Shift Invariant

In this paper, we investigate the effectiveness of a financial time-series forecasting strategy which exploits the mul- tiresolution property of the wavelet transform. A financial series is decomposed into an over complete, shift invariant scale-related representation. In transform space, each individual wavelet series is modeled by a separate multilayer perceptron (MLP). To better utilize the detailed information in the lower scales of wavelet coef- ficients (high frequencies) and general (trend) information in the higher scales of wavelet coefficients (low frequencies), we applied the Bayesian method of automatic relevance determination (ARD) to choose short past windows (short-term history) for the inputs to the MLPs at lower scales and long past windows (long-term history) at higher scales. To form the overall forecast, the indi- vidual forecasts are then recombined by the linear reconstruction property of the inverse transform with the chosen autocorrelation shell representatio...

Image blur and noise are difficult to avoid in many situations and can often ruin a photograph. We present a novel image deconvolution algorithm that deblurs and denoises an image given a known shift-invariant blur kernel. Our algorithm uses local color statistics derived from the image as a constraint in a unified framework that can be used for deblurring, denoising, and upsampling. A pixel's color is required to be a linear combination of the two most prevalent colors within a neighborhood of the pixel. This two-color prior has two major benefits: it is tuned to the content of the particular image and it serves to decouple edge sharpness from edge strength. Our unified algorithm for deblurring and denoising out-performs previous methods that are specialized for these individual applications. We demonstrate this with both qualitative results and extensive quantitative comparisons that show that we can out-perform previous methods by approximately 1 to 3 DB.

We present the nonsubsampled contourlet transform and its application in image enhancement. The nonsubsampled contourlet transform is built upon nonsubsampled pyramids and nonsubsampled directional filter banks and provides a shiftinvariant directional multiresolution image representation. Existing methods for image enhancement cannot capture the geometric information of images and tend to amplify noises when they are applied to noisy images since they cannot distinguish noises from weak edges. In contrast, the nonsubsampled contourlet transform extracts the geometric information of images, which can be used to distinguish noises from weak edges. Experimental results show the proposed method achieves better enhancement results than a wavelet-based image enhancement method.

Automatic network road extraction from high resolution remotely sensed images has been under study by computer scientists for over 30 years. In fact, Conventional methods to create and update road information rely heavily on manual work and therefore are very expensive and time consuming. This paper presents an efficient and computationally fast method to extract road from very high resolution images automatically. We propose in this paper a new approach for following roads path based on a quaternionic wavelet transform insuring a good local space-frequency analysis with very important directional selectivity. In fact, the rich phase information given by this hypercomplex transform overcomes the lack of shift invariance property shown by the real discrete wavelet transform and the poor directional selectivity of both real and complex wavelet transform.

We propose orthogonal Fourier-Mellin moments, which are more suitable than Zernike moments, for scaleand rotation-invariant pattern recognition. The new orthogonal radial polynomials have more zeros than do the Zernike radial polynomials in the region of small radial distance. The orthogonal Fourier-Mellin moments may be thought of as generalized Zernike moments and orthogonalized complex moments. For small images, the description by the orthogonal Fourier-Mellin moments is better than that by the Zernike moments in terms of image-reconstruction errors and signal-to-noise ratio. Experimental results are shown.

This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the time-frequency plane. A straightforward algorithm is described for the construction of minimal-length perfect reconstruction filters with a specified number of vanishing moments; whereas, in the nonredundant rational case, no such algorithm is available. The algorithm is based on matrix spectral factorization. The analysis/synthesis functions (discrete-time wavelets) can be very smooth and can be designed to closely approximate the derivatives of the Gaussian function.

and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a 'scale-space representation', in wphich a given image is represented by a one-dimensiond family of images representing that image on vurious levels of inner spatial scale. A n early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family o f scaled pariial differential operators: the Gaussian kernel (the lowest order, rescaling operator) and its linear partirzl derivatives. They enable local image analysis through the detection of local differential structure in a robust wuy, while at the same time capturing global features through the extra scale degree of freedom. In this paper we show why the operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale. cartesiarl differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles J h r i n d in mammalian frontend visual systems.

