Kernel RX Research Articles

Efficient Nonlinear RX Anomaly Detectors

Current anomaly detection (AD) algorithms are typically challenged by either accuracy or efficiency. More accurate nonlinear detectors are typically slow and not scalable. In this letter, we propose two families of techniques to improve the efficiency of the standard kernel Reed-Xiaoli (KRX) method for AD by approximating the kernel function with either the data-independent random Fourier features or the data-dependent basis with the Nyström approach. We compare all methods for both real multi- and hyperspectral images. We show that the proposed efficient methods have a lower computational cost, and they perform similar to (or outperform) the standard KRX algorithm thanks to their implicit regularization effect. Last but not least, the Nyström approach has an improved power of detection.

Nonparametric Spectral-Spatial Anomaly Detection

Due to abundant spectral information contained in the hyperspectral images, they are suitable data for anomalous targets detection. The use of spatial features in addition to spectral ones can improve the anomaly detection performance. An anomaly detector, called nonparametric spectral-spatial detector (NSSD), is proposed in this work which utilizes the benefits of spatial features and local structures extracted by the morphological filters. The obtained spectral-spatial hypercube has high dimensionality. So, accurate estimates of the background statistics in small local windows may not be obtained. Applying conventional detectors such as Local Reed Xiaoli (RX) to the high dimensional data is not possible. To deal with this difficulty, a nonparametric distance, without any need to estimate the data statistics, is used instead of the Mahalanobis distance. According to the experimental results, the detection accuracy improvement of the proposed NSSD method compared to Global RX, Local RX, weighted RX, linear filtering based RX (LF-RX), background joint sparse representation detection (BJSRD), Kernel RX, subspace RX (SSRX) and RX and uniform target detector (RX-UTD) in average is 47.68%, 27.86%, 13.23%, 29.26%, 3.33%, 17.07%, 15.88%, and 44.25%, respectively.

Fast Real-Time Kernel RX Algorithm Based on Cholesky Decomposition

Real-time processing has attracted wide attention in hyperspectral anomaly detection. The traditional local real-time kernel RX detector (LRT-KRXD) is still with some computational limitations, which lower the processing speed and even damage the detection due to the matrix singularity. In this letter, we present LRT-KRXD based on Cholesky decomposition (LRT-KRXD-CD). First, the derivation of kernel covariance matrices is computationally expensive in KRXD, while each two adjacent matrices contain almost identical content. To remove the repeated computation, a recursive strategy for these kernel covariance matrices is used. Second, the kernel covariance matrix is symmetric positive definite after adding a diagonal matrix with small scale. With this property, Cholesky decomposition and linear system solving can be utilized to address the problem of inverse matrix. In this case, the detection of LRT-KRXD-CD becomes robust and its processing speed is improved as well. Experimental results on two hyperspectral images substantiate the effectiveness of LRT-KRXD-CD.

Modified Kernel RX Algorithm Based on Background Purification and Inverse-of-Matrix-Free Calculation

The kernel RX detector (KRXD) has better performance than the RX algorithm in anomaly detection (AD). However, it generally suffers from two challenges: 1) it is more prone to background contamination by anomalous pixels and noise in local statistics since the local AD is normally implemented for KRXD to relieve high computational complexity in global AD and 2) the inverse of the kernelized background covariance matrix is usually rank deficient. Accordingly, this letter proposes a Gaussian background purification approach according to background data samples probability distribution and an inverse-of-matrix-free method based on kernel PCA to address the above problems, respectively. The experimental results indicate that the improved KRXD overcomes both the difficulties and procures preferable effects.

A Weighted Spatial-Spectral Kernel RX Algorithm and Efficient Implementation on GPUs.

The kernel RX (KRX) detector proposed by Kwon and Nasrabadi exploits a kernel function to obtain a better detection performance. However, it still has two limits that can be improved. On the one hand, reasonable integration of spatial-spectral information can be used to further improve its detection accuracy. On the other hand, parallel computing can be used to reduce the processing time in available KRX detectors. Accordingly, this paper presents a novel weighted spatial-spectral kernel RX (WSSKRX) detector and its parallel implementation on graphics processing units (GPUs). The WSSKRX utilizes the spatial neighborhood resources to reconstruct the testing pixels by introducing a spectral factor and a spatial window, thereby effectively reducing the interference of background noise. Then, the kernel function is redesigned as a mapping trick in a KRX detector to implement the anomaly detection. In addition, a powerful architecture based on the GPU technique is designed to accelerate WSSKRX. To substantiate the performance of the proposed algorithm, both synthetic and real data are conducted for experiments.

