Daichi Kitahara

Algebraic phase unwrapping along the real axis: extensions and stabilizations

The unwrapped phase of a complex function is defined with a line integral of the gradient of the arctangent of the ratio of the real and imaginary parts of the function. The phase unwrapping, which is a problem to reconstruct the unwrapped phase of an unknown complex function from its finite observed samples, has been a key for estimating useful physical quantity in many signal and image processing applications. In the light of the functional data analysis, it is natural to estimate first the unknown complex function by a certain piecewise complex polynomial and then to compute the exact unwrapped phase of the piecewise complex polynomial with the algebraic phase unwrapping algorithms (Yamada et al. in IEEE Trans Signal Process 46(6), 1639–1664, 1998; Yamada and Bose in IEEE Trans Circuits Syst I Fundam Theory Appl 49(3), 298–304, 2002; Yamada and Oguchi in Multidimens Syst Signal Process 22(1–3), 191–211, 2011). In this paper, we propose several useful extensions and numerical stabilizations of the algebraic phase unwrapping along the real axis which was established originally in Yamada and Oguchi (Multidimens Syst Signal Process 22(1–3), 191–211, 2011). The proposed extensions include (i) removal of a certain critical assumption premised in the original algebraic phase unwrapping, and (ii) algebraic phase unwrapping for a pair of bivariate polynomials. Moreover, in order to resolve certain numerical instabilities caused by the coefficient growth in an inductive step in the original algorithm, we propose to compute directly a certain subresultant sequence without passing through the inductive step. The extensive numerical experiments exemplify the notable improvement, in the performance of the algebraic phase unwrapping, made by the proposed numerical stabilization.

Algebraic Phase Unwrapping Based on Two-Dimensional Spline Smoothing Over Triangles

Two-dimensional (2D) phase unwrapping is an estimation problem of a continuous phase function, over a 2D domain, from its wrapped samples. In this paper, we propose a novel approach for high-resolution 2D phase unwrapping. In the first step - SPline Smoothing (SPS), we construct a pair of the smoothest spline functions which minimize the energies of their local changes while interpolating, respectively, the cosine and the sine of given wrapped samples. If these functions have no common zero over the domain, the proposed estimate of the continuous phase function can be obtained by algebraic phase unwrapping in the second step - Algebraic Phase Unwrapping (APU). To avoid the occurrence of common zeros in SPS due to phase noise in the observed wrapped samples, we also propose a denoising step - Denoising by Selective Smoothing (DSS) - as preprocessing, which selectively smooths unreliable wrapped samples by using convex optimization. The smoothness of the proposed unwrapped phase function is guaranteed globally over the domain without losing any desired consistency with all reliable wrapped samples. Numerical experiments for terrain height estimation demonstrate the effectiveness of the proposed 2D phase unwrapping scheme.

Algebraic phase unwrapping over collection of triangles based on two-dimensional spline smoothing

Phase unwrapping is a reconstruction problem of the continuous phase function from its finite wrapped samples. Especially the two-dimensional phase unwrapping has been a common key for estimating many crucial physical information, e.g, the surface topography measured by interferometric synthetic aperture radar. However almost all two-dimensional phase unwrapping algorithms are suffering from either the path dependence or the excess smoothness of the estimated result. In this paper, to guarantee the path independence and the appropriate smoothness of the estimated result, we present a novel algebraic approach by combining the ideas in the algebraic phase unwrapping with techniques for a piecewise polynomial interpolation of two-dimensional finite data sequence.

Algebraic phase unwrapping for functional data analytic estimations—Extensions and stabilizations

The phase unwrapping, which is a problem to reconstruct the continuous phase function of an unknown complex function from its finite observed samples, has been a key for estimating useful physical quantity in many signal and image processing applications. In the light of the functional data analysis, it is natural to estimate first the unknown complex function by a certain piecewise complex polynomial and then to compute the exact unwrapped phase of the piecewise complex polynomial with the algebraic phase unwrapping algorithms. In this paper, we propose several useful extensions and numerical stabilization of the algebraic phase unwrapping along the real axis. The proposed extensions include (i) removal of a certain critical assumption premised in the original algebraic phase unwrapping, and (ii) algebraic phase unwrapping for a pair of bivariate polynomials. Moreover, in order to resolve certain numerical instabilities caused by the coefficient growth in an inductive step in the original algorithm, we propose to compute directly a certain subresultant sequence without passing through the inductive step.

A virtual resampling technique for algebraic two-dimensional phase unwrapping

Two-dimensional (2D) phase unwrapping is a reconstruction problem of a continuous phase, defined over 2D-domain, from its wrapped samples. In our previous work, we presented a two-step phase unwrapping algorithm which first constructs, as the real and imaginary parts of a complex function, a pair of piecewise polynomials having no common zero over the domain, then estimates the unwrapped phase by applying the algebraic phase unwrapping. In this paper, we propose a preprocessing of the above algorithm for avoiding the appearance of zeros of the complex function in the first step. The proposed preprocessing is implemented by a convex optimization and resampling, and its effectiveness is shown in a terrain height estimation by the interferometric synthetic aperture radar.

Algebraic phase unwrapping with self-reciprocal polynomial algebra

Algebraic phase unwrapping gives the exact expression of the unwrapped phase of a complex polynomial. However, in computation of a Sturm sequence, there exist numerical instabilities due to coefficient growth. In this paper, we refine algebraic phase unwrapping by modifying the Sturm sequence with the newly defined self-reciprocal polynomial division. The proposed refinement enables us to compute the unwrapped phase, without suffering from the coefficient growth, by using the self-reciprocal subresultant which is newly defined as the determinant of a certain matrix. Numerical experiments show that algebraic phase unwrapping is greatly stabilized by the proposed method.

A fast dual iterative algorithm for convexly constrained spline smoothing

This paper proposes fast iterative algorithms for solving convexly constrained spline smoothing through a characterization of solutions in a primal-dual space. In view of achievements for fast implementations of spline interpolation, the update of the proposed algorithm is designed as the composition of solving a spline interpolation problem and computing the projection onto the constraint set. In addition, the update of the proposed algorithm is performed in an efficient dimensional space having the same size as given observations. These desired properties significantly reduce the computational cost in the update, which is demonstrated by a numerical example.

Two-dimensional positive spline smoothing and its application to probability density estimation

Spline is a piecewise polynomial and has been widely used for interpolation and smoothing of observed data. In this paper, with the use of the sufficient condition, derived by Heß and Schmidt, for the nonnegativity of bivariate splines on square grid, we propose two-dimensional positive spline interpolation/smoothing on square grid for estimation of positive continuous functions. Moreover, we newly derive a sufficient condition for the nonnegativity on triangular grid and propose positive spline interpolation/smoothing on triangular grid. Then we estimate a two-dimensional probability density function (PDF) from its histogram by using the idea of the positive spline smoothing. Numerical experiments show the effectiveness of the newly derived sufficient condition and the proposed PDF estimator.

Probability density function estimation by positive quartic C 2 -spline functions

Spline is a continuous function piecewise-defined by polynomials and is widely used for interpolation and smoothing of observed data. In 1994, Heβ and Schmidt proposed a positive quartic C2-spline interpolation for estimation of a non-negative and twice continuously differentiable function. In this paper, first we generalize the positive quartic C2-spline interpolation to the positive quartic C2-spline smoothing. Then we propose two estimation methods of a probability density function from its histogram by extending the ideas of the positive quartic C2-spline interpolation and smoothing. Finally numerical experiments show the effectiveness of the proposed methods.