4.2.2 Brightness interpolation

• Assume that the planar transformation has been accomplished, and new point co-ordinates $(x',y')$ were obtained.

• The position of the pixel transformed does not in general fit the discrete grid of the output image.

• Values on the integer grid are needed.

• Each pixel value in the output image can be obtained by brightness interpolation of some neighboring non-integer samples, transformed from the input image.

• The brightness interpolation problem is usually expressed in a dual way (by determining the brightness of the original point in the input image that corresponds to the point in the output image lying on the discrete raster).

• Assume that we wish to compute the brightness value of the pixel $(x',y')$ in the output image where $x'$ and $y'$ lie on the discrete grid.

• The co-ordinates of the point $(x, y)$ in the original image can be obtained by inverting the transformation

  $$(x,y) = T^{-1} (x',y').$$ (4.34)

  
  

• In general the real co-ordinates $(x, y)$ after inverse transformation do not fit the input image discrete grid, and so brightness is not known.

• To get the brightness value of the point $(x, y)$ the input image is re-sampled or interpolated:

  $$f_n (x,y) = \sum_{l = -\infty}^{\infty} \sum_{k = -\infty}^{\infty} g_s (l \Delta x, k \Delta y) h_n (x - l \Delta x, y - k \Delta y)$$ (4.35)

  
  

  where $f_{n}(x,y)$ is the result of interpolation and $h_{n}$ is the interpolation kernel. $n$ distinguishes different interpolation methods.

• Usually, a small neighborhood is used, outside which $h_{n}$ is zero.