Structured covariance principal component analysis for real-time onsite feature extraction and dimensionality reduction in hyperspectral imaging

Jaime Zabalza, Jinchang Ren, Jie Ren, Zhe Liu, and Stephen Marshall

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Jaime Zabalza, Jinchang Ren, Jie Ren, Zhe Liu, and Stephen Marshall, "Structured covariance principal component analysis for real-time onsite feature extraction and dimensionality reduction in hyperspectral imaging," Appl. Opt. 53, 4440-4449 (2014)

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Abstract

Presented in a three-dimensional structure called a hypercube, hyperspectral imaging suffers from a large volume of data and high computational cost for data analysis. To overcome such drawbacks, principal component analysis (PCA) has been widely applied for feature extraction and dimensionality reduction. However, a severe bottleneck is how to compute the PCA covariance matrix efficiently and avoid computational difficulties, especially when the spatial dimension of the hypercube is large. In this paper, structured covariance PCA (SC-PCA) is proposed for fast computation of the covariance matrix. In line with how spectral data is acquired in either the push-broom or tunable filter method, different implementation schemes of SC-PCA are presented. As the proposed SC-PCA can determine the covariance matrix from partial covariance matrices in parallel even without prior deduction of the mean vector, it facilitates real-time data analysis while the hypercube is acquired. This has significantly reduced the scale of required memory and also allows efficient onsite feature extraction and data reduction to benefit subsequent tasks in coding and compression, transmission, and analytics of hyperspectral data.

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Method	Data Matrix/Vector Used	Covariance Matrix (Partial)	Covariance Matrix (Full)
Conventional PCA	$P = [\begin{matrix} p_{1} & p_{2} & \dots & p_{H W} \end{matrix}] \in R^{B \times H W}$	$C = P P^{T} \in R^{B \times B}$
Pixel-based SC-PCA	$p_{n} \in R^{B \times 1}$	$C_{n}^{(pixel)} = p_{n} p_{n}^{T}, n \in [1, H W]$	$C^{(pixel)} = \sum_{n = 1}^{H W} C_{n}^{(pixel)}$
Row-based SC-PCA	$P_{h}^{(R)} \in R^{B \times W}$	$C_{h}^{(Row)} = P_{h}^{(R)} {[P_{h}^{(R)}]}^{T}, h \in [1, H]$	$C^{(Row)} = \sum_{h = 1}^{H} C_{h}^{(Row)}$
Column-based SC-PCA	$P_{w}^{(C)} \in R^{B \times H}$	$C_{w}^{(Col)} = P_{w}^{(C)} {[P_{w}^{(C)}]}^{T}, w \in [1, W]$	$C^{(Col)} = \sum_{w = 1}^{W} C_{w}^{(Col)}$
Band-based SC-PCA	$P_{b}^{(B)} \in R^{H \times W}$	$C^{(Band)} (i, j) = vec (P_{b = i}^{(B)}) {[vec (P_{b = j}^{(B)})]}^{T}, i, j \in [1, B]$

Method	Indian Pines	Pavia UA	Botswana A
PCA	$80.24 \pm 0.31$	$96.93 \pm 0.20$	$94.34 \pm 1.67$
SC-PCA/P	$80.24 \pm 0.31$	$96.93 \pm 0.20$	$94.34 \pm 1.67$
SC-PCA/R	$80.24 \pm 0.31$	$96.93 \pm 0.20$	$94.34 \pm 1.67$
SC-PCA/C	$80.24 \pm 0.31$	$96.93 \pm 0.20$	$94.34 \pm 1.67$
SC-PCA/B	$80.24 \pm 0.31$	$96.93 \pm 0.20$	$94.34 \pm 1.67$

Method	Matrices Size	Indian Pines	Pavia UA	Botswana A
PCA	$B \times H W$	33,640	18,540	26,100
SC-PCA/P	$B \times 1$	1.6	0.83	1.2
SC-PCA/R	$B \times W$	232	124	348
SC-PCA/C	$B \times H$	232	124	87
SC-PCA/B	$1 \times H W$	168	180	180

Method	Multiplications	Additions
PCA	$B + B^{2} H W$	$B (2 H W - 1) + B^{2} (H W - 1) + 0$
SC-PCA/P	$B + B^{2} H W$	$B (2 H W - 1) + 0 + B^{2} (H W - 1)$
SC-PCA/R	$B + B^{2} H W$	$B (2 H W - 1) + B^{2} H (W - 1) + B^{2} (H - 1)$
SC-PCA/C	$B + B^{2} H W$	$B (2 H W - 1) + B^{2} (H - 1) W + B^{2} (W - 1)$
SC-PCA/B	$B + B^{2} H W$	$B (2 H W - 1) + B^{2} (H W - 1) + 0$

Method	Loop (k) Complexity		Loop (k)	After Acquisition
Method	Mult.	Add.	Loop (k)	Mult.	Add.
SC-PCA/P	$B^{2}$	$3 B^{2}$	HW	$2 B^{2}$	$B^{2}$
SC-PCA/R	$B^{2} W$	$3 B^{2} W$	H	$2 B^{2}$	$B^{2}$
SC-PCA/C	$B^{2} H$	$3 B^{2} H$	W	$2 B^{2}$	$B^{2}$
SC-PCA/B	$H W (2 k - 1) + 1$	$H W (2 k + 1) - 2 k$	B	0	0

Structured covariance principal component analysis for real-time onsite feature extraction and dimensionality reduction in hyperspectral imaging

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Applied Optics