Note for Reidentification by Relative Distance Comparison

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Reidentification by Relative Distance Comparison

Challenge:

• large visual appearance changes caused by variations in view angle, lighting, background clutter, and occlusion
• 之前的大部分算法寻找独特的视觉特征。但寻找在数据规模大、现实条件不同的数据集中能够保持鲁棒性的视觉特征仍然十分困难。
• 在不同条件下，有些特征比其他特征更重要，更稳定，使用l1-Norm等普遍采用的标准的距离评估方法并不合适，因为它们会等权重地对待所有特征。

In order to find a correc match Given a query image of a person:

• First, a feature representation is computed from both the query and each of the gallery images.
• Second, the distance between each pair of potential matches is measured

Solution(part 1):

• given a set of features extracted from each person image, we seek to quantify and differentiate these features by learning the optimal distance measure that is most likely to give correct matches.
• In essence, images of each person in a training set form a class.
• This learning problem can be framed as a distance learning problem which always searches for a distance that minimizes intraclass distances while maximizing interclass distances.

Question:

• the person reidentification problem has four characteristics

  • The intraclass variation can be large and, more importantly, can vary significantly for different classes
  • The interclass variation also varies drastically across different pairs of classes and there are often severe overlaps between classes in a feature space
  • In order to capture the large intra and intervariations, the number of classes is necessarily large
  • Annotating a large number of matched people across camera views is not only tedious, but also inherently limited in its usefulness

• the data are inherently undersampled for building a representative class distribution

• a learning model could easily be overfitted and/or be intractable if it is learned by minimizing intraclass distance and maximizing interclass distance simultaneously by brute-force

Solution(part 2):

• formulate the problem as a relative distance comparison (RDC) problem
• the model aims to learn an optimal distance in the sense that for a given query image, the true match is desired to be ranked higher than the wrong matches among the gallery image set
• not easily biased by large variations across many undersampled classes as it aims to seek an optimized individual comparison between any two data points rather than comparison among data distribution boundaries or among clusters of data
• Furthermore, in order to alleviate the large space complexity (memory usage cost) and the local optimum learning problem due to the proposed iterative algorithm for solving high-order nonlinear optimization criterion, we develop an ensemble RDC in this work

Details:

Proposed Relative Distance Comparison Learning

给出训练集\(Z={\{(\mathbf{z_i},y_i)\}}^N_{i=1}\),其中\(\mathbf{z_i}\)是表示一个视图中一个人的多维特征向量，\(y_i\)是对呀的类标签（人的ID）。

定义集合\(O_i=\{O_i = (x^p_i, x^n_i)\}\),其中\(x^p_i\)为两个相同类别样本的差异向量，\(x^n_i\)为两个不同类别样本的差异向量

\[ x=d(\mathbf{z,z'}),\quad \mathbf{z,z'} \in R^q\]

其中d是作用在矩阵每个元素上的差异函数。

给定\(O\)，距离函数\(f\)以差异向量作为输入，通过相对距离比较的方式进行学习，从而使得

\[ f(x^p_i) < f(x^n_i)\]

为了描述这个优化目标，并且让它可以求导，令

\[C_{f}\left(\mathbf{x}_{i}^{p}, \mathbf{x}_{i}^{n}\right)=\left(1+\exp \left\{f\left(\mathbf{x}_{i}^{p}\right)-f\left(\mathbf{x}_{i}^{n}\right)\right\}\right)^{-1}\]

假定the events of distance comparison between a relevant pair and a related irrelevant pair are independent，优化目标成为

\[\min _{f} r(f, O),\quad r(f, O)=-\log \left(\prod_{O_i} C_{f}\left(\mathbf{x}_{i}^{p}, \mathbf{x}_{i}^{n}\right)\right)\]

令\(f\)为马氏距离，其中M为半正定矩阵。问题转化为学习M。
\[f(\mathbf{x})=\mathbf{x}^{T} \mathbf{M} \mathbf{x}, \quad \mathbf{M} \succeq 0\]

