Dear Researchers,
I had done some image rotation experiments using Matlab but I received good results in Bilinear and Bicubic Interpolation approximately higher than 93%, one the other hand, results in Nearest Neighbour Interpolation is not good approximately lower than 63%. I want to know why I have good results in Bilinear and Bicubic rather than Nearest Neighbour? Please telling clearly in mathematically.
Regards,
Ehsan Nowroozi
Bicubic and bilinear perform interpolations (including subpixel level) using neighbor pixel values to fill in the 'new' pixel locations exposed due to rotation. In nearest neighbors, you just fill in the exposed locations based on the nearest neighbor pixel value, which is akin to repeating a pixel value across its neighbors. This can be modeled as convolution with different filters (step for NN, /ramp, FIR filters for bil/bic). In the frequency domain, the FIR filters have a much better response (no aliasing) compared to the step filter. Visually, this translates to a smooth appearance using bilinear/bicubic versus 'jaggies' using NN
Bicubic and bilinear perform interpolations (including subpixel level) using neighbor pixel values to fill in the 'new' pixel locations exposed due to rotation. In nearest neighbors, you just fill in the exposed locations based on the nearest neighbor pixel value, which is akin to repeating a pixel value across its neighbors. This can be modeled as convolution with different filters (step for NN, /ramp, FIR filters for bil/bic). In the frequency domain, the FIR filters have a much better response (no aliasing) compared to the step filter. Visually, this translates to a smooth appearance using bilinear/bicubic versus 'jaggies' using NN
Bilinear and Bicubic use a mathematic fitting function to achieve the sub-pixel accuracy.
Math examples:
https://en.wikipedia.org/wiki/Curve_fitting
Nearest Neighbour just copies the nearest pixel. No function fitting and no sub-pixel accuracy.
Another way to explain is, that you try to interpolate using polynoms of different degree. Nearest neighbor can be seen as degree 0, bilinear as degree 1 and so one. Similar to a Taylor approximation, the approximation is more accurate the higher the order.
However, this is just intuition because you can never know for sure how a pixel in between looks like unless you observe it. Assuming a smooth gradient between pixels just proved to be quite close to reality.
The issue is with the interpolation operation used to calculate new pixel values. The nature of the interpolation kernel required depends on the degree of correlation between pixels in the image. For the mathematics of interpolation, see
A. W. Doerry, E. Bishop, J. Miller, V. Horndt, D. Small, “Designing interpolation kernels for SAR data resampling,” SPIE 2012 Defense, Security & Sensing Symposium, Radar Sensor Technology XVI, Vol. 8361, Baltimore MD, 23-27 April 2012.
"The nature of the interpolation kernel required depends on the degree of correlation between pixels in the image" (Armin Walter Doerry)
In simple words, higher order interpolation (bicubic, sinc, ...) can be a good solution for smooth images, but on images that contain sharp edges, they can produce artifacts (over- and undershoot near the edges). Even nearest neighbor can be the winner on rare occasions, for example when you have a completely black and white (non grayscale) image.
The reason is that the interpolation method tries to reconstruct the original, continuous image from the discretized values, i.e. from the pixels. Therefore the best method for a given problem has to use qualitatively similar functions to the continuous "reality" that is discretized on the image.
Three comments
1. "nearest neighbor" does not have any theoretical foundation, it is just based on the intuitive idea of similarity for pixels that are close geographically. Hence it might appear to work well for one data set and very poorly for another.
2. In contrast Bilinear and Bicubic are special cases of interpolation using Radial Basis functions (although really intended for point data not for pixels). Do a search on Google for "Radial Basis function interpolation"
3. You give percentages but do not explain what these represent, if somehow these are supposed to imply relative accuracy that is misleading. The only way you can quantify "accuracy in interpolation is if you know the real values you are supposedly generating by interpolation but in that case you wouldn't need to use interpolation
If you really want to rotate an image you should focus on transforming the coordinates of the pixels (by a rotation) rather than by interpolation
http://paulbourke.net/texture_colour/imageprocess/
http://dsp.stackexchange.com/questions/18265/bicubic-interpolation
http://www.cambridgeincolour.com/tutorials/image-interpolation.htm
http://stackoverflow.com/questions/15176972/bi-cubic-interpolation-algorithm-for-image-scaling
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