Metric Space (X,d) $d:X \times X \rightarrow \Re ^{+}$

X is set, not necessarily vector space

$x, y, z \in X$

1. $d(x,y)=d(y,x)$
2. $d(x,z)\leq d(x,y)+d(y,z)$
3. $d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y)$

If X is vector space, metric can be induced by the norm $||\cdot||$.

$d(x,y)=||y-x||$

Norm is defined as follows

$||\cdot||: X \rightarrow \Re ^{+}$

1. $|x| \geq 0, ||x||=0 \Leftrightarrow x=0$
2. $||\alpha x||=|\alpha | ||x||$
3. $||x+y|| \leq ||x|| + ||y||$

Defining metric, we can measure similarity of elements of set X.

Example of metric

1. Minkowski Metric $\left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}$
2. Riemannian Metric $D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})}$
3. Tanimoto metric $D(S_1, S_2) = \frac {|S_1|+|S_2|-2|S_1 \bigcap S_2| }{|S_1|+|S_2|-|S_1 \bigcap S_2|}$
4. Procrustes metric $D(p,\bar p)= min_{R,T} \sum_{i=1}^n {\begin{Vmatrix} Rp_i+T-\bar p_i \end{Vmatrix}} _{L^2}$, R: Rotation, T: Translation

## Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.