对矩阵M作特征分解，

\[\mathbf{M}=\mathbf{A} \mathbf{\Lambda} \mathbf{A}^{T}=\mathbf{W} \mathbf{W}^{T}, \quad \mathbf{W}=\mathbf{A} \mathbf{\Lambda}^{\frac{1}{2}}\]

其中\(\mathbf{A}\)由正交特征向量构成，而\(\mathbf{\Lambda}\)由对应特征值构成

令\(\mathbf{W}=(\mathbf{w}_{1}, \ldots, \mathbf{w}_{l}, \ldots, \mathbf{w}_{L})\)

问题转化为

\[\min _{\mathbf{W}} r(\mathbf{W}, O), \text { s.t. } \quad \mathbf{w}_{i}^{T} \mathbf{w}_{j}=0, \forall i \neq j\]

\[
r(\mathbf{W}, O)=\sum_{O_{i}} \log \left(1+\exp \left\{\left\|\mathbf{W}^{T} \mathbf{x}_{i}^{p}\right\|^{2}-\left\|\mathbf{W}^{T} \mathbf{x}_{i}^{n}\right\|^{2}\right\}\right)
\]

上式即 relative distance comparisong for person reidentification

An Iterative Optimization Algorithm

• 初值：
  • \(O_i=\{O_i = (x^p_i, x^n_i)\},\quad \epsilon \gt 0\)
  • \(\mathbf{w}_{0} \longleftarrow \mathbf{0}, \quad \tilde{\mathbf{w}}_{0} \longleftarrow \mathbf{0}\)
  • \(\mathbf{x}_{i}^{s, 0} \longleftarrow \mathbf{x}_{i}^{s}, s \in\{p, n\}, O^{0} \longleftarrow O\)

• 第\(l\)次迭代：

  • 令优化目标中的项

  \[a_{i}^{l+1}=\exp \left\{\sum_{j=0}^{l}\left\|\mathbf{w}_{j}^{T} \mathbf{x}_{i}^{p, j}\right\|^{2}-\left\|\mathbf{w}_{j}^{T} \mathbf{x}_{i}^{n, j}\right\|^{2}\right\}\]

  其中\(\mathbf{x}_{i}^{p, l}，\mathbf{x}_{i}^{n, l}\)为第\(l\)次迭代的差别向量,定义为

  \[\mathbf{x}_{i}^{s, \ell}=\mathbf{x}_{i}^{s, l-1}-\tilde{\mathbf{w}}_{l-1} \tilde{\mathbf{w}}_{l-1}^{T} \mathbf{x}_{i}^{s, l-1}, \quad s \in\{p, n\}, i=1, \ldots,|O|\]

  其中\(l \ge 1\)并且\(\tilde{\mathbf{w}}_{l-1} = \mathbf{w}_{l-1} / \|\mathbf{w}_{l-1}\|\)

  （个人理解，相当于一个动量）

  • 计算\(\mathbf{x}_{i}^{p, l+1}，\mathbf{x}_{i}^{n, l+1}\)，得到新的\(O^{l+1}\)

  梯度下降法最小化目标

  \[\mathbf{w}_{l+1}=\arg \min _{\mathbf{w}} r_{l+1}\left(\mathbf{w}, \mathbf{O}^{l+1}\right)\]

  其中

  \[r_{l+1}(\mathbf{w}, \mathbf{O}^{l+1})=\sum_{O_{i}^{l+1}} \log (1+a_{i}^{l+1} \exp \{\|\mathbf{w}^{T} \mathbf{x}_{i}^{p, l+1}\|^{2}-\|\mathbf{w}^{T} \mathbf{x}_{i}^{n, l+1}\|^{2}\})\]

  \(a^{l+1}_i\)的存在考虑上一次迭代（上一组数据）的影响

  注意到\(\mathbf{w}_{l-1}^{T} \mathbf{x}_{i}^{s, l}=0\)，过早的迭代样本不会影响到下一次的\(w\)

• 出口：

\[r_{l}\left(\mathbf{w}_{l}, O^{l}\right)-r_{l+1}\left(\mathbf{w}_{l+1}, O^{l+1}\right)<\varepsilon\]

ENSEMBLE LEARNING FOR LARGE SCALE COMPUTATION