Fukunaga-Koontz transform for small sample size problems

In this paper, we propose the Fukunaga-Koontz Transform (FKT) as applied to small-sample size (SSS) problems and formulate a feature scatter matrix based equivalent of the FKT. We establish the classical linear discriminant analysis (LDA) analogy of the FKT and apply it to a SSS situation. We demonstrate the significant computational savings and robustness associated with our approach using a multi-class face detection proble

L1-norm unsupervised Fukunaga-Koontz transform

Article number 107942The Fukunaga-Koontz transform (FKT) is a powerful supervised feature extraction method used in twoclass recognition problems, particularly when the classes have equal mean vectors but different covariance matrices. The present work proves that it is also possible to perform the FKT in an unsupervised manner, sparing the need for labeled data, by using a variant of L1-norm Principal Component Analysis (L1-PCA) that minimizes the L1-norm in the feature space. Rigorous proof is given in the case of data drawn from a mixture of Gaussians. A working iterative algorithm based on gradient-descent in the Stiefel manifold is put forward to perform L1-norm minimization with orthogonal constraints. A number of numerical experiments on synthetic and real data confirm the theoretical findings and the good convergence characteristics of the proposed algorithm

Fukunaga-Koontz feature transformation for statistical structural damage detection and hierarchical neuro-fuzzy damage localisation

Piotr Omenzetter and Simon Hoell’s work on this paper within the Lloyd’s Register Foundation Centre for Safety and Reliability Engineering at the University of Aberdeen was supported by Lloyd’s Register Foundation. The Foundation helps to protect life and property by supporting engineering-related education, public engagement and the application of research.Peer reviewedPostprin

Chernoff Dimensionality Reduction-Where Fisher Meets FKT

Well known linear discriminant analysis (LDA) based on the Fisher criterion is incapable of dealing with heteroscedasticity in data. However, in many practical applications we often encounter heteroscedastic data, i.e., within-class scatter matrices can not be expected to be equal. A technique based on the Chernoff criterion for linear dimensionality reduction has been proposed recently. The technique extends well-known Fisher\u27s LDA and is capable of exploiting information about heteroscedasticity in the data. While the Chernoff criterion has been shown to outperform the Fisher\u27s, a clear understanding of its exact behavior is lacking. In addition, the criterion, as introduced, is rather complex, making it difficult to clearly state its relationship to other linear dimensionality reduction techniques. In this paper, we show precisely what can be expected from the Chernoff criterion and its relations to the Fisher criterion and Fukunaga-Koontz transform. Furthermore, we show that a recently proposed decomposition of the data space into four subspaces is incomplete. We provide arguments on how to best enrich the decomposition of the data space in order to account for heteroscedasticity in the data. Finally, we provide experimental results validating our theoretical analysis

Fast Compressed Automatic Target Recognition for a Compressive Infrared Imager

Many military systems utilize infrared sensors which allow an operator to see targets at night. Several of these are either mid-wave or long-wave high resolution infrared sensors, which are expensive to manufacture. But compressive sensing, which has primarily been demonstrated in medical applications, can be used to minimize the number of measurements needed to represent a high-resolution image. Using these techniques, a relatively low cost mid-wave infrared sensor can be realized which has a high effective resolution. In traditional military infrared sensing applications, like targeting systems, automatic targeting recognition algorithms are employed to locate and identify targets of interest to reduce the burden on the operator. The resolution of the sensor can increase the accuracy and operational range of a targeting system. When using a compressive sensing infrared sensor, traditional decompression techniques can be applied to form a spatial-domain infrared image, but most are iterative and not ideal for real-time environments. A more efficient method is to adapt the target recognition algorithms to operate directly on the compressed samples. In this work, we will present a target recognition algorithm which utilizes a compressed target detection method to identify potential target areas and then a specialized target recognition technique that operates directly on the same compressed samples. We will demonstrate our method on the U.S. Army Night Vision and Electronic Sensors Directorate ATR Algorithm Development Image Database which has been made available by the Sensing Information Analysis Center

Sobre las propiedades discriminativas del análisisen componentes principales basado en la norma L1

El análisis de componentes principales (PCA) basado en la norma L1 es una técnica cada vez más popular para el análisis de datos multivariantes. Como idea intuitiva se utiliza que, para las direcciones en las que la nube se extiende por el espacio, las proyecciones de esos puntos han de tener una gran varianza. Este criterio es muy efectivo, pero tiene el inconveniente de que la varianza es un estadístico poco robusto: si los datos están contaminados con valores atípicos (outliers), las estimaciones de la varianza tendrán un gran error. Como solución, se ha propuesto sustituir la varianza por el promedio del valor absoluto de las proyecciones. Esta técnica resultante es lo que se ha denominado PCA basado en la norma L1 o L1-PCA, consiguiendo algoritmos muy robustos. Esta Tesis demuestra que un vínculo entre L1-PCA y la transformada de Fukunaga-Koontz (FKT, del inglés Fukunaga-Koontz transform). En su formulación original, L1-PCA proyecta los datos de manera que maximiza, en promedio, el valor absoluto de las proyecciones. De esta forma, se consiguen resultados similares al PCA tradicional. Ahora bien, manteniendo el valor absoluto como función objetivo, pero cambiando maximizar por minimizar, L1-PCA proporciona un resultado equivalente al que se obtiene mediante FKT. La importancia práctica de este resultado es que la FKT estándar es una técnica supervisada, es decir, para estimar los parámetros de la transformación, requiere un conjunto de datos de entrenamiento pertenecientes a cada una de las clases correctamente etiquetados. Por el contrario, minimizar el valor absoluto puede llevarse a cabo de manera totalmente no supervisada, haciendo innecesarios por ello los datos de entrenamiento. De esta forma, se ofrece una alternativa completamente novedosa para el cálculo de la FKT. Esto abre nuevas líneas de investigación en el área del aprendizaje automático o 'machine learning'

dbcsp: User-friendly R package for Distance-Based Common Spacial Patterns

Common Spatial Patterns (CSP) is a widely used method to analyse electroencephalography (EEG) data, concerning the supervised classification of the activity of brain. More generally, it can be useful to distinguish between multivariate signals recorded during a time span for two different classes. CSP is based on the simultaneous diagonalization of the average covariance matrices of signals from both classes and it allows the data to be projected into a low-dimensional subspace. Once the data are represented in a low-dimensional subspace, a classification step must be carried out. The original CSP method is based on the Euclidean distance between signals, and here we extend it so that it can be applied on any appropriate distance for data at hand. Both the classical CSP and the new Distance-Based CSP (DB-CSP) are implemented in an R package, called dbcsp