[컴선설] Lec 02 Spline

1. Vector (Linear) Space of Function

• Spline function : Piecewise polynomials.
• B-spline function : Basis spline. A subset of splines
• $P_2(x) = a_0 + a_1x + a_2x^2$
• ${\bf v}(a_0, a_1, a_2) \in \R^3$

2. Polynomial Vector Space

• Polynomial vector space is closed to addition and scalar multiplication
• $\forall {\bf v, w} \in \R^3, \alpha{\bf v}+\beta {\bf w} ={\bf z}\in \R^3$

3. Basis Vector

• like basis in vectorspace, basis in polynomial vectorspace is
• ${\bf e_1} = 1, {\bf e_2} = x, {\bf e_3} = x^2$

4. Linear independence between function

• Same rule applies on the function
• if $f_1(x) = x, f_2(x) = 1+x, f_3(x) = 3x$
• $\text{span} (f_1, f_2, f_3) = \alpha_1f_1 + \alpha_2 f_2 + \alpha_3 f_3$
• Nontrivial solution $\alpha_1 = -3\alpha_3, \alpha_2 = 0$ exists, $f_1, f_2, f_3$ are linearly dependent

5. Vector Space of Polynomial function

Basis can be omitted when expressing.

Check linear independence of functions

• Example !

6. Spline (Piecewise Polynomial) Vector Space

• degree : Highest order term (최고차 항의 수)
• order : Number of terms (dim), 항의 갯수 = degree + 1
• $C^{-1}$ : discontinuous
• $C^0$ : continuous
• $C^1$ : first derivative is continuous
• $C^2$ : second derivative is continuous

Total sum of numbers of dimension : total dimension

apply constraints ($C^1$ at t=1 etc..)

number of pivot terms : number of constraints, express without pivot term

7. Spline Application

• Example !
  1. Perform dimensional analysis : calculate DOF
  • Sum of dimension of each spline
  • minus number of constraints
  • Determine which kind of problem to be solved : approximation, interpolation
    1. Solve constraints and POC conditions
    2. In case of Approximation, use $A^TA{\bf x} = A^T{\bf b}$

(Spline interpolation/approximation)
1. Perform dimensional analysis
2. Get equations of constraints : i.e. b term are zero
3. compute nullspace equations in order to fit constraints
4. And then apply POC constraints