*2.2. SBS-CEM Algorithm*

To accelerate the task of target detection using CEM, some researchers choose to calculate local detection by using a partial set of pixel vectors instead of all data sample vectors [6,29,30]. For instance, Yang [21] proposed an FPGA-based implementation of SBS-CEM by using a new matrix inversion method to perform the correlation operation and the inversion operation simultaneously.

#### 2.2.1. Principle of the SBS-CEM Algorithm

Unlike the classical CEM algorithm, SBS-CEM takes the inverse of the correlation matrix of the *K*-group pixel vectors to replace the entire pixel vectors. More specifically, SBS-CEM can be described as follows. 

$$\mathbf{R}\_n = (1/K) \left[ \sum\_{i=n-K}^{n-1} \mathbf{x}\_i \mathbf{x}\_i^T \right] \tag{5}$$

$$\mathbf{S}\_n = \left[\sum\_{i=n-K}^{n-1} \mathbf{x}\_i \mathbf{x}\_i^T\right] \tag{6}$$

Now, **S**−<sup>1</sup> *n* is the inverse of the correlation matrix of the *K*-group pixel vectors. The Sherman-Morrison formula is used to derive the following two formulas.

$$\mathbf{C}^{-1} = \left(\mathbf{S}\_n + \mathbf{x}\_n \mathbf{x}\_n^T\right)^{-1} = \mathbf{S}\_n^{-1} - \frac{\mathbf{S}\_n^{-1} \mathbf{x}\_n \mathbf{x}\_n^T \mathbf{S}\_n^{-1}}{\mathbf{x}\_n^T \mathbf{S}\_n^{-1} \mathbf{x}\_n + 1} \tag{7}$$

$$\mathbf{S}\_{n+1}^{-1} = \left(\mathbf{C} - \mathbf{x}\_{n-K} \mathbf{x}\_{n-K}^{\mathrm{T}}\right)^{-1} = \mathbf{C}^{-1} - \frac{\mathbf{C}^{-1} \mathbf{x}\_{n-K} \mathbf{x}\_{n-K}^{\mathrm{T}} \mathbf{C}^{-1}}{\mathbf{x}\_{n-K}^{\mathrm{T}} \mathbf{C}^{-1} \mathbf{x}\_{n-K} - 1} \tag{8}$$

Based on this streaming framework, the inverse matrix can be updated by using Equations (7) and (8). When applying the Sherman-Morrison formula, the initial value of **S**−<sup>1</sup> 0 should be set. Let **S**−<sup>1</sup> 0 = *β* · **I**; then **S***K*+<sup>1</sup> can be expressed as:

$$\mathbf{S}\_{K+1} = (1/\boldsymbol{\beta}) \cdot \mathbf{I} + \mathbf{x}\_1 \mathbf{x}\_1^\mathrm{T} + \mathbf{x}\_2 \mathbf{x}\_2^\mathrm{T} + \dots + \mathbf{x}\_K \mathbf{x}\_K^\mathrm{T} \tag{9}$$

Among them, the matrix (1/*β*) · **I** does not affect the performance of the detector. On the contrary, it makes the detection results be more stable [31]. The detection equation of the SBS-CEM algorithm is then derived as: 

$$\text{SBS} - \text{CEM}\left(\mathbf{x}\right) = \frac{K\left(\mathbf{x}^{\text{T}}\mathbf{S}^{-1}\mathbf{d}\right)}{K\left(\mathbf{d}^{\text{T}}\mathbf{S}^{-1}\mathbf{d}\right)} = \frac{\mathbf{x}^{\text{T}}\mathbf{S}^{-1}\mathbf{d}}{\mathbf{d}^{\text{T}}\mathbf{S}^{-1}\mathbf{d}}\tag{10}$$

Since the pixel to be detected is located in the middle of the window, SBS-CEM can also be expressed as:

$$SBS-CEM\left(\mathbf{x}\_{n-K/2}\right) = \frac{\mathbf{x}\_{n-K/2}^{\mathrm{T}}\mathbf{S}\_n^{-1}\mathbf{d}}{\mathbf{d}^{\mathrm{T}}\mathbf{S}\_n^{-1}\mathbf{d}}\tag{11}$$
