Image Warpper

Bezier Curve Evaluation Algorithm

A bézier line is an arc that is strictly based on a set number of points instead of on an ellipse. A bézier curve uses at least four points to draw on.
Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces

Linear Bézier curves

Given distinct points $P_0$ and $P_1$, a linear Bézier curve is simply a straight line between those two points. The curve is given by
$$B ( t ) = P_0 + t ( P_1-P_0) = (1 - t)P_0 + tP_1 , 0 \leq t \leq 1$$

The coeficients, $b_i$ are the control points or Bézier points and together with the basis function $B_{i,n}(t)$determine the shape of the curve. Lines drawn between consecutive control points of the curve form the control polygon. A cubic Bezier curve together with its control polyg`on is shown in Fig. 1.1. Bezier curves have the following properties:

Geometry invariance property: Partition of unity property of the Bernstein polynomial assures the invariance of the shape of the Bézier curve under translation and rotation of its control points

![avatar](./Bezier curve.png)

Fig. 1.1.

$Slerp(p, q, t)$, linear interplate two control points $p$ and $q$,

inline CPoint CBezier::lerp( CPoint & p, CPoint & q, double t )
{
    CPoint r = p * ( 1- t) + q * t;
    return r;
};

$deBoor(C0, C1, C2, C3, t)$, compute a point on the Bezier curve at parametert

inline CPoint CBezier::deBoor( CPoint & c0, CPoint & c1, CPoint & c2, CPoint & c3, double t )
{
    //Modify this procedure 
    CPoint C01 = lerp( c0, c1, t );
    CPoint C12 = lerp( c1, c2, t ); 
    CPoint C23 = lerp( c2, c3, t );
    CPoint C012 = lerp( C01, C12, t );
    CPoint C123 = lerp( C12, C23, t );
    CPoint C0123 = lerp( C012, C123, t );
    return C0123;
};

Bezier Surface Evaluation Algorithm

Given a control net $C[4][4]$, and $2D$ parameters $uv$, evaluate the point on the Bezier surface constructed from the control net at the parameters $uv$, the algorithm evaluate the point p[k] on four Bezier curves controlled by points m_control_net$[k][0]$, m_control_net$[k][1]$, m_control_net$[k][2]$, m_control_net$[k][3]$, then evaluate the point on the Bezier curve controlled by $D[0], D[1], D[2], uv[3]$.

inline CPoint CBezier::evaluate( CPoint2 uv )
{
    //Modify this procedure
    CPoint D[4];
    for (int k = 0; k < 4; k++)
    {
        D[k] = deBoor(m_control_net[k][0], m_control_net[k][1], m_control_net[k][2], m_control_net[k][3], uv[0]);
    }
    CPoint r = deBoor( D[0], D[1], D[2], D[3], uv[1] );
    return r;
};

Image Warpping

The image is embedded in the unit square, the unit square is deformed by a nonlinear mapping $\phi :S \rightarrow \Re^2 $, where $S$ is the unit square. The mapping is modeled as Bezier surface, controlled by the control net.

$$\phi(u, v) = \sum_{i=0}^3\sum_{j=0}^3B_3^i(u)B_3^j(v)C_{ij},$$
where ${C_{ij} }$ form the control net,$B_3^i$ is the Bernstein polynomial,

$$B_3^i(t) = C_3^i(1 − t)^i t^{3−i} ,$$