[学习笔记] CS131 Computer Vision: Foundations and Applications：Lecture 2 颜色和数学基础

大纲

what is color?

• The result of interaction between physical light in the environment and our visual system.
• A psychological property of our visual experiences when we look at objects and lights, not a physical property of those objects or lights.

 Human encoding of color

Color Spaces

• linear space: RGB/CIE XYZ
• nolinear space: HSV

Use of color in computer vision:

• color histogram for indexing and retrieval
• skin detection
• nude people detection
• image segmentation and retrieval
• build apperance models for tracking
• ...

Linear Algebra Primer: Vectors and Matrix

1. 向量

列向量：$v in R^{n*1} v = egin{bmatrix} v_1 \ v_2\ cdot \ cdot \ cdot \ v_n end{bmatrix}$

行向量：$v^T in R^{1*n} v^T = [v_1 v_2 ... v_n]$  (T转置运算符)

向量使用：点的空间表示；表示数据，没有空间意义，但是计算仍然有意义

2. 矩阵

矩阵运算：addition, scaling

矩阵范数：

one norm：$||x||_1 = sum_{i=1}^n |x_i| $

two norm：$||x||_2 = sqrt{sum_{i=1}^n x_i^2}

infinity norm： $||x||_inf = max |x_i|$

general P norm：||x||_p = (sum_{i=1}^n x_i^p)^1/p$

matrix norm：||A||_F = sqrt{sum_{i=1}^m sum_{j = 1}^n A_ij^2 = sqrt{tr(A^TA)}$

矩阵的秩：

• $det(AB) = det(BA)$
• $det(A^-1) = frac{1}{det(A)}$
• $det(A^T) = det(A)$
• $det(A) = 0$ 当且仅当$A$是奇异的

矩阵的迹：对角元素的和

特殊矩阵： 

• 单位矩阵(Identity Matrix)：对角元素为0，其他元素为1
• 对角矩阵(diagonal matrix)：非对角元素为0
• 对称矩阵(Symmetric Matrix)：$A^T = A$
• 反对称矩阵(Skew-symmetric Matrix) $A^T = -A$