1.外在的高维与本质的低维
基于模型的方法
跟踪
初始化的代价
Role of Manifold:
作为任务中的约束
HMM组合线性模型以近似非线性
Linear Bilinear and Multi-linear Models:
能否对整个configuration space的分解？
1. PCA SVD
2.bilinear multi-linear tensor analysis HOSVD
目标是动态的，能否用Linear Embedding分解出形状信息？
deformation and self-occlusion 导致流形的非线性、扭曲
在流形上极远点，可能在Euclidean visual input上很近
因此， PCA MDS等不能正确发现其流形。
Nonlinear Dimensionality Reduction and Decomposition of Orthogonal Factors:
embedding of nonlinear manifolds based on local structure of the manifold.
2 categories:
spectral-embedding approaches : Isomap LLE Laplacian eigenmaps Manifold Charting
construct an affinity matrix between data points using data dependent kernels, which reflect local manifold
kernel based learning , in particular KPCA
statistical approaches:
Biological Motivation:
2.2 Learning a Simple Motion manifold
2.2.1 Case Study : The Gait Manifold
通过非线性降维，可以得到一欧氏空间内的嵌入
但被嵌入的空间最少可有几维？
与视角有关， 前、后视只有2， 而侧则要3
2.2.2 Learning the Visual Manifold:Generative Model
如何利用来学习移动物体的表达，以支持合成、姿态恢复、重建、跟踪
y_t = T_a \gamma{(x_t ; a)}
由body configuration 到 image presentation, 再经 global geometric transformation
!! 身体的configuration落在流形之上，从而受其约束，为合理状态
1. 在嵌入流形上，用函数表示，或用exemplar； 或用HMM EM建模等
2. 嵌入流形如何与visual input space映射？
问题： recover body configuration from input, ie. 学习 R^d –> R^e 但这是不可行的，因为 visual input is very high dimensional so learning such mapping would required a large number of sample to interpolate. 而且， inherent ambiguity in 2D data
改为: 学习从embedding space 到 visual input space , i.e. , in a generative manner，再加一个 mechanism to directly solve for the inverse mapping
3 Y={y_ui \in R^d, i=1,…,N} X={x_I \in R^e, i=1,…,2N} f^k : R^e –>R f^k( x ) = p^k(x) + \sum_{i=1}^{N}{w_i^k \phi (|x – x_i|)}
basis function常用的选择: thin-plate spline: \phi (u)= u^2 log(u) multiquadric: \phi( u) \ root{(U^2 + c^2)} Guassian: e^{-cu^2} biharmonic: \phi ( u) = u and triharmonic \phi ( u) = u^3
f(x) = B \Phi(x)
保证orthogonality and to make the problem well posed: \sum_{i=1}^{N}{w_I p_j(x_i)} =0, j =1,…,m
2.2.3 Solving for the Embedding Coordinates
Given a new input y \in R^d , find x \in R^e ,by solving for the inverse mapping.
x^* = \argmin_{x}{\Left | y – B \Phi {x} \|}
Least Squre
B = USV^T \Phi(x) = V\hat S U^T y
2.2.4 Synthesis , Recovery and Reconstruction
2.3 Adding More Variability : Factoring out the Style
Any input image is a function of many aspects such as person body structure, appearance, viewpoint illumination, and body configuration, etc.
1. Consider silhouette only. Adding a variable describing people shape variability.
Aim to learn a decomposable generative model that explicitly decomposes the following two factors:
1. Content
2.Style
y_t^s = \gamma(x_t^c; a, b^s)
1. 同一人的风格不变，可照前
2.不同人间，建立线性模型
2.4 Style Adaptive Tracking: Bayesian Tracking on a Manifold