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Nonlinear scalespace should be based on a hierarchical statistical model of the image intensity function. This model should contain an explicit representation of the multiscale structure of edges and corners. Using this model we can have a non-ad-hoc basis for computing the parameters we need to determine how much smoothing we should do at points that appear to be edge points. We also have a basis for computing the apparent error in our scalespace calculations. Hierarchical statistical modeling is a technique that can be applied to other problems in low-level vision, but in this introductory paper we just present the application of our scalespace theory to image smoothing.

In this paper, we present an analytical approach in obtaining the probability density function (pdf) of the random decision variable Y, formed at the output of power-cubic all-optical nonlinear preprocessor followed by the photodetector. Our approach can be used to accurately evaluate the performance of ultrafast pulse detection in the presence of Gaussian noise. Through rigorous Monte-Carlo simulation, the accuracy of widely used Gaussian approximation of decision variable Y is refuted. However, in this paper we show that the so called Log-Pearson type-3 probability density function (LP3 pdf) is an excellent representation for the decision variable Y . Three distinguishable parameters of the LP3 pdf are obtained through analytical derivation of three moments of the decision variable Y . Furthermore, toward a more realistic model, in addition to ASE Gaussian noise, the effects of shot and thermal noises are also included. Finally, using the presented analytical approach, it is shown that power-cubic preprocessor outperforms its quadratic counterparts, i.e., Second Harmonic Generation (SHG) and Two Photon Absorption (TPA) devices, in high power regime where shot and thermal noises can be neglected.

In this paper, we present an analytical approach in obtaining the probability density function (pdf) of the random decision variable Y, formed at the output of power-cubic all-optical nonlinear preprocessor followed by the photodetector. Our approach can be used to accurately evaluate the performance of ultrafast pulse detection in the presence of Gaussian noise. Through rigorous Monte-Carlo simulation, the accuracy of widely used Gaussian approximation of decision variable Y is refuted. However, in this paper we show that the so called Log-Pearson type-3 probability density function (LP3 pdf) is an excellent representation for the decision variable Y . Three distinguishable parameters of the LP3 pdf are obtained through analytical derivation of three moments of the decision variable Y . Furthermore, toward a more realistic model, in addition to ASE Gaussian noise, the effects of shot and thermal noises are also included. Finally, using the presented analytical approach, it is shown that power-cubic preprocessor outperforms its quadratic counterparts, i.e., Second Harmonic Generation (SHG) and Two Photon Absorption (TPA) devices, in high power regime where shot and thermal noises can be neglected.