Spatial smoothing is a widely used preprocessing scheme for direction-of-arrival (DOA) estimation of more than one source from a single snapshot, although the effective array aperture gets reduced by this process. In this paper we propose a preprocessing scheme applicable for DOA estimation algorithms that exploit the shift invariance property of the array steering matrix and call it spatial smoothing with improved aperture (SSIA). SSIA, when applied to a noise corrupted data vector, improves the effective array aperture significantly as opposed to conventional spatial smoothing. Simulations confirm the significant performance gain provided by SSIA in conjunction with Unitary ESPRIT.

A shift-invariant non-negative tensor factorisation algorithm for musical source separation is proposed which generalises previous work by allowing each source to have its own parameters rather a fixed set of parameters for all sources. This allows independent control of the number of allowable notes, number of harmonics and shifts in time for each source. This increased flexibility allows the incorporation of further information about the sources and results in improved separation and resynthesis of the separated sources.

This paper presents a new implementation of the YAST algorithm for principal and minor subspace tracking. YAST was initially derived from the Subspace Projection (SP) algorithm by C.E. Davila, which was known for its exceptional convergence rate, compared to other classical principal subspace trackers. The novelty in the YAST algorithm was the lower computational cost (linear if the data correlation matrix satisfies a so-called shift-invariance property), and the extension to minor subspace tracking. However, the original implementation of the YAST algorithm suffered from a numerical stability problem (the subspace weighting matrix slowly loses its orthonormality). We thus propose in this paper a new implementation of YAST, whose stability is established theoretically and tested via numerical simulations. This algorithm combines all the desired properties for a subspace tracker: remarkably high convergence rate, lowest steady state error, linear complexity, and numerical stability regarding the orthonormality of the subspace weighting matrix.

Clinical ultrasound images are often perceived as difficult to interpret due to image blurring and speckle inherent in the ultrasound imaging. But the image quality can be improved by deconvolution using an estimate of the point-spread function. However, it is difficult to obtain a sufficiently accurate estimate of the point-spread function in vivo because of the unknown properties of the soft tissue in clinical applications. Local variations in the speed of sound and attenuation change the pulse and beam shape. These in turn affect the point-spread function. The purpose and novelty of this paper is therefore to explore the sensitivity of a state-of-the-art deconvolution algorithm to uncertainty in the point-spread function. The point-spread function in our restoration algorithm is made shift invariant in the lateral dimension but shift dependent in the axial direction, and is modelled to match a 128-element 1D linear array often found in clinical use. We present simulated and in vitro sensitivity analyses of two-dimensional deconvolution while varying six parameters on which the point-spread function depends. Uncertainty in the ultrasound machine is analysed by varying the axial depths of lateral and elevational foci alongside height and width of transducer elements. Sensitivity to tissue influence is investigated by varying the speed of sound and frequency-dependent attenuation of the electro-mechanical impulse response. The results are analysed both quantitatively and in terms of the perceived image quality. First, the assessment of deconvolution using the logarithmic image amplitude is found to be a better indicator of the perceived improvement in the restoration. Secondly, the two most critical parameters for two-dimensional deconvolution are discovered to be the lateral focus and the speed of sound, because the success of deconvolution is perceived primarily in terms of deblurring. We also observed similar patterns for the simulation and in vitro experiment. Finally, we show that it is possible to restore in vivo ultrasound images using an assumed point-spread function and hence conclude that an exact point-spread function is not necessary for enhancing ultrasound image quality by deconvolution.

In this paper, we investigate the effectiveness of a financial time-series forecasting strategy which exploits the multiresolution property of the wavelet transform. A financial series is decomposed into an over complete, shift invariant scale-related representation. In transform space, each individual wavelet series is modeled by a separate multilayer perceptron (MLP). To better utilize the detailed information in the lower scales of wavelet coefficients (high frequencies) and general (trend) information in the higher scales of wavelet coefficients (low frequencies), we applied the Bayesian method of automatic relevance determination (ARD) to choose short past windows (short-term history) for the inputs to the MLPs at lower scales and long past windows (long-term history) at higher scales. To form the overall forecast, the individual forecasts are then recombined by the linear reconstruction property of the inverse transform with the chosen autocorrelation shell representation, or by another perceptron which learns the weight of each scale in the prediction of the original time series. The forecast results are then passed to a money management system to generate trades. Compared with previous work on combining wavelet techniques and neural networks to financial time-series, our contributions include 1) proposing a three-stage prediction scheme; 2) applying a multiresolution prediction which is strictly based on the autocorrelation shell representation, 3) incorporating the Bayesian technique ARD with MLP training for the selection of relevant inputs; and 4) using a realistic money management system and trading model to evaluate the forecasting performance. Using an accurate trading model, our system shows promising profitability performance. Results comparing the performance of the proposed architecture with an MLP without wavelet preprocessing on 10-year bond futures indicate a doubling in profit per trade ($AUD1753:$AUD819) and Sharpe ratio improvement of 0.732 versus 0.367, as well as significant improvements in the ratio of winning to loosing trades, thus indicating significant potential profitability for live trading.

