Interpolation Based method for Directional Derivative Calculation in Image Processing 1.0
Directional derivative calculation can be done by using a number of kernels in image processing. Most of them are fixed and can only use in single direction. This article attempts to discuss and use a geometrical model to as a solution for this situation.
Considering the gradient calculation in image processing, since it is discrete domain, calculations are done by forward and backward differences. Therefore co-efficients of pixels can directly use for built a kernel which can convolve with a given image to determine the gradient.
Consider two pixels f(x,y) and f(xr,yr) of an image which are in r direction with |r| distance. Then the directional derivative can be written as;
CASE 1 : 0≤θ≤45o
This minimised to,
Therefore we can finalise the kernel K for 0≤θ≤45o,
By using the same approach, we can conclude the kernels for 0≤θ≤180o as;
RESULTS FOR ROTATION INVARIENT IMAGE
Figure 1:Derivative Angle and Sum of Directional Derivatives