spectral based tangential feedback controllers this way new local boundary stabilization results of navier stokes equations are obtained over v barbu i lasiecka r triggiani tangential boundary stabilization of navier stokes equations memoir ams to appear

connecting spatial and frequency domains for the quaternion fourier transform the quaternion fourier transform qft is an important tool in multi dimensional data analysis in particular for the study of color images an important problem when applying the qft is the mismatch between the spatial and frequency domains the convolution of two quaternion signals does not map to the pointwise product of their qft images the recently defined mustard convolution behaves nicely in the frequency domain but complicates the corresponding spatial domain analysis the present paper analyses in detail the correspondence between classical convolution and the new mustard convolution in particular an expression is derived that allows one to write classical convolution as a finite linear combination of suitable mustard convolutions this result is expected to play a major role in the further development of quaternion image processing as it yields a formula for the qft spectrum of the classical convolution

manifold ranking based matrix factorization for saliency detection saliency detection is used to identify the most important and informative area in a scene and it is widely used in various vision tasks including image quality assessment image matching and object recognition manifold ranking mr has been used to great effect for the saliency detection since it not only incorporates the local spatial information but also utilizes the labeling information from background queries however mr completely ignores the feature information extracted from each superpixel in this paper we propose an mr based matrix factorization mrmf method to overcome this limitation mrmf models the ranking problem in the matrix factorization framework and embeds query sample labels in the coefficients by incorporating spatial information and embedding labels mrmf enforces similar saliency values on neighboring superpixels and ranks superpixels according to the learned coefficients we prove that the mrmf has good generalizability and develops an efficient optimization algorithm based on the nesterov method experiments using popular benchmark data sets illustrate the promise of mrmf compared with the other state of the art saliency detection methods