A filtered backprojection (FBP) reconstruction algorithm for attenuated fan-beam projections has been derived based on Novikov's inversion formula. The derivation uses a common transformation between parallel-beam and fan-beam coordinates. The filtering is shift-invariant. Numerical evaluation of the FBP algorithm is presented as well. As a special application, we also present a shiftinvariant FBP algorithm for fan-beam SPECT reconstruction with uniform attenuation compensation. Several other fan-beam reconstruction algorithms are also discussed. In the attenuation-free case, our algorithm reduces to the conventional fan-beam FBP reconstruction algorithm. Keywords Novikov's inversion formula; Fan-beam projections; FBP algorithm This paper is organized as follows. Section II reviews Novikov's inversion formula for parallelbeam geometry. Section III describes the derivation of a fan-beam FBP reconstruction

Classical dictionary learning algorithms (DLA) allow unicomponent signals to be processed. Due to our interest in two-dimensional (2D) motion signals, we wanted to mix the two components to provide rotation invariance. So, multicomponent frameworks are examined here. In contrast to the well-known multichannel framework, a multivariate framework is first introduced as a tool to easily solve our problem and to preserve the data structure. Within this multivariate framework, we then present sparse coding methods: multivariate orthogonal matching pursuit (M-OMP), which provides sparse approximation for multivariate signals, and multivariate DLA (M-DLA), which empirically learns the characteristic patterns (or features) that are associated to a multivariate signals set, and combines shiftinvariance and online learning. Once the multivariate dictionary is learned, any signal of this considered set can be approximated sparsely. This multivariate framework is introduced to simply present the 2D rotation invariant (2DRI) case. By studying 2D motions that are acquired in bivariate real signals, we want the decompositions to be independent of the orientation of the movement execution in the 2D space. The methods are thus specified for the 2DRI case to be robust to any rotation: 2DRI-OMP and 2DRI-DLA. Shift and rotation invariant cases induce a compact learned dictionary and provide robust decomposition. As validation, our methods are applied to 2D handwritten data to extract the elementary features of this signals set, and to provide rotation invariant decomposition.

Shift invariance associated with good directional selectivity is important for the use of a wavelet transform (WT) in many fields of image processing. Generally, complex wavelet transforms, e.g., the double-tree complex WT (DTCWT), have these useful properties. In this paper, we propose the use of a recently introduced implementation of such a WT, namely, the hyperanalytic WT (HWT), in association with filtering techniques already used with the discrete WT (DWT). The result is a very simple and fast image denoising algorithm. Some simulation results and comparisons prove the performance obtained using the new method.

A vision based approach for calculating accurate 3D models of the objects is presented. Generally industrial visual inspection systems capable of accurate 3D depth estimation rely on extra hardware tools like laser scanners or light pattern projectors. These tools improve the accuracy of depth estimation but also make the vision system costly and cumbersome. In the proposed algorithm, depth and dimensional accuracy of the produced 3D depth model depends on the existing reference model instead of the information from extra hardware tools. The proposed algorithm is a simple and cost effective software based approach to achieve accurate 3D depth estimation with minimal hardware involvement. The matching process uses the well-known coarse to fine strategy, involving the calculation of matching points at the coarsest level with consequent refinement up to the finest level. Vector coefficients of the wavelet transform-modulus are used as matching features, where wavelet transform-modulus maxima defines the shift invariant high-level features with phase pointing to the normal of the feature surface. The technique addresses the estimation of optimal corresponding points and the corresponding 2D disparity maps leading to the creation of accurate depth perception model.

