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Scratch Mark Calculation and Analysis

English|中文

Since SVG does not directly support hyperbolic segments, our goal is to generate an SVG scratch path using cubic Bézier curves.

1. Mapping Height to Curves

  • Height Value Calculation
    The height of each point is determined by the pixel value from the depthImage and subtracting a baseline height zeroHeight.

$$ depth = depthImage(y, x) - zeroHeight $$

  • Curvature
    Curvature is mapped by translating the height value to the image width. Points with a height greater than zero will produce an upward curve, while points with a negative height will curve downward. The formula for curvature is:

$$ curvature = \frac{depth \times imageWidth}{bFactor} $$

  • Offset
    Since cubic functions do not pass through the origin at the vertex, we substitute ( t = 0.5 ) into the following equation to obtain an offset:

$$ offset = \frac{(1 + 3 \times aFactor) \times curvature}{4} $$

  • Original Function

Original Function

  • After Compensation

After Compensation

2. Bézier Curve Control Points

We use a cubic Bézier curve to define the path, which consists of a start point, an endpoint, and two control points.

  • Origin (Compensation)

$$ \mathbf{C} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 1 \\ offset \end{bmatrix} $$

  • Start and End Points

$$ \mathbf{P_0} = C + curvature \times \begin{bmatrix} - 1 \\ - 1 \end{bmatrix} $$

$$ \mathbf{P_3} = C + curvature \times \begin{bmatrix} 1 \\ - 1 \end{bmatrix} $$

  • Control Points

$$ \mathbf{P_1} = C + curvature \times \begin{bmatrix} - 1 \\ - 1 \end{bmatrix} \times aFactor $$

$$ \mathbf{P_2} = C + curvature \times \begin{bmatrix} 1 \\ - 1 \end{bmatrix} \times aFactor $$

3. Bézier Curve Equation

The mathematical equation for a cubic Bézier curve is:

$$ \mathbf{P}(t) = (1-t)^3 \mathbf{P_0} + 3(1-t)^2 t \mathbf{P_1} + 3(1-t) t^2 \mathbf{P_2} + t^3 \mathbf{P_3} \quad【1】 $$

  • ( t ) is the parameter on the curve, where ( t ) ranges from 0 to 1.
  • When ( t = 0 ), the point is at the start point ( P_0 ).
  • When ( t = 1 ), the point is at the endpoint ( P_3 ).
  • The control points ( P_1 ) and ( P_2 ) determine the shape and curvature of the curve.

4. SVG Path Generation

For each point, a cubic Bézier curve is generated and added to the SVG path:

$$ \text{} $$

  • M (Move To): Move to the start point $( P_0 )$
  • C (Cubic Bézier Curve): Define control points and endpoint $( P_1, P_2, P_3 )$