Piecewise Cubic Polynomial Research Articles

Using filter bank property to simplify the calculations of Empirical Mode Decomposition

The Empirical Mode Decomposition (EMD) is a method to analyze non-linear non-stationary signals. EMD was proved to behave like a dyadic filter bank when operating on a broadband signal. This means that the frequency content of the Intrinsic Mode Functions (IMFs) calculated decreases from one IMF to the following one. As the frequency contents of the signal decreases, the number of points needed to represent them and the calculations they require should decrease similar to what happens in the Fast Wavelet Transform. The sifting operation of the original EMD algorithm does not benefit from this property. It evaluates the upper and lower envelopes for all data points regardless of the decrease in frequency content that occurs from one IMF to the other. In this work, we present the Computationally Adaptive Empirical Mode Decomposition (CAEMD). This method approximates data using a piecewise-cubic polynomial (PCP) having a minimal number of knots. The sifting and all the steps of the EMD are modified to be applicable to a PCP. After the calculation of each IMF, the PCP is reapproximated to one with fewer knots. Hence, this method benefits from the frequency decrease between IMFs and simplifies the required calculations and reduces the memory needed to store the IMFs. The proposed method was implemented and has demonstrated an improvement in execution time and required storage when compared to EMD. This method, although it involves approximation, can be set to produce a negligible reconstruction error comparable to the rounding error obtained from the EMD method.

3D light fields based foreground segmentation in static scenes

To address the problem of extracting foreground objects from complex scenes,a 3D light field based method for foreground segmentation of static scenes was presented. Firstly,the method relies on densely sampled 3D light field formed by stacking a sequence of images captured at different viewpoints. These viewpoints were equally spaced along a linear path so that object trace was smooth in epipolar plan image( EPI).Secondly,line segment detection( LSD) method was performed to extract edges and compute the corresponding depth in EPI,based on which a depth map of the scene was obtained through a fast local depth interpolation algorithm. Piecewise cubic Hermite interpolating polynomial( PCHIP) algorithm proved to have desirable results in this phase. Finally,the recovered dense depth information was exploited to facilitate foreground segmentation. A threshold approach was used to separate different objects in scenes. Preliminary experimental results show that our method is able to estimate the correct relative spatial relation of multiple objects and our proposed foreground segmentation method reduces over-segmentation effects existed in traditional methods based on region clustering and mathematical morphology.

X-Splines of odd degree

Recently Clenshaw and Negus [3] and Behforooz, Papamichael, and Worsey [2] introduced a number of cubic X-splines in the class of all C’ piecewise-cubic polynomial interpolatory functions which present several practical advantages in comparison with the conventional cubic splines. A detailed discussion of these advantages can be found in [2, 3, 5,8]. More recently, Papamichael and Worsey [6] defined a class of C2 quintic Xsplines and investigated convergence and smoothness properties of some Xsplines in this class. In this paper we define X-splines of degree 2n + 1 for a positive integer n. More precisely, let A be an arbitrary partition of a compact interval I= Ca, bl,