In this paper, we propose a blind copy move image forgery detection method using dyadic wavelet transform (DyWT). DyWT is shift invariant and therefore more suitable than discrete wavelet transform (DWT) for data analysis. First we decompose the input image into approximation (LL1) and detail (HH1) subbands. Then we divide LL1 and HH1 subbands into overlapping blocks and measure the similarity between blocks. The key idea is that the similarity between the copied and moved blocks from the LL1 subband should be high, while the one from the HH1 subband should be low due to noise inconsistency in the moved block. We sort pairs of blocks based on high similarity using the LL1 subband and high dissimilarity using the HH1 subband. Using thresholding, we obtain matched pairs from the sorted list as copied and moved blocks. Experimental results show the effectiveness of the proposed method over competitive methods using DWT and the LL1 or HH1 subbands only.

In this paper, a shift-invariant filtered backprojection cone-beam image reconstruction algorithm is derived, based upon Katsevich's general inversion scheme, and validated for the source trajectory of two concentric circles. The source trajectory is complete according to Tuy's data sufficiency condition and is used as the basis for an exact image reconstruction algorithm. The algorithm proceeds according to the following steps. First, differentiate the cone-beam projection data with respect to the detector coordinates and with respect to the source trajectory parameter. The data are then separately filtered along three different orientations in the detector plane with a shift-invariant Hilbert kernel. Eight different filtration groups are obtained via linear combinations of weighted filtered data. Voxel-based backprojection is then carried out from eight sets of view angles, where separate filtered data are backprojected from each set according to the backprojection sets' associated filtration group. The algorithm is first derived for a scanning configuration consisting of two concentric and orthogonal circles. By performing an affine transformation on the image object, the developed image reconstruction algorithm has been generalized to the case where the two concentric circles are not orthogonal. Numerical simulations are presented to validate the reconstruction algorithm and demonstrate the dose advantage of the equal weighting scheme.

This paper reviews design considerations along with measurement results pertinent to amorphous silicon (a-Si:H) thin-film transistor (TFT) drive circuits for active matrix organic light-emitting diode displays, and follows from work presented earlier (A. Nathan et al., 2002), (A. Nathan et al., 2003)

Recently, shift-invariant tensor factorisation algorithms have been proposed for the purposes of sound source separation of pitched musical instruments. However, in practice, existing algorithms require the use of log-frequency spectrograms to allow shift invariance in frequency which causes problems when attempting to resynthesise the separated sources. Further, it is difficult to impose harmonicity constraints on the recovered basis functions. This paper proposes a new additive synthesis-based approach which allows the use of linear-frequency spectrograms as well as imposing strict harmonic constraints, resulting in an improved model. Further, these additional constraints allow the addition of a source filter model to the factorisation framework, and an extended model which is capable of separating mixtures of pitched and percussive instruments simultaneously.

The Banach space ℓ 1 (Z) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ℓ 1 (Z) weak *continuous. This is equivalent to making the natural convolution multiplication on ℓ 1 (Z) separately weak *-continuous and so turning ℓ 1 (Z) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C 0 (K) (for countable locally compact K) and ℓ 1 (Z) is c 0 (Z). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak *-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c 0. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of Z. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c 0. * Supported by NSF grants DMS0856148 and DMS0556013.

A novel technique for despeckling the medical ultrasound images using lossy compression is presented. The logarithm of the input image is first transformed to the multiscale wavelet domain. It is then shown that the subband coefficients of the log-transformed ultrasound image can be successfully modeled using the generalized Laplacian distribution. Based on this modeling, a simple adaptation of the zero-zone and reconstruction levels of the uniform threshold quantizer is proposed in order to achieve simultaneous despeckling and quantization. This adaptation is based on: 1) an estimate of the corrupting speckle noise level in the image; 2) the estimated statistics of the noise-free subband coefficients; and 3) the required compression rate. The Laplacian distribution is considered as a special case of the generalized Laplacian distribution and its efficacy is demonstrated for the problem under consideration. Context-based classification is also applied to the noisy coefficients to enhance the performance of the subband coder. Simulation results using a contrast detail phantom image and several real ultrasound images are presented. To validate the performance of the proposed scheme, comparison with two two-stage schemes, wherein the speckled image is first filtered and then compressed using the state-of-the-art JPEG2000 encoder, is presented. Experimental results show that the proposed scheme works better, both in terms of the signal to noise ratio and the visual quality.

