Equating the Two Intensity-Curvature Terms, Before and After Interpolation, Attempting to Obtain Resilient Interpolation: Trivariate Linear Interpolation Function

Equating the Two Intensity-Curvature Terms, Before and After Interpolation, Attempting to Obtain Resilient Interpolation: Trivariate Linear Interpolation Function

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch012
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As well accepted in literature it should be almost impossible to reconstruct the signal intensity value at time-space location where the sampling instrument has not recorded the signal. This happens because sampling is a discrete process, while instead signals (as those of biological nature among others) are continuous. Indeed this mismatch between nature and humanly invented devices gives immediate explanation as to why interpolation is a signal reconstruction technique that has quite substantial and general relevance. This chapter introduces the novel concept of resilient interpolation for the case of the trivariate linear function while continuing the treatise from Chapter VII where the concept was presented for the case of the bivariate linear function. The mathematical process is presented in full and equations are given to characterize resilient interpolation. Additionally, a discussion is provided to the reader to further clarify the methodology employed and the relevant implications. Overall, the intent of this purely theoretical chapter is that of attempting to further improve the approximation capabilities of the trivariate linear interpolation function while making use of the knowledge provided with the SRE-based methodological approach and also introducing a further conceptualization. The intent is therefore once again a more accurate signal reconstruction to be obtained through a revised form of the classic interpolation function. Details of the meaning behind the concepts herein presented indicate that resilient interpolation is such that the intensity-curvature content before and after interpolation remains unchanged.
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The Mathematical Procedure

From equations (11) and (10) of Chapter IV it can be written that the intensity curvature terms before (Eo), and after interpolation (EIN) are:EIN = EIN(x, y, z) = - ωf Hxyz (x, y, z) [ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ](1) Eo = Eo(x, y, z) = - f (0, 0, 0) x y z ωfxy + θyz + θzx ] (2) Where through equations (9) through (12) of Chapter III it is posited that:

θxy = θyx = - [ f(0, 0, 0) – f(1, 0, 0) – f(0, 1, 0) + f(1, 1, 0) ] / ωf(3) θxz = θzx = - [ f(0, 0, 0) – f(1, 0, 0) – f(0, 0, 1) + f(1, 0, 1) ] / ωf(4) θyz = θzy = - [ f(0, 0, 0) – f(0, 1, 0) – f(0, 0, 1) + f(0, 1, 1) ] / ωf(5) ωf = [ f(1, 1, 1) - f(1, 1, 0) - f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) - f(0, 0, 0) ] (6)

Let us proceed by posing Eo(x, y, z) = EIN(x, y, z), and solving the resulting equation in the unknown f(0, 0, 0). The problem herein addressed can be formulated in the following terms.

For the given re-sampling location what is the true pixel (signal) intensity value? Can the model interpolation function calculate that signal intensity value that is the one that would be sampled? It is reasonable to attempt to answer the question by posing in equality the two intensity curvature terms which is like assuming that there exists a pixel (signal) intensity value that remains the same before and after interpolation. This pixel (signal) intensity value is not the one we start from (before interpolation) nor is the value that it would be obtained through re-sampling the signal through the model interpolation function at the given placement.

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