Explain about gray level interpolation.
The distortion correction equations yield non integer values for x' and y'. Because the distorted image g is digital, its pixel values are defined only at integer coordinates. Thus using non integer values for x' and y' causes a mapping into locations of g for which no gray levels are defined. Inferring what the gray-level values at those locations should be, based only on the pixel values at integer coordinate locations, then becomes necessary. The technique used to accomplish this is called gray-level interpolation.
The simplest scheme for gray-level interpolation is based on a nearest neighbor approach.
This method, also called zero-order interpolation, is illustrated in Fig. 6.1. This figure shows
(A) The mapping of integer (x, y) coordinates into fractional coordinates (x', y') by means of following equations
and
x' = c1x + c2y + c3xy + c4
y' = c5x + c6y + c7xy + c8
(B) The selection of the closest integer coordinate neighbor to (x', y'); and
(C) The assignment of the gray level of this nearest neighbor to the pixel located at (x, y).
Fig. 6.1 Gray-level interpolation based on the nearest neighbor concept.
Although nearest neighbor interpolation is simple to implement, this method often has the drawback of producing undesirable artifacts, such as distortion of straight edges in images of high resolution. Smoother results can be obtained by using more sophisticated techniques, such as cubic convolution interpolation, which fits a surface of the sin(z)/z type through a much larger number of neighbors (say, 16) in order to obtain a smooth estimate of the gray level at any
desired point. Typical areas in which smoother approximations generally are required include 3- D graphics and medical imaging. The price paid for smoother approximations is additional computational burden. For general-purpose image processing a bilinear interpolation approach that uses the gray levels of the four nearest neighbors usually is adequate. This approach is straightforward. Because the gray level of each of the four integral nearest neighbors of a non integral pair of coordinates (x', y') is known, the gray-level value at these coordinates, denoted v(x', y'), can be interpolated from the values of its neighbors by using the relationship
v (x', y') = ax' + by' + c x' y' + d
where the four coefficients are easily determined from the four equations in four unknowns that can be written using the four known neighbors of (x', y'). When these coefficients have been determined, v(x', y') is computed and this value is assigned to the location in f{x, y) that yielded the spatial mapping into location (x', y'). It is easy to visualize this procedure with the aid of Fig. 6.1. The exception is that, instead of using the gray-level value of the nearest neighbor to (x', y'), we actually interpolate a value at location (x', y') and use this value for the gray-level assignment at (x, y).