To accelerate the training of kernel machines, we propose to map the input data to a randomized low-dimensional feature space and then apply existing fast linear methods. Our randomized features are designed so that the inner products of the transformed data are approximately equal to those in the feature space of a user specified shift-invariant kernel. We explore two sets of random features, provide convergence bounds on their ability to approximate various radial basis kernels, and show that in large-scale classification and regression tasks linear machine learning algorithms that use these features outperform state-of-the-art large-scale kernel machines.

We present a new fast algorithm for Recursive Least-Squares (RLS) adaptive filtering that uses displacement structure and subsampled updating. The FSU VrF algorithm is based on the Fast Transversal Filter (vr~) algorithm, which exploits the shift invariance that is present in the RLS adaptation of a FIR filter. The FTF algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the Levinson algorithm. In the subsampled updating approach, we accumulate the rotation matrices over some time interval before applying them to the filters. It turns out that the successive rotation matrices themselves can be obtained from a Schur type algorithm which, once properly initialized, does not require inner products. The various convolutions that thus appear in the algorithm are done using the Fast Fourier Transform (EFT). For relatively long filters, the computational complexity of the new algorithm is smaller than the one of the well-known LMS algorithm, rendering it especially suitable for applications such as acoustic echo cancellation.

Shift and rotation invariance properties of linear time-frequency representations are investigated. It is shown that among all linear time-frequency representations, only the short-time Fourier transform (STFT) family with the Hermite-Gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the Wigner distribution (WD). By extending the time-bandwidth product (TBP) concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is defined. For mono-component signals, it is shown that GTBP provides a rotation independent measure of compactness. Similar to the TBP optimal STFT, the GTBP optimal STFT that causes the least amount of increase in the GTBP of the signal is obtained. Finally, a linear canonical decomposition of the obtained GTBP optimal STFT analysis is presented to identify its relation to the rotationally invariant STFT.

A denoising scheme is presented suitable for magnetic exploration data. It is based on an empirical study and combines the shift invariant denoising method with the use of a variable threshold, which is calculated in each cycle of the shift invariant algorithm. The calculation of each threshold is based on the wavelet transformation of random noise with the same standard deviation as in the data. The scheme is particularly useful in "archaeological geophysics" where it is important to produce results that mostly resemble the plane view of the buried objects.

We present an algorithm for multilinear decomposition that allows for arbitrary shifts along one modality. The method is applied to neural activity arranged in the three modalities space, time, and trial. Thus, the algorithm models neural activity as a linear superposition of components with a fixed time course that may vary across either trials or space in its overall intensity and latency. Its utility is demonstrated on simulated data as well as actual EEG, and fMRI data. We show how shift-invariant multilinear decompositions of multiway data can successfully cope with variable latencies in data derived from neural activitya problem that has caused degenerate solutions especially in modeling neuroimaging data with instantaneous multilinear decompositions. Our algorithm is available for download at www.erpwavelab.org.

The paper aims to establish a new sharp Burkholder-type maximal inequality in Lp for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed leading to an exponential inequality for maximum of partial sums. As an application we present an invariance principle for partial sums of certain maps of Bernoulli shifts processes.

Abstract:-Real valued wavelets are used widely in signal processing applications. Although complex valued wavelets exists, but rarely used. Complex wavelet transform provides important phase information of the signal and it is almost shift-invariant. Due to these added advantages it can be very much useful for signal processing applications. This paper explores various properties of Daubechies complex wavelet transform. It shows that the nature of the complex wavelet coefficients does not change at multiple levels. This ...

The 2-band discrete wavelet transform (DWT) provides an octave-band analysis in the frequency domain, but this might not be 'optimal' for a given signal. The discrete wavelet packet transform (DWPT) provides a dictionary of bases over which one can search for an optimal representation (without constraining the analysis to an octave-band one) for the signal at hand. However, it is well known that both the DWT and the DWPT are shift-varying. Also, when these transforms are extended to 2-D and higher dimensions using tensor products, they do not provide a geometrically oriented analysis. The dual-tree complex wavelet transform (DT-CWT), introduced by Kingsbury, is approximately shift-invariant and provides directional analysis in 2-D and higher dimensions. In this paper, we propose a method to implement a dual-tree complex wavelet packet transform (DT-CWPT), extending the DT-CWT as the DWPT extends the DWT. To find the best complex wavelet packet frame for a given signal, we adapt the basis selection algorithm by Coifman and Wickerhauser, providing a solution to the basis selection problem for the DT-CWPT. Lastly, we show how to extend the 2-band DT-CWT to an M -band DT-CWT (provided that M = 2 b ) using the same method.