Real-Time Anomaly Detection Based on a Fast Recursive Kernel RX Algorithm

Abstract: Real-time anomaly detection has received wide attention in remote sensing image processing because many moving targets must be detected on a timely basis. A widely-used anomaly detection algorithm is the Reed-Xiaoli (RX) algorithm that was proposed by Reed and Yu. The kernel RX algorithm proposed by Kwon and Nasrabadi is a nonlinear version of the RX algorithm and outperforms the RX algorithm in terms of detection accuracy. However, the kernel RX algorithm is computationally more expensive. This paper presents a novel real-time anomaly detection framework based on the kernel RX algorithm. In the kernel RX detector, the inverse covariance matrix and the estimated mean of the background data in the kernel space are non-causal and computationally inefficient. In this work, a local causal sliding array window is used to ensure the causality of the detection system. Using the matrix inversion lemma and the Woodbury matrix identity, both the inverse covariance matrix and estimated mean can be recursively derived without extensive repetitive calculations, and, therefore, the real-time kernel RX detector can be implemented and processed pixel-by-pixel in real time. To substantiate its effectiveness and utility in real-time anomaly detection, real hyperspectral data sets are utilized for experiments.

A Novel Cluster Kernel RX Algorithm for Anomaly and Change Detection Using Hyperspectral Images

The Reed–Xiaoli (RX) algorithm has been widely used as an anomaly detector for hyperspectral images. Recently, kernel RX (KRX) has been proven to yield high performance in anomaly detection and change detection. In this paper, we present a generalization of the KRX algorithm. The novel algorithm is called cluster KRX (CKRX), which becomes KRX under certain conditions. The key idea is to group background pixels into clusters and then apply a fast eigendecomposition algorithm to generate the anomaly detection index. Both global and local versions of CKRX have been implemented. Application to anomaly detection using actual hyperspectral images is included. In addition to anomaly detection, the CKRX algorithm has been integrated with other prediction algorithms for change detection. Spatially registered visible and near-infrared hyperspectral images collected from a tower-based geometry have been used in the anomaly and change detection studies. Receiver operating characteristics curves and actual computation times were used to compare different algorithms. It was demonstrated that CKRX has comparable detection performance as KRX, but with much lower computational requirements.

Problematic Projection to the In-Sample Subspace for a Kernelized Anomaly Detector

We examine the properties and performance of kernelized anomaly detectors, with an emphasis on the Mahalanobis-distance-based kernel RX (KRX) algorithm. Although the detector generally performs well for high-bandwidth Gaussian kernels, it exhibits problematic (in some cases, catastrophic) performance for distances that are large compared to the bandwidth. By comparing KRX to two other anomaly detectors, we can trace the problem to a projection in feature space, which arises when a pseudoinverse is used on the covariance matrix in that feature space. We show that a regularized variant of KRX overcomes this difficulty and achieves superior performance over a wide range of bandwidths.

Collaborative Representation for Hyperspectral Anomaly Detection

In this paper, collaborative representation is proposed for anomaly detection in hyperspectral imagery. The algorithm is directly based on the concept that each pixel in background can be approximately represented by its spatial neighborhoods, while anomalies cannot. The representation is assumed to be the linear combination of neighboring pixels, and the collaboration of representation is reinforced by l 2 -norm minimization of the representation weight vector. To adjust the contribution of each neighboring pixel, a distance-weighted regularization matrix is included in the optimization problem, which has a simple and closed-form solution. By imposing the sum-to-one constraint to the weight vector, the stability of the solution can be enhanced. The major advantage of the proposed algorithm is the capability of adaptively modeling the background even when anomalous pixels are involved. A kernel extension of the proposed approach is also studied. Experimental results indicate that our proposed detector may outperform the traditional detection methods such as the classic Reed-Xiaoli (RX) algorithm, the kernel RX algorithm, and the state-of-the-art robust principal component analysis based and sparse-representation-based anomaly detectors, with low computational cost.

Decision fusion for dual-window-based hyperspectral anomaly detector

In hyperspectral anomaly detection, the dual-window-based detector is a widely used technique that employs two windows to capture nonstationary statistics of anomalies and back- ground. However, its detection performance is usually sensitive to the choice of window sizes and suffers from inappropriate window settings. In this work, a decision-fusion approach is pro- posed to alleviate such sensitivity by merging the results from multiple detectors with different window sizes. The proposed approach is compared with the classic Reed-Xiaoli (RX) algorithm as well as kernel RX (KRX) using two real hyperspectral data. Experimental results demonstrate that it outperforms the existing detectors, such as RX, KRX, and multiple-window-based RX. The overall detection framework is suitable for parallel computing, which can greatly reduce computational time when processing large-scale remote sensing image data. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or repro- duction of this work in whole or in part requires full attribution of the original publication, including its DOI. (DOI: 10.1117/1.JRS.9.097297) tional probability density functions under the two hypotheses (without and with anomaly) are assumed to be Gaussian. The solution turns out to be an adaptive Mahalanobis distance between the pixel under test and the local background. It is preferred to use local background to capture nonstationary statistics, and its advantage of using a global background covariance matrix has been demonstrated in the literature. 11-13 The RX detector has become the benchmark of anomaly detection algorithms in HSI. Obviously, the key to success is an appropriate estimate of a local background covariance matrix for effective background suppression. An adaptive RX detector employs a dual-window strategy: the inner window is slightly larger than the pixel size, the outer window is even larger than the inner one, and only the samples in the outer region (i.e., between the frames of inner and outer windows) are used to estimate the background covariance matrix to avoid the use of the potential anomalous pixels. Intuitively, the number of pixels in the outer region (related to the sizes of inner and outer windows) should be more than the number of bands so that the resulting covariance matrix can be full-rank for inverse matrix operation. However, even when the covariance matrix is ill-rank, its inversion can still be computed by several strategies, such

