research effort to build a complete filter bank theory. Necessary and sufficient conditions for decomposing a signal in subsampled components with a filtering scheme, and recovering the same signal with an inverse transform, were established by Smith and Barnwell . Daubechies  designed univariate two-channel perfect reconstruction filterbanks having finite impulse response (FIR) corresponding to univariate orthonormal wavelet having compact support and vanishing moments. According to the theory of subband filtering and multiresolutionanalysis, in the two channel case, a filter bank is the vector (m 0 , m 1 , m ˜ 0 , m ˜ 1 ) with
It is known that there is a close relation between filterbanks and wavelets. A lot of discrete wavelet transforms can be implemented as filterbanks, and perfect reconstructed filterbanks may generate wavelets, provided that they satisfy some regularity condition. The condition for a filter bank being capable to generate wavelet bases is very complicated, as indicated in . However, a necessary condition for a filter can be used to generate a scaling function is that it must have some zeros , at the point of z 1 . If this condition is satisfied, the filter bank may be capable to generate a multiresolutionanalysis and a wavelet basis .
This paper explored the underlying mathematical framework of the one-dimensional multiresolutionanalysis based on the nonequally spaced samples environment. The study shows that the scaling and wavelet functions are not respectively given by dilating and translating one unique prototype function as in the traditional case. The specification of the multiresolution spaces involves the construction of the corresponding scaling and wavelet bases. As shown in previous work, the orthonormalization procedure has aﬀected the regularity of the piecewise polynomial scaling and wavelet functions. This paper has proposed a new basis orthonormalization procedure which (i) reduces the number of the wavelets to a single one per interval (depending on the localization of the samples) and (ii) satisfies the continuity conditions of the scaling and wavelet functions on the considered knot sequence. Moreover the orthogonal decomposition is implemented using filterbanks.
The analysis of signals by filterbanks requires minim- ally filtering techniques using Fourier analysis , which use complex sinusoids as basis functions. How- ever, a difficulty that has often been raised with this ap- proach is that, because of the infinite extent of the basis functions, any time-local information is spread out over the whole frequency axis . Under such constraints, the wavelet basis is a set of functions that can represent signals with good resolution in both time and frequency domains. The wavelet transform is well defined within the multiresolution framework, which allows signal ana- lysis in several scales. Wavelets are characterized by time locality, allowing an efficient capture of transient behav- ior in a signal. Furthermore, the time-frequency reso- lution trade-off, provided by the multiresolutionanalysis, enables a better signal representation over the
(infinite impulse response) transfer functions can be used for generating the overall system. The selection of the filter type depends on the criteria at hand. An FIR filter easily achieves a strictly linear-phase response, but requires a larger number of operations per output sample when compared with an equal magnitude response IIR filter. The linear-phase FIR filter is an adequate choice when the waveform of the signal has to be preserved. An advantage of the multirate design approach is the ability of improving significantly the efficiency of FIR filters thus making them very desirable in practice [11-15].
(prefiltered) to produce reasonable values for the coe ﬃ cients of the multiscaling function at the finest scale . A num- ber of preprocessing techniques have been proposed for this purpose. Repeated row (RR) and approximation (AP) pre- processings are the two most widely used techniques . In RR preprocessing, the rows of the input vector to the filter- bank are obtained by scaling the first row consisting of the signal samples. This preprocessing increases the total num- ber of samples leading to an oversampling of the original sig- nal. AP preprocessing, which is based on the approximation properties of continuous-time wavelets, on the other hand, yields a critically sampled representation.
The first step towards facial expression analysis using images was taken by Suwa, Sugie and Fujimora . But the pioneering work of Mase and Pentland  was the one to inspire other researchers in this field. Surveys of the work done in this field till the last decade is found in the works of Fasel et. al and Pantic [21, 44]. A glance, from now, at the previous works broadly divides the fields into two parts; (a) Static-image based recognition (b) Image sequence or video based recognition. Static image based researches rely on peak or apex of the expressions and process still images, and hence, they miss subtle details of expressions. But, these methods are faster and simpler compared to their image sequence or video based counterparts that can capture the dynamics of expressions. In spite of this dichotomy, all facial expression recognition approaches abide by a generic framework.
the actual frequency characteristics of the image. Some compression schemes exploit the strong directional features in fingerprint images caused by ridges and valleys. A scanning procedure following dominant ridge direction has shown to improve lossless coding results as compared to JPEG-LS and PNG . A wavelet footprint repre- sentation characterizing efficiently singular structures (corresponding to ridges)  that delivers better results as compared to the SPIHT algorithm was proposed by Sudhakar et al.. Contourlets , ,  and contourlet packets  are used to ex- ploit directional information which also results in PSNR improvements as compared to classical algorithms. Recently, fingerprint image compression schemes that use ge- netic algorithm , ,  to generate wavelet and scaling coefficients for each level of decomposition have also been proposed. Multiresolutionanalysis tools have been successfully applied to fingerprint image compression in the last two decades; we propose two new fingerprint image compression techniques based on wave atoms decomposition:
The multiresolution approach takes into account the requirement for invariance over scale by representing the image at several dierent spatial resolutions. This necessarily introduces an element of redundancy, but this is a relatively small price to pay for the advantages it aords. In this technique, the original image is sub-sampled to create a set of images known as a pyramid , starting with the original image, and with each successive image at a lower resolution (usually half) than its predecessor. Originally, most pyramids were built in a quad-tree structure, with each sub-sampled pixel having four children, and each child only one parent. In this case the parents grey level would be set to the average value of its four children. However, Burt and Adelson  proposed the use of a Gaussian weighting function to replace the quad- tree approach, in which for a 5 by 5 lter each parent has 25 children, and each child 9 parents. To state this more precisely, if w(m;n) is a Gaussian weighting function, and g l (i;j) the image at level l of the pyramid, then the construction of the pyramid
-band filter bank . In this approach all the filters of analysis and synthesis section are obtained by cosine modulation of single linear phase prototype low pass filter which normally has linear phase and a finite length impulse response as shown in Figure 1. Let H(z) be the transfer function of the prototype filter. It is given as:
suboptimal, as has been shown in some later pa- pers. However, they have made several impor- tant observations concerning the properties of low-delay filterbanks that apply to PR biorthog- onal filterbanks as well as to QMF banks. First, it is not advisable to design filterbanks with a very small delay compared to the filter orders, because after a certain filter order for the same overall delay, larger orders result only in a neg- ligible improvement in the performance of the filter bank. Second, additional constraints are usually necessary in the transition bands of the filters due to the possible artifacts occurring in these bands. Abdel-Rahem, El-Guibaly, and Antoniou presented in 9 ] two approaches for
Auditory filters of different shapes have been trained discriminatively for robust speech recognition . Fil- ter banks can also be trained discriminatively using Fisher discriminant analysis (FDA) method . In recent years, deep neural networks (DNN) have achieved significant success in the field of audio processing and recogni- tion because of its advantages in discriminative fea- ture extraction. Standard filterbanks computed in the time domain have been simulated using unsupervised convolutional restricted Boltzmann machine(ConvRBM) . The speech recognition performance of ConvRBM features is improved compared to the Mel-frequency cep- strum coefficients (MFCCs), and the relative improve- ments are 5% on TIMIT test set and 7% on WSJ0 database using GMM-HMM systems. Discriminative frequency fil- ter banks can also be learned together with the recogni- tion error using a time-convolutional layer and a temporal pooling layer over the raw waveform . The results in  show that the filter size and pooling operation play an important role in the performance improvement, but the temporal convolutional operation is time-consuming.
After linear phase and PR being imposed on a filter bank, the remaining degrees of freedom can be used for gain optimiza- tion (see (6)), or more importantly, to achieve subjectively good performance. It is obvious that the more degrees of freedom that can be exploited towards a given optimization criterion, the better. The correspondence between subjective criteria and simple mathematical criteria, as used presently, is rather poor. Typically, filterbanks are designed to mini- mize the mean square error (MSE) after signal decompres- sion for a given source statistics and quantization scheme. Furthermore, encapsulating subjective performance criteria into a set of mathematical equations which can be incorpo- rated into an overall optimization criterion is warranted.
(10) and (15)) for these three filterbanks are given in Table 2. For comparison purposes, we also consider four filterbanks produced using methods previously proposed by others, with three being quincunx and one being separable. The first two quincunx filterbanks are constructed using the technique of , and are henceforth referred to by the names KS1 and KS2. The third quincunx filter bank is the so-called (6, 2) fil- ter bank proposed in , which we henceforth refer to by the name G62. The one separable filter bank considered herein is the well-known 9/7 filter bank employed in the JPEG-2000 standard . Some important characteristics of the various filterbanks are shown in Table 3. The OPT1 filter bank was designed using Algorithm 1 with two lifting steps. The next two filterbanks, referred to as OPT3 and OPT4, were de- signed using Algorithm 2 with three or more lifting steps, and thus, the desired vanishing-moment conditions are only guaranteed to be met approximately (i.e., the moments in question are only guaranteed to be close to zero). For each of these two filterbanks, the order of the largest nonzero moment (of those in question) is shown in the rightmost column of Table 3. The frequency responses of the analysis and synthesis lowpass filters are shown in Figures 7, 8, and 9. Since the highpass filter frequency responses are simply mod- ulated versions of the lowpass ones, the former have been omitted here due to space constraints. The primal scaling and wavelet functions are illustrated in Figures 10, 11, and 12.
basis functions, the most attractive and distinctive prop- erty of B-splines are that they are compactly supported and can be analytically formulated in an explicit form. Most importantly, they form a multiresloution analysis (MRA) (Chui, 1992). B-splines are unique, among many com- monly used basis functions, because they simultaneously possess the three remarkable properties, namely compactly supported, analytically formulated and multiresolution anal- ysis oriented, among many popular basis functions. These splendid properties make B-splines remarkably appropriate for nonlinear dynamical system modelling. The most com- monly used B-splines are those of orders 1 to 4, which are shown in Table 1.
With recent advances in remote sensing technology, many change detection methodologies have been developed. From simple image differencing algorithms to more advanced regression algorithms, these techniques help to map the extent of change in our environment. In this study, potential uses of correspondence analysis (CA) are explored for performing change detection. Results are compared to known techniques (i.e, principal component analysis (PCA), NDVI differencing, etc.) to assess its performance. Even though CA is a widely known technique among ecologists and statisticians, its application to remote sensing is new. Similar to PCA, it transforms data into different coordinate systems so that the first component accounts for most of the information within the original data (Carr and Matanawi, 1999). The biggest difference between PCA and CA is that CA uses chi-square distance as opposed to Euclidean distance (Ludwig and Reynolds, 1988). The following is an exploration of the potential uses of CA in change detection studies.
An analog filter is a filter which operates on continuous- time signals. It is designed with various electronic devices and components. It is an electronic circuit which is used to modify the signal in a predefined way that is applied at the input of this filter circuit. Output of this circuit depends on the characteristics of the filter. Analog filters are basic building block of signal processing circuits. Analog filters are used in audio signal processing, multichannel transmission, telephony, radio, television, stereo systems, conditioning of electrical power supply  , eliminating harmonics in an electronic circuits, noise elimination circuits, etc.
Filter bank based multicarrier (FBMC) system is an alternative technique to overcome the limitations in OFDM. The main characteristics of FBMC system is that it provides better spectral shaping of the sub bands which in turn simplifies the equalization of inter- carrier and inter-symbol interference when no cyclic prefix is used. This system is developed from the multirate filterbanks in transmultiplexer configuration . The IFFT and FFT block in OFDM systems are replaced by synthesis and analysis filters of filter bank theory in FBMC system. Cosine Modulated Filterbanks (CMFB) uses filters of larger length and provides smaller side lobes and better stop band attenuation than Discrete Fourier transform (DFT) based systems . The analysis and synthesis filters are cosine modulated versions of low pass prototype filter and therefore the design of entire filter bank reduces to the design of a single prototype filter. Here the prototype filter for CMFB is designed using window method with Kaiser Window as the window function. Kaiser window offers a simple means of designing low pass FIR filters and can have a direct control over the stop-band attenuation of the prototype filter .