We present a new, doubly fast algorithm for recursive least-squares (RLS) adaptive filtering that uses displacement structure and subsampled-updating. The fast subsampled-updating stabilized fast transversal filter (FSU SFTF) algorithm is mathematically equivalent to the classical fast transversal filter (FTF) algorithm. The FTF algorithm exploits the shift invariance that is present in the RLS adaptation of an FIR filter. The FTF algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the Levinson algorithm. In the subsampled-updating approach, we accumulate the rotation matrices over some time interval before applying them to the filters. It turns out that the successive rotation matrices themselves can be obtained from a Schur-type algorithm that, once properly initialized, does not require inner products. The various convolutions that appear in the algorithm are done using the fast Fourier transform (FFT). The resulting algorithm is doubly fast since it exploits FTF and FFT's. The roundoff error propagation in the FSU SFTF algorithm is identical to that in the SFTF algorithm: a numerically stabilized version of the classical FTF algorithm. The roundoff error generation, on the other hand, seems somewhat smaller. For relatively long filters, the computational complexity of the new algorithm is smaller than that of the well-known LMS algorithm, rendering it especially suitable for applications such as acoustic echo cancellation.

In this paper, we present a new method for incorporating global shift invariance in support vector machines. Unlike other approaches which incorporate a feature extraction stage, we first scale the image and then classify it by using the modified support vector machines classifier. Shift invariance is achieved by replacing dot products between patterns used by the SVM classifier with the maximum cross-correlation value between them. Unlike the normal approach, in which the patterns are treated as vectors, in our approach the patterns are treated as matrices (or images). Cross- correlation is computed by using computationally efficient techniques such as the fast Fourier transform. The method has been tested on the ORL face database. The tests indicate that this method can improve the recognition rate of an SVM classifier.

If f: C -> P^n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation t(z)=z+c, then f is periodic with period c. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This qumbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation and to solve a linear ordinary second order q-difference equation.

Considering human visual system and characteristics of images, a novel image fusion strategy is presented for panchromatic high resolution image and multispectral image in nonsubsampled contourlet transform (NSCT) domain. The NSCT can give an asymptotic optimal representation of edges and contours in image by virtue of its characteristics of good multiresolution, shiftinvariance, and high directionality. An intensity component addition strategy based on LHS transform is introduced into NSCT domain to preserve spatial resolution and color content. Experiments show that the fusion method proposed can improve spatial resolution and keep spectral information simultaneously, and that there are improvements both in visual effects and quantitative analysis compared with the traditional principle component analysis (PCA) method, intensity-hue-saturation (IHS) transform technique, wavelet transform weighted fusion method, corresponding wavelet transform-based fusion method, and contourlet transform-based fusion method.

We introduce a framework for construction of non-separable multivariate splines that are geometrically tailored for general sampling lattices. Voronoi splines are B-spline-like elements that inherit the geometry of a sampling lattice from its Voronoi cell and generate a lattice-shift-invariant spline space for approximation in . The spline spaces associated with Voronoi splines have guaranteed approximation order and degree of continuity. By exploiting the geometric properties of Voronoi polytopes and zonotopes, we establish the relationship between Voronoi splines and box splines which are used for a closed-form characterization of the former. For Cartesian lattices, Voronoi splines coincide with tensor-product B-splines and for the 2-D hexagonal lattice, the proposed approach offers a reformulation of hex-splines in terms of multi-box splines. While the construction is for general multidimensional lattices, we particularly characterize bivariate and trivariate Voronoi splines for all 2-D and 3-D lattices and specifically study them for body centered cubic and face centered cubic lattices.

Dual-tree complex wavelet transform (DTCWT) is a shift invariant transform with limited redundancy. Complex quadrature signals are dual channel signals obtained from the systems employing quadrature demodulation. An example of such signals is quadrature Doppler signal obtained from blood flow analysis systems. Prior to processing Doppler signals using the DTCWT, directional flow signals must be obtained and then two separate DTCWT applied, increasing the computational complexity. In order to decrease computational complexity, a modified DTCWT algorithm is proposed. A comparison between the new transform and the phasing-filter technique is presented. The results show that the proposed method gives the same output as the phasing-filter method and the computational complexity for processing quadrature signals using DTCWT is greatly reduced.