A modified kernel-RX algorithm for anomaly detection in hyperspectral images

The use of kernel-based methods has recently become a popular approach to many hyperspectral image applications such as anomaly detection (AD). A well-known kernel-based anomaly detector in hyperspectral images is the Kernel-Reed-Xiaoli (KRX) algorithm. This study aims at improving the performance of the original KRX. To meet this purpose, a modified version of the KRX that assumes a spherical covariance matrix for the background class is proposed. Performance comparison is accomplished using two real hyperspectral data sets that contain subpixel and multipixel targets. Experimental results demonstrate that the use of the modified KRX, compared to the original KRX, improves the detection and computational efficiency for AD.

Anomaly detection based on a parallel kernel RX algorithm for multicore platforms

Anomaly detection is an important task for hyperspectral data exploitation. A standard approach for anomaly detection in the literature is the method developed by Reed and Yu, also called RX algorithm. It implements the Mahalanobis distance, which has been widely used in hyperspectral imaging applications. A variation of this algorithm, known as kernel RX (KRX), consists of applying the same concept to a sliding window centered around each image pixel. KRX is computationally very expensive because, for every image pixel, a covariance matrix and its inverse has to be calculated. We develop an efficient implementation of the kernel RX algorithm. Our proposed approach makes use of linear algebra libraries and further develops a parallel implementation optimized for multi-core platforms, which is a well known, inexpen- sive and widely available high performance computing technology. Experimental results for two hyperspectral data sets are provided. The first one was collected by NASA's airborne visible infra-red imaging spectrometer (AVIRIS) system over the World Trade Center (WTC) in New York, five days after the terrorist attacks, and the second one was collected by the hyperspectral digital image collection experiment (HYDICE). Our anomaly detection accuracy, evaluated using receiver operating characteristics (ROC) curves, indicates that KRX can significantly out- perform the classic RX while achieving close to linear speedup in state-of-the-art multi-core platforms. © 2012 Society of Photo-Optical Instrumentation Engineers (SPIE). (DOI: 10.1117/1.JRS .6.061503)

Kernel wavelet-Reed–Xiaoli: an anomaly detection for forward-looking infrared imagery

This paper describes a new kernel wavelet-based anomaly detection technique for long-wave (LW) forward-looking infrared imagery. The proposed approach called kernel wavelet-Reed-Xiaoli (wavelet-RX) algorithm is essentially an extension of the wavelet-RX algorithm (combination of wavelet transform and RX anomaly detector) to a high-dimensional feature space (possibly infinite) via a certain nonlinear mapping function of the input data. The wavelet-RX algorithm in this high-dimensional feature space can easily be implemented in terms of kernels that implicitly compute dot products in the feature space (kernelizing the wavelet-RX algorithm). In the proposed kernel wavelet-RX algorithm, a two-dimensional wavelet transform is first applied to decompose the input image into uniform subbands. A number of significant subbands (high-energy subbands) are concatenated together to form a subband-image cube. The kernel RX algorithm is then applied to this subband-image cube. Experimental results are presented for the proposed kernel wavelet-RX, wavelet-RX, and the classical constant false alarm rate (CFAR) algorithm for detecting anomalies (targets) in a large database of LW imagery. The receiver operating characteristic plots show that the proposed kernel wavelet-RX algorithm outperforms the wavelet-RX as well as the classical CFAR detector.

Anomaly Detection for Hyperspectral Images Based on Robust Locally Linear Embedding

In this paper, anomaly detection in hyperspectral images is investigated using robust locally linear embedding (RLLE) for dimensionality reduction in conjunction with the RX anomaly detector. The new RX-RLLE method is implemented for large images by subdividing the original image and applying the RX-RLLE operations to each subset. Moreover, from the kernel view of LLE, it is demonstrated that the RX-RLLE is equivalent to introducing a locally linear embedding (LLE) kernel into the kernel RX (KRX) algorithm. Experimental results indicate that the RX-RLLE has good anomaly detection performance and that RLLE has superior performance to LLE and principal component analysis (PCA) for dimensionality reduction in the application of anomaly detection.