Sampling for shift-invariant and wavelet subspaces

This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

Sampling theory is one of the most powerful results in signal analysis. The objective of sampling is to reconstruct a signal from its samples. Walter extended the Shannon sampling theorem to wavelet subspaces. In this paper we give a further characterization on some shift-invariant subspaces, especially the closed subspaces on which the sampling theorem holds. For some shift-invariant subspaces with sampling property, the sampling functions are explicitly given.

Let M be a shift invariant subspace in the vector-valued Hardy space \({H_{E}^{2}(\mathbb{D})}\). The Beurling–Lax–Halmos theorem says that M can be completely characterized by \({\mathcal{B}(E)}\)-valued inner function \({\Theta}\). When \({E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}\) is the Hardy space on the bidisk \({H^{2}(\mathbb{D}^2)}\). Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function \({\Theta(z)}\) for two well-known invariant subspaces in \({H^{2}(\mathbb{D}^{2})}\). This paper generalizes the \({\Theta(z)}\) by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of \({M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}\) when M is an invariant subspace in \({H^{2}(\mathbb{D}^{2})}\). Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.

An analysis of wavelet frame based scattered data reconstruction

The Shift-Invariant Subspaces in L1(R)

We consider dual frames generated by actions of countable discrete groups on a Hilbert space. Module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group. This has applications to shift-invariant spaces and wavelet theory. One of the main findings in this paper is that whenever a shift-invariant subspace in $L^2(\mathbb {R}^n)$ has compactly supported dual frame generators, then it also has compactly supported bi-orthogonal generators. The crucial part in the proof is a theorem by Swan that states that every finitely generated projective module over the Laurent polynomials in $n$ variables is free.

When searching for landmines using vehicular-mounted sensors, it is important that the ground locations of the detected mines be accurately determined. This is useful for data association when one has multiple looks at a mine by a single sensor or if one uses multiple sensors. This knowledge is necessary for neutralizing the detected mines or at least marking them for avoidance. Factors that contribute to errors in geo-location include sensor error, inaccurate knowledge of the vehicle position and/or attitude, and incomplete knowledge about the terrain being searched. This paper addresses the problem of incomplete terrain knowledge and its effects on geo-location accuracy. It presents different techniques for modelling the terrain and reducing the geo-location errors. A variety of uneven surfaces have been simulated. Random noise has been added to the basic surfaces to simulate the terrain roughness. A terrain model is then derived based on measurements one could obtain from a coarse scan of the field of view using a ranging laser. Two types of sensors are considered for the landmine detection: Synthetic Aperture Forward-Looking Radar (SAFLR) and Camera Type Sensors such as Infrared (IR). Relationships between the terrain unevenness and resulting geo-location errors are presented. The geo-location errors are worse for the Camera Type Sensor, but they can be significant for SAFLR also. The results of this analysis demonstrate that there may be significant geo-location errors for situations where the terrain is not so smooth, e.g. off-road searches, and that the problem can be alleviated via better knowledge of the terrain. If the field of view is planar or almost planar, even coarse-range scanning can improve the geo-location accuracy. For more complex surfaces, finer scanning may be required (preliminary results on this topic were presented at the SPIE (International Society for Optical Engineering) Conference, Orlando, FL, April 2001 (G. Cook, S. Jakkidi, S. Kansal, C. Amazeen and K. Sherbondy, “Impact of uneven terrain on geo-locations errors for mines detected via vehicular mounted sensors”, in Proceedings of SPIE, 2001, pp. 594–605.)). The computed geo-location errors, and the conclusions drawn as to the effectiveness of the different models are presented in the paper.

The structure of shift–modulation invariant spaces: The rational case

Cheng-An Lin, a biomedical engineer at Chung Yuan Christian University (CYCU) in Taiwan, received the 2010 Young Investigator Award from the International Society for Optical Engineering (SPIE) on 25 January.

I am pleased to welcome you to this timely and informative conference organized by The International Society for Optical Engineering (SPIE). As Conference Chair, I amvery pleased, but not at all surprised, by your interest and enthusiastic response: Over 300 participants are contributing over 210 technical papers to six topical conferences.

There are two kinds of RRP: (1) invertible ones, such as global Fourier transform (FT), local wavelet transform (WT), and adaptive wavelet transform (AWT); and (2) non-invertible ones, e.g. ICA including the global principle component analysis (PCA). The invertible FT and WT can be related to the non-invertible ICA when the continuous transforms are approximate din discrete matrix-vector operations. The landmark accomplishment of ICA is to obtain, by unsupervised learning algorithm, the edge-map as image feature ayields, shown by Helsinki researchers using fourth order statistics of nyields -- Kurosis K(uyields), and derived from information- theoretical first principle is augmented by the orthogonality property of the DWT subband used necessarily for usual image compression. If we take the advantage of the subband decorrelation, we have potentially an efficient utilization of a pari of communication channels if we could send several more mixed subband images through the pair of channels.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

An optimal quadratic filter is developed to detect a target signature distorted by a linear transformation group (e.g. translations, scaling, etc.). The filter is constructed from the invariant subspaces of the transformation group. This guarantees an equal response to all target images. A maximum signal-to-clutter ratio is achieved with a weighted combination of invariant subspaces. Examples are discussed for some one-parameter subgroups of the projective transformation.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Backward shift operator and finite dimensional de Branges Rovnyak spaces in the ball

In target recognition based on high range resolution profile (HRRP), the major challenge is to extract invariant and robust features from the measured high dimension range profile samples. Focused on the reduction of dimensionality, a novel algorithm for extracting HRRP features is proposed in this paper, and the Legendre moments with orthogonal and translation-invariant properties was adopted. The proposed algorithm utilizes the average HRRP in aspect-frame, and then preprocesses the HRRP with amplitude l1 normalization. The Legendre moments are calculated with mapping range values into orthogonal definition domain to provide invariant feature vectors. Meanwhile, spectrum analysis (SA), which can be regard as first-order approximation of relax algorithm, is adopted to reconstruct the real HRRP obtained by inverse fast fourier transform (IFFT) in order to desensitize moments in low SNR environment. Several experiments with four different targets measured data demonstrate that Legendre moments features with flexible maximum order to be chosen have significant advantages over central moments. SA can provide a more robust recognition performance in low signal noise ratio (SNR) environment as 5 dB compared with those of HRRP directly obtained by IFFT.

Approximation from shift-invariant spaces by generalized sampling formulas