Abstract. In this paper the characterization as convolution operators of filters sending finite energy signals to bounded signals is used to prove several theoretical results concerning the distance between the ideal filter and the spaces of physically realizable filters. Both the analog and the digital cases are studied and the formulas for the distance and the angle between the filters in each case are also given.

Let $\Sigma$ be a finite alphabet, $\Omega=\Sigma^{\mathbb{Z}^{d}}$ equipped with the shift action, and $\mathcal{I}$ the simplex of shift-invariant measures on $\Omega$. We study the relation between the restriction $\mathcal{I}_n$ of $\mathcal{I}$ to the finite cubes $\{-n,...,n\}^d\subset\mathbb{Z}^d$, and the polytope of "locally invariant" measures $\mathcal{I}_n^{loc}$. We are especially interested in the geometry of the convex set $\mathcal{I}_n$ which turns out to be strikingly different when $d=1$ and when $d\geq 2$. A major role is played by shifts of finite type which are naturally identified with faces of $\mathcal{I}_n$, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of $\mathcal{I}_n$, although in dimension $d\geq 2$ there are also extreme points which arise in other ways. We show that $\mathcal{I}_n=\mathcal{I}_n^{loc}$ when $d=1$, but in higher dimension they differ for $n$ large enough. We also show ...

Dual-tree complex wavelet transform (DTCWT) is a shift invariant transform with limited redundancy. Complex quadrature signals are dual channel signals obtained from the systems employing quadrature demodulation. An example of such signals is quadrature Doppler signal obtained from blood flow analysis systems. Prior to processing Doppler signals using the DTCWT, directional flow signals must be obtained and then two separate DTCWT applied, increasing the computational complexity. In order to decrease computational complexity, a modified DTCWT algorithm is proposed. A comparison between the new transform and the phasing-filter technique is presented. The results show that the proposed method gives the same output as the phasing-filter method and the computational complexity for processing quadrature signals using DTCWT is greatly reduced.

This paper reviews design considerations along with measurement results pertinent to amorphous silicon (a-Si:H) thin-film transistor (TFT) drive circuits for active matrix organic light-emitting diode displays, and follows from work presented earlier (A. Nathan et al., 2002), (A. Nathan et al., 2003)

An approach for designing a critically sampled FIR rational rate filter bank satisfying conditions such as perfect reconstruction, regularity orders and good frequency selectivity is proposed. This approach is used to design a rational discrete wavelet transform with dilation factor of 3/2 and satisfies a shift invariant property approximately. The rational filter bank design problem is transformed to an equivalent uniform M-channel filter bank design. The perfect reconstruction (PR) condition gives rise to a set of non-linear conditions between the synthesis and analysis filter coefficients while regularity gives rise to a set of linear constraints. The design problem is solved using a constrained nonlinear optimisation program in which the PR and regularity are the constraints to an objective function defined as sum of the squared of the difference between the magnitude responses of the designed analysis/synthesis filters and the ideal brick-wall responses. The optimisation program used for the design is a simple linear-search built-in program in Matlab. The obtained filter bank designs show excellent frequency selectivity and their iterations are approximately shift invariant.

This paper addresses the problem of estimating the shape parameters of seismic wavefields in linear array for the separation of waves by exploiting the diversity of the polarization state and propagation velocity between wavefields. We propose a method based on the shiftinvariance properties of multiple wavefields impinging on a uniform linear array of sensors to separate different waves, by jointly estimating the velocity and polarization parameters. Furthermore, the surface waves are removed using a spatial filter specifically designed to exploit velocity and polarization of different waves. We introduce a model for wideband polarized signals received by an array of three-components sensors used as a framework for implementing the proposed algorithms. Examples on simulated and experimental data illustrate the applicability of the proposed methods.

Recently Bolotnikov and Rodman characterized finite dimensional backward shift invariant subspaces of the Arveson space. In this note we study finite dimensional backward shift invariant Hilbert spaces contractively included in the Arveson space. To that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions.