Karhunen-Loeve Transform Research Articles

The Modified Oja-RLS Algorithm, Stochastic Convergence Analysis and Application for Image Compression

The results of theoretical analysis for stochastic convergence of the modified Oja-RLS learning rule are presented. The rule is used to find Karhunen Loeve Transform. Based on this algorithm, an image compression scheme is developed by combining approximated 2D KLT transform and JPEG standard quantization and entropy coding stages. Though 2D KLT transform is of higher complexity than 2D DCT, the resulting PSNR quality of reconstructed images is better even by 2[dB].

Lipreading from color video

We have designed and implemented a lipreading system that recognizes isolated words using only color video of human lips (without acoustic data). The system performs video recognition using "snakes" to extract visual features of geometric space, Karhunen-Loeve transform (KLT) to extract principal components in the color eigenspace, and hidden Markov models (HMM's) to recognize the combined visual features sequences. With the visual information alone, we were able to achieve 94% accuracy for ten isolated words.

Echo components for aspect-independent classification

Monostatic, backscattered data from a finite, ribbed cylinder are shown. These data from a nonspherical scatterer exhibit a great deal of angular dependence. This angular dependence complicates the design of a classifier which uses templates to test the similarity of a test signal with the known scatterer. The scattering data have previously been shown to primarily consist of components generated by specular reflection, elastic waves, and Bloch waves. The angular dependence of this data set is shown to be due to fading of each component from a physical mechanism, as well as the changing interference of the various components as their relative timing changes. The separation of the echoes into components is shown to decrease the angular dependence. Finally, it is shown that a modest number of templates can be derived from the angular-dependent components through the use of the Karhunen–Loeve transformation and the assumption of equal statistical weighting of each aspect angle. The ka range considered extends from approximately 3 to 10. [Work supported by ONR.]

Dynamic characteristics analysis of forcing jet by Karhunen-Loeve transformation

The snapshot method is introduced to approximate the coherent structures of planar forcing jet flow. The numerical simulation of flow field is simulated by discrete vortex method. With snapshot method we could treat the data efficiently and approximate coherent structures inhered in the planer jet flow. By forcing the jet at a sufficient amplitude and at a well-chosen frequency, the paring can be controlled in the region of the jet. Finally we expressed the underlying coherent structures of planar jet flow in the minimum number of modes by Karhunen-Loeve transformation in order to understand jet flow and to make the information storage and management in computers easier.

Region-Based Coding of Color Images Using Karhunen–Loeve Transform

In this paper we consider a number of extensions to current adaptive compression methods for color images. One, we use clustering or segmentation procedures to determine self-similar image regions. Two, for each such region we use a Karhunen–Loeve compression method to model the important spatiochromatic information. Three, we employ linear prediction to encode the resultant eigenimages. Finally, comparisons are made with current methods and improvements are demonstrated particularly for low bit-rate coding of color textured images such as those that occur in aerial photography.

Speech recognition using continuous density hidden Markov models and the orthogonalizing Karhunen–Loeve transformations

A recognition system comprises a feature extractor for extracting a feature vector x from an input speech signal, and a recognizing section for defining continuous density Hidden Markov Models of predetermined categories k as transition network models each having parameters of transition probabilities p(k,i,j) that a state Si transits to a next state Sj and output probabilities g(k,s) that a feature vector x is output in transition from the state Si to one of the states Si and Sj, and recognizing the input signal on the basis of similarity between a sequence X of feature vectors extracted by the feature extractor and the continuous density HMMs. Particularly, the recognizing section includes a memory section for storing a set of orthogonal vectors φ m (k,s) provided for the continuous density HMMs, and a modified CDHMM processor for obtaining each of the output probabilities g(k,s) for the continuous density HMMs in accordance with corresponding orthogonal vectors φ m (k,s).

Multispectral data compression using bidirectional interband prediction

The introduction of high spatial and spectral resolution sensors on-board remote-sensing spacecraft has increased, by orders of magnitude, the data rates which need to be sustained on the down-link or cross-link transmission channels. Since these channels are severely limited in capacity, the need arises to perform on-board compression to reduce the volume of data which would need to be down-linked. This paper discusses the development and refinement of a low complexity lossy spectral/spatial compression method which provides high compression ratios at low levels of distortion. The developed techniques uses pixels in adjacent bands to predict the intensity of pixels in the band being compressed via a simple linear prediction model. This prediction method when combined with a low-distortion discrete cosine transform (DCT) block coding method yields performance comparable to block-adaptive Karhunen-Loeve Transform (KLT)-DCT methods without incurring the complexity penalty of the KLT. The methods' performance suffers under misregistration. A fractional-pixel interpolation enhancement to the basic technique significantly improves the performance in the case of misregistered bands.

Relative Karhunen-Loeve transform

The Karhunen-Loeve transform (KLT) provides the best approximation for a stochastic signal under the condition that its rank is fixed. It has been successfully used for data compression in communication. However, since the KLT does not consider noise, its ability to suppress noise is very poor. For the optimum linear data compression in the presence of noise, we propose the concept of a relative Karhunen-Loeve transform (RKLT). It minimizes the sum of the mean squared error between the original signal and its approximation and the mean squared error caused by a noise under the condition that its rank is fixed. We also provide another type of RKLT. It minimizes the same sum under the condition that its rank is not greater than a fixed integer. Since the former type of RKLT does not always exist, we provide a necessary and sufficient condition under which it does exist. We also provide their general forms. The advantage of RKLTs is illustrated through computer simulations.

A signal subspace approach for speech enhancement

A comprehensive approach for nonparametric speech enhancement is developed. The underlying principle is to decompose the vector space of the noisy signal into a signal-plus-noise subspace and a noise subspace. Enhancement is performed by removing the noise subspace and estimating the clean signal from the remaining signal subspace. The decomposition can theoretically be performed by applying the Karhunen-Loeve transform (KLT) to the noisy signal. Linear estimation of the clean signal is performed using two perceptually meaningful estimation criteria. First, signal distortion is minimized while the residual noise energy is maintained below some given threshold. This criterion results in a Wiener filter with adjustable input noise level. Second, signal distortion is minimized for a fixed spectrum of the residual noise. This criterion enables masking of the residual noise by the speech signal. It results in a filter whose structure is similar to that obtained in the first case, except that now the gain function which modifies the KLT coefficients is solely dependent on the desired spectrum of the residual noise. The popular spectral subtraction speech enhancement approach is shown to be a particular case of the proposed approach. It is proven to be a signal subspace approach which is optimal in an asymptotic (large sample) linear minimum mean square error sense, assuming the signal and noise are stationary. Our listening tests indicate that 14 out of 16 listeners strongly preferred the proposed approach over the spectral subtraction approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficiency of shape-adaptive 2-D transforms for coding of arbitrarily shaped image segments

We introduce a formula to compute an optimum 2-D shape-adaptive Karhunen-Loeve transform (KLT) suitable for coding pels in arbitrarily-shaped image segments. The efficiency of the KLT on a 2-D AR(1) process is used to benchmark two other shape-adaptive transforms described in literature. It is shown that the optimum KLT significantly outperforms the well known shape-adaptive DCT method introduced by Gilge et al. (1989) for coding Segments of arbitrary shape in intraframe coding mode. A statistical transform gain close to the Gilge-method can be achieved with a shape-adaptive DCT algorithm introduced by Sikora and Makai (see Proc. Workshop Image Anal. Image Coding, Berlin, FRG, Nov. 1993) which is implemented with much lower complexity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Surface roughness classification using pattern recognition theory

Pattern recognition theory is introduced to perform rough surface classification. The light intensity distribution scattered from a rough surface is defined as the pattern vector of the rough surface. Using the Karhunen-Loeve transformation, the pattern vector is transformed into a feature vector for classification. To achieve a maximum separability, a modified method is proposed. The feasibility and effectiveness of the proposed method is demonstrated by computer simulation results.

High-speed closest codeword search algorithms for vector quantization

One of the most serious problems for vector quantization is the high computational complexity involved in searching for the closest codeword through a codebook in both codebook design and encoding phases. In this paper, based on the assumption that the distortion is measured by the squared Euclidean distance, two high-speed search methods will be proposed to speed up the search process. The first one uses the difference between the mean values of two vectors to reduce the search space. The second is to find the Karhunen-Loeve transform (KLT) for the distribution of the set of training vectors and then applies the partial distortion elimination method to the transformed vectors. Experimental results show that the proposed methods can reduce lots of mathematical operations.

Optimally adaptive transform coding

The optimal linear block transform for coding images is well known to be the Karhunen-Loeve transformation (KLT). However, the assumption of stationarity in the optimality condition is far from valid for images. Images are composed of regions whose local statistics may vary widely across an image. While the use of adaptation can result in improved performance, there has been little investigation into the optimality of the criterion upon which the adaptation is based. In this paper we propose a new transform coding method in which the adaptation is optimal. The system is modular, consisting of a number of modules corresponding to different classes of the input data. Each module consists of a linear transformation, whose bases are calculated during an initial training period. The appropriate class for a given input vector is determined by the subspace classifier. The performance of the resulting adaptive system is shown to be superior to that of the optimal nonadaptive linear transformation. This method can also be used as a segmentor. The segmentation it performs is independent of variations in illumination. In addition, the resulting class representations are analogous to the arrangement of the directionally sensitive columns in the visual cortex.

A new transform with symmetrical coding performance for Markov (1) signals

Based on the DCT and DST-II, we present a new transform that provides a symmetrical performance for positively and negatively correlated Markov (1) signals (first-order stationary Markov process). A fast computational algorithm is presented. A sufficient condition of symmetrical transforms is also provided, and the idea of constructing the discrete cosine-sine transform (DCST) is then generalized to the Karhunen-Loeve transform (KLT) of any Markov (1) signal. >

Neural network approaches to image compression

This paper presents a tutor a overview of neural networks as signal processing tools for image compression. They are well suited to the problem of image compression due to their massively parallel and distributed architecture. Their characteristics are analogous to some of the features of our own visual system, which allow us to process visual information with much ease. For example, multilayer perceptions can be used as nonlinear predictors in differential pulse-code modulation (DPCM). Such predictors have been shown to increase the predictive gain relative to a linear predictor. Another active area of research is in the application of Hebbian learning to the extraction of principal components, which are the basis vectors for the optimal linear Karhunen-Loeve transform (KLT). These learning algorithms are iterative, have some computational advantages over standard eigendecomposition techniques, and can be made to adapt to changes in the input signal. Yet another model, the self-organizing feature map (SOFM), has been used with a great deal of success in the design of codebooks for vector quantization (VQ). The resulting codebooks are less sensitive to initial conditions than the standard LBG algorithm, and the topological ordering of the entries can be exploited to further increasing the coding efficiency and reducing the computational complexity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Practical transform coding of multispectral imagery

We present an implementable three dimensional terrain adaptive transform based bandwidth compression technique for multispectral imagery. The algorithm exploits the inherent spectral and spatial correlations in the data. The compression technique is based on Karhunen-Loeve transformation for spectral decorrelation followed by the standard JPEG algorithm for coding the resulting spectrally decorrelated eigen images. The algorithm is conveniently parameterized to accommodate reconstructed image fidelities ranging from near-lossless at about 5:1 CR to visually lossy beginning at about 30:1 CR. The novelty of this technique lies in its unique capability to adaptively vary the characteristics of the spectral correlation transformation as a function of the variation of the local terrain. The spectral and spatial modularity of the algorithm architecture allows the JPEG to be replaced by a alternate spatial coding procedure. The significant practical advantage of this proposed approach is that it is based on the standard and highly developed JPEG compression technology.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Image compression with a dynamic autoassociative neural network

Image compression using neural networks has been attempted with some promise. Among the architectures, feedforward backpropagation networks (FFBPN) have been used in several attempts. Although it is demonstrated that using the mean quadratic error function is equivalent to applying the Karhunen-Loeve transformation method, promise still arises from directed learning possibilities, generalization abilities and performance of the network once trained. In this paper we propose an architecture and an improved training method to attempt to solve some of the shortcomings of traditional data compression systems based on feedforward neural networks trained with backpropagation—the dynamic autoassociation neural network (DANN). The successful application of neural networks to any task requires proper training of the network. In this research, this issue is taken as the main consideration in the design of DANN. We emphasize the convergence of the learning process proposed by DANN. This process provides an escape mechanism, by adding neurons in a random state, to avoid the local minima trapping seen in traditional PFBPN. Also, DANN's training algorithm constrains the error for every pattern to an allowed interval to balance the training for every pattern, thus improving the performance rates in recognition and generalization. The addition of these two mechanisms to DANN's training algorithm has the result of improving the final quality of the images processed by DANN. The results of several tasks presented to DANN-based compression are compared and contrasted with the performance of an FFBPN-based system applied to the same task. These results indicate that DANN is superior to FFBPN when applied to image compression.

Comparison of colour transformations for image segmentation

A statistical model for the colour variation of pixels contained in a uniform colour region of an image is proposed. A comparison of the colour transformations is performed using the correlation measure between colour axes. The result shows that the best transformation is the variable KL (Karhunen-Loeve) transformation of the colour model or its modifications.

Comparison of transform coding techniques for two-dimensional arbitrarily shaped images

Envisioned advanced multimedia video services include arbitrarily shaped (AS) image segments as well as regular rectangular images. Image segments of the TV weather report produced by the chromo-key technique [1] and image segments produced by video analysis and image segmentation [2–4] are typical examples of AS image segments. This paper explores efficient intraframe transform coding techniques for general two-dimensional (2D) AS image segments, treating the traditional rectangular images as a special case. In particular, we focus on the transform coding of the partially defined image blocks along the boundary of the AS image segments. We recognize two different approaches — thebrute force transform coding approach and theshape-adaptive transform coding approach. The former fills the uncovered area with the optimal redundant data such that the resulting transform spectrum is compact. A simple but efficient mirror image extension technique is proposed. Once augmented into full image blocks, these boundary blocks can be processed by traditional block-based transform techniques like the popular discrete cosine transform (DCT). In the second approach, we change either the transform basis or the coefficient calculation process adaptively based on the shape of the AS image segment. We propose an efficientshape-projected problem formulation to reduce the dimension of the problem. Existing coding algorithms, such as the orthogonal transform by Gilge [5] and the iterative coding by Kaup and Aach [6], can be interpreted intuitively. We also propose a new adaptive transform based on the same principle as that used in deriving the DCT from the optimal Karhunen-Loeve transform (KLT). We analyze the tradeoff relationship between compression performance, computational complexity, and codec complexity for different coding schemes. Simulation results show that complicated algorithms (e.g., iterative, adaptive) can improve the quality by 5–10 dB at some computational or hardware cost. Alternatively, the simple mirror image extension technique improves the quality by 3–4 dB without any overheads. The contributions of this paper lie in efficient problem formulations, new transform coding techniques, and numerical tradeoff analyses.

A facial feature communications interface for the non-vocal

A viable communications interface for the nonvocal handicapped that is a by-product of recent face recognition research is described. The system uses either the Karhunen-Loeve transform (KLT) or the discrete cosine transform (DCT) to encode video camera images on a subject's face. The encoding differentiates among three expressions: tongue out which is classified as 'yes'; mouth open, which is classified as 'no'; and mouth closed, which is classified as 'null'. A review of the theory of the KLT and DCT is included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>