121024775
Arbitrarily smooth orthogonal nonseparable wavelets in R 2
For each $r\in\N$, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of the construction is a scaling function that satisfies a dilation equation with special coefficients and a special dilation matrix M: the coefficients are aligned along two adjacent rows, and $\detwo$. We prove that if $\M^2=\pm 2I$, e.\,g., %%$\M=\ourmtx$ or $\M=\qsymtx$, $M=({0\,\,2 \atop 1\,\,0})$ or $M=({1\,\,\phantom{-}1 \atop 1\,\,-1})$, then the smoothness of the wavelets improves asymptotically by $1-\frac{1}{2}\log_23 \approx 0.2075$ when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.
14649490
Generation of finite tight frames by Householder transformations
Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ℝn or ℂn only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ℝn or ℂn, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?
7385162
On the use of high-order ambiguity function for multi-component polynomial phase signals
Abstract Nonstationary signals appear often in real-life applications and many of them can be modeled as polynomial-phase signals (PPS). High-order ambiguity function (HAF) was first introduced to estimate the parameters of a single-component PPS, but has not been widely used for multi-component PPS because of its nonlinearity. Multi-component PPS arise for example, in Doppler radar applications when multiple objects are tracked simultaneously. In this paper, we show that HAF is virtually additive for multi-component PPS and suggest an algorithm to estimate their parameters. Numerical examples are presented to illustrate the theories.
2456151
Subdivision Schemes and Refinement Equations with Nonnegative Masks
We consider the two-scale refinement equation f(x)=@?^N"n"="0c"nf(2x-n) with @?"nc"2"n=@?"nc"2"n"+"1=1 where c"0, c"N 0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all c"n>=0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 00} is 1. We prove the conjecture for a large class of refinement equations.
1536242
Substitution Delone Sets 

Abstract. Substitution Delone set families are families of Delone sets X =(X1, . . ., Xn) which satisfy the inflation functional equation
 X_i = \bigvee_{j=1}^m ( \mbox{\phvr A} (X_j) + {\cal D}_$ij$ ) , \qquad 1 \le i \le m, 
 in which A is an expanding matrix, i.e., all of the eigenvalues of A fall outside the unit circle. Here the Dij are finite sets of vectors in Rd and V denotes union that counts multiplicity. This paper characterizes families X=(X1, . . ., Xn) that satisfy an inflation functional equation, in which each Xi is a multiset (set with multiplicity) whose underlying set is discrete. It then studies the subclass of Delone set solutions, and gives necessary conditions on the coefficients of the inflation functional equation for such solutions X to exist. It relates Delone set solutions to a narrower subclass of solutions, called self-replicating multi-tiling sets, which arise as tiling sets for self-replicating multi-tilings.

17444206
The Uniformity of Non-Uniform Gabor Bases
There have been extensive studies on non-uniform Gabor bases and frames in recent years. But interestingly there have not been a single example of a compactly supported orthonormal Gabor basis in which either the frequency set or the translation set is non-uniform. Nor has there been an example in which the modulus of the generating function is not a characteristic function of a set. In this paper, we prove that in the one dimension and if we assume that the generating function g(x) of an orthonormal Gabor basis is supported on an interval, then both the frequency and the translation sets of the Gabor basis must be lattices. In fact, the Gabor basis must be the “trivial” one in the sense that |g(x)|=cχΩ(x) for some fundamental interval of the translation set. We also give examples showing that compactly supported non-uniform orthonormal Gabor bases exist in higher dimensions.
8367818
Segment Hough Transform -- a Novel Hough-based Algorithm for Curve Detection
A new Hough-based algorithm termed segment Hough transform (SHT) is proposed to detect the curve in the binary image. The segment-transform, which is the main idea of the novel approach, makes the curve detection very efficient. Thanks to the chain code termed angle chain code (ACC) as well as the modified direction measurement, the segmentation operation is also considerably fast and precise. Compared with the conventional Hough transform (CHT), our algorithm is dramatically faster (by over 20 times) And when the background is complex, the SHT is about four times faster than the random Hough transform (RHT), Furthermore, the proposed algorithm inherits the high robustness from CHT. In some situations, it is even more accurate. We implement the novel algorithm in an embedded system to estimate passengers flow on the bus. The detection rate is over 95% which indicates it is quite suitable for real-time application.
1878790
Unsupervised Discovery of Action Classes
In this paper we consider the problem of describing the action being performed by human figures in still images. We will attack this problem using an unsupervised learning approach, attempting to discover the set of action classes present in a large collection of training images. These action classes will then be used to label test images. Our approach uses the coarse shape of the human figures to match pairs of images. The distance between a pair of images is computed using a linear programming relaxation technique. This is a computationally expensive process, and we employ a fast pruning method to enable its use on a large collection of images. Spectral clustering is then performed using the resulting distances. We present clustering and image labeling results on a variety of datasets.
5597681
An Ensemble Approach to Learning to Rank
In recent years, 'learning to rank' is a focused approach for information retrieval, which can learn the ranking order given by experts and construct a uniform model to rank for new query. But in practice user queries vary in large diversity, it makes a single learned ranker not representative. Therefore, we propose an ensemble approach to 'learning to rank,' in which a lower generalization error can be gotten by generating a set of rankers and leveraging these rankers for the final prediction. Moreover, two strategies of creating multiple base rankers are proposed to make the ensemble more effective for information retrieval. The experiment results on two real world datasets indicate that the proposed approach can outperform the original 'learning to rank' methods significantly.
452800
Rough Derivatives in Rough Function Model and Their Application
On the theory and application of rough derivatives in rough function model, the notion of generalized rough functions is defined to improve the definition of higher order rough derivatives Pawlak proposed, and analyze the functional features of roughly derived functions and higher order roughly derived functions. The rough extremum definition of discrete functions is given, Fermat theorem and Rolle theorem of roughly smooth discrete functions are proposed and proved, which complete and perfect the theoretical foundation of rough derivatives application in rough function model. The concepts that are rough monotone and rough convexity of discrete functions are defined. By comparing with the derivative application of real continuous function, a series of theorems are put forward which are the relation theorem of rough derivatives and rough monotone, the two sufficient conditions of rough extrema, the two relation theorems of rough derivatives and rough convexity. Moreover, some new results are achieved such as the sufficient condition of rough smoothing of discrete functions, etc.
39329211
Correction to "Attribute Clustering for Grouping, Selection, and Classification of Gene Expression Data"