Narrow Results

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τ_i, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τ_i is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τ_i to operators on the Hilbert space L²(μ). Instead, we show that it is possible to realize the maps τ_i quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫_X|φ|² dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and

We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

The eyes are important facial landmarks, both for image normalization due to their relatively constant interocular distance, and for post processing due to the anchoring on model-based schemes. This paper introduces a novel approach for the eye detection task using optimal wavelet packets for eye representation and Radial Basis Functions (RBFs) for subsequent classification ("labeling") of facial areas as eye versus non-eye regions. Entropy minimization is the driving force behind the derivation of optimal wavelet packets. It decreases the degree of data dispersion and it thus facilitates clustering ("prototyping") and capturing the most significant characteristics of the underlying (eye regions) data. Entropy minimization is thus functionally compatible with the first operational stage of the RBF classifier, that of clustering, and this explains the improved RBF performance on eye detection. Our experiments on the eye detection task prove the merit of this approach as they show that eye images compressed using optimal wavelet packets lead to improved and robust performance of the RBF classifier compared to the case where original raw images are used by the RBF classifier.

This paper is concerned with the case of an exogenous system in which a model is required to forecast a periodic output time series using a causal input. A novel approach is developed in which the wavelet packet transform is taken of both the dependent time series and causal input. This results in two sets of basis dictionaries and requires two bases to be chosen. It is proposed that the best bases to choose are those which maximize the mutual information. Input selection is then implemented by eliminating those coefficients of the selected input basis with low mutual information. As an example, a model is constructed to forecast short-term electrical demand.

It is well known that the Shannon Sampling Theorem allows us to fully recover a continuous-time bandlimited signal from its digital samples, as long as the sampling rate to be chosen is not smaller than the Nyquist frequency. This theory applies to all bandlimited signals, which may or may not occupy the entire frequency band. Hence, it is intuitively convincing that for continuous-time signals, such as those in speech, that do not fully utilize the entire frequency intervals, less digital samples are required for their full recovery. Current techniques in sub-band coding are used for achieving this goal. The objective of this paper is to present a wavelet theory for establishing the mathematical foundation of this sub-band coding approach. A wavelet packet decomposition of the signal provides the optimal sub-band coding bit-rate by using the Shannon wavelet library introduced in this paper.

Faces often produce inconsistent features under different lighting conditions. Classifying lighting effects within a face image is therefore the first crucial step of building a lighting invariant face recognition system. This paper presents a hybrid system to classify face images based on the lighting effects present in the image. The theories of multivariate discriminant analysis and wavelet packets transform are utilised to develop the proposed system. An extensive set of face images of different poses, illuminated from different angles, are used to train the system. The performance of the proposed system is evaluated by conducting experiments on different test sets and by comparing its results against those of some existing counterparts.

It is well known that the sensitivity to translations and orientations is a major drawback in 2D discrete wavelet transform (DWT). In this paper, we have proposed an effective scheme for rotation invariant adaptive wavelet packet transform. During decomposition, the wavelet coefficients are obtained by applying a polar transform (PT) followed by a row-shift invariant wavelet packet decomposition (RSIWPD). In the first stage, the polar transform generates a row-shifted image and is adaptive to the image size to achieve complete and minimum sampling rate. In the second stage, the RSIWPD is applied to the row-shifted image to generate rotation invariant but over completed subbands of wavelet coefficients. In order to reduce the redundancy and computational complexity, we adaptively select some subbands to decompose and form a best basis representation with minimal information cost with respect to an appropriate information cost function. With this best basis representation, the original image can be reconstructed easily by applying a row-shift invariant wavelet packet reconstruction (RSIWPR) followed by an inverse polar transform (IPT). In the experiments, we study the application of this representation for texture classification and achieve 96.5% classification accuracy.

Wavelet packets are subsets of a multiresolution analysis and retain many of the orthogonality, smoothness and localization properties of their parent wavelets. In this paper, we study the construction of p-wavelet packets associated with multiresolution p-analysis defined by Farkov for L²(ℝ⁺). The collection of all dilations and translations of the wavelet packets defines the general wavelet packets and is an overcomplete system.

This paper describes a study on evaluation of mental stress using adaptive wavelet packets filtering techniques. For mental stress to become chronic or if it remains for longer periods of time, then it can cause or trigger diseases such as hypertension, diabetes, insomnia, depression etc. There is a need for objective assessment of stress and develop counter conditions to relieve the body from stress. We have designed a wireless wearable sensor platform that can be used to capture a vast range of biomedical signals including heart rate, skin conductance and respiration rates. As stress is a reactionary phenomenon and its response vary from person to person, so an adaptive wavelet filtering packet is proposed to minimize intra subjective variations. The filtering technique has two folded effects; it reduces the classification error and it provides tuning to adapt it to users requirements. The wearable sensor platform is based on ease of use and a subject can be monitored for days and changes in his/her physiological conditions are recorded. The main advantage in monitoring stress is that if required in needs than necessary actions can be taken to remove stressful conditions.

In this paper, it is proved that under some conditions, wavelet packet basis of $L^2(ℝ)$ can be used as a tool for the uniform approximation of an $M$-times ($M>0$) continuously differentiable and square integrable function $f$. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function $f$ at lower levels are uniformly reliable and they uniformly approach zero as $j\to \infty$ are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in $L^2(ℝ)$ is given and its error analysis has been discussed.

Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal.158 (1998) 209–241] introduced the Nonuniform multiresolution analysis (NUMRA) whose translation set is not a group. Farkov [Orthogonal p-wavelets on ℝ₊, in Proc. Int. Conf. Wavelets and Splines (St. Petersburg State University, St. Petersburg, 2005), pp. 4–26] studied multiresolution analysis (MRA) on positive half line and constructed associated wavelets. Meenakshi et al. [Wavelets associated with Nonuniform multiresolution analysis on positive half line, Int. J. Wavelets, Multiresolut. Inf. Process.10(2) (2011) 1250018, 27pp.] studied NUMRA on positive half line and proved the analogue of Cohen's condition for the NUMRA on positive half line. We construct the associated wavelet packets for such an MRA and study its properties.

Solar, galactic and magnetospheric energetic particles can create a serious radiation danger during space activities and affect astronauts and equipment, up to and including loss. Anomalous changes in the dynamics of cosmic rays occur during periods of nonstationary solar phenomena and heliospheric disturbances. At the moment, the problem of identifying and forecasting anomalies in cosmic rays is open.

We propose an automated method for detecting and identifying non-periodic variations in cosmic rays, based on a combination of nonlinear adaptive schemes and deep learning. The Autoencoder paradigm and orthogonal package decompositions are used. The effectiveness of the method is confirmed by numerical calculations using data from neutron monitors of a network of ground stations.

In the paper, the notion of univariate orthogonal waveletpackets has been generalized. First, the definition for orthogonal vector-valued wavelet packets with scale = a (2 ≤ a ∈ Z) is given. A procedure for constructing the orthogonl vector-valued wavelet packets is presented. Second, the properties of thevector-valued wavelet packets are studied. Finally, a new orthogonal basis of space L²(R ,C^s) is derived from the orthogonal vector-valued wavelet packets.