Generalized principal component analysis (GPCA)
Generalized principal component analysis (GPCA)
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DOI：
10.1109/cvpr.2003.1211411
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发表时间：
2003-06
期刊：
影响因子：
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通讯作者：
R. Vidal;Yi Ma;S. Sastry
中科院分区：
文献类型：
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作者：
R. Vidal;Yi Ma;S. Sastry
我们针对从样本数据点估计线性子空间混合的问题提出一种代数几何方法，即所谓的广义主成分分析（GPCA）问题。在无噪声的情况下，我们表明GPCA等同于对一个齐次多项式进行因式分解，该多项式的次数是子空间的数量，其因子（根）代表每个子空间的法向量。我们推导出子空间数量\(n\)的一个公式，并使用线性代数技术为因式分解问题提供一种解析解。当且仅当\(n\leq4\)时，该解是闭式的。在有噪声的情况下，我们将GPCA表述为一个约束非线性最小二乘问题，并推导出一个最优函数，利用标准非线性优化技术可直接从该函数恢复子空间。我们将GPCA应用于计算机视觉中的运动分割问题，即从二维图像估计运动模型混合的问题。
We propose an algebraic geometric approach to the problem of estimating a mixture of linear subspaces from sample data points, the so-called generalized principal component analysis (GPCA) problem. In the absence of noise, we show that GPCA is equivalent to factoring a homogeneous polynomial whose degree is the number of subspaces and whose factors (roots) represent normal vectors to each subspace. We derive a formula for the number of subspaces n and provide an analytic solution to the factorization problem using linear algebraic techniques. The solution is closed form if and only if n /spl les/ 4. In the presence of noise, we cast GPCA as a constrained nonlinear least squares problem and derive an optimal function from which the subspaces can be directly recovered using standard nonlinear optimization techniques. We apply GPCA to the motion segmentation problem in computer vision, i.e. the problem of estimating a mixture of motion models from 2D imagery.