Explain Forward warping algorithm for transforming an image f(x) into an image g(x) through the parametric transform x= h(x)

Explain Forward warping algorithm for transforming an image f(x) into an image g(x) through the parametric transform x= h(x)

Answer:-

In image processing and computer vision, geometric transformations are used to remap pixel coordinates from a source image f(x) to a destination image g(x) using a transformation function. The forward warping algorithm is a technique used to implement such transformations based on the parametric mapping:

x' = h(x)

Here, x represents coordinates in the source image, and h(x) is the parametric function (such as affine or projective) that maps them to new coordinates x' in the destination image.

Definition of Forward Warping

In the forward warping approach, each pixel x from the source image f(x) is transformed using the function h(x), and the resulting transformed pixel x' = h(x) is placed into the destination image g(x').

Algorithm Steps

1. For each pixel coordinate x in the input image f(x):
   • Compute the transformed coordinate: x' = h(x)
   • Assign the intensity value: g(x') = f(x)
2. Repeat this for all pixels in f(x)

Problems with Forward Warping

While conceptually simple, forward warping suffers from several practical issues:

• Holes in the output image: Because output coordinates x' may not align with integer pixel locations, many output pixels may remain empty or “undefined”.
• Overlaps or collisions: Multiple source pixels may map to the same destination pixel, leading to ambiguity unless averaging or blending is used.

Dealing with Holes and Overlaps

To mitigate holes and overlaps, two common strategies are used:

• Rounding + Interpolation: Round x' to the nearest integer and optionally use interpolation to distribute values to nearby pixels.
• Splatting: Distribute the source pixel’s value to nearby destination pixels using a kernel (e.g., Gaussian splat).

Comparison with Backward Warping

Unlike forward warping, backward warping avoids holes by mapping destination pixels x' back to source coordinates x = h-1(x'), and interpolating values from f(x). This makes backward warping more commonly used in practice.

Applications

Forward warping is used in:

• Texture mapping in graphics
• Image mosaicking (if handled with splatting)
• Physical simulations of pixel movement