Interpolation Of Sequences Research Articles

Measurement Interpolation Methods for Dual One-Way Ranging Systems

A dual one-way ranging method is designed to minimize the oscillator noise effect in intersatellite microwave ranging systems by combining two one-way phase measurements. This system requires the measurement timing synchronization of the two measurements. The interpolation of the two measurements toward a common epoch resolves this requirement but causes interpolation error. To reduce the interpolation error, a new method is proposed, which modifies the interpolation sequence. Simulation data based on the Gravity Recovery and Climate Experiment mission are tested with the new method, and it demonstrates the advantage over a conventional interpolation method. Experiments with the flight data agree with the simulation results. In addition to the interpolation error reduction, this method partlyevaluates the interpolation errorlevel in the flight data processing.

Refolding Planar Polygons

This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against self-intersection is independent from a distance metric that determines the overall character of the interpolation sequence. This decoupled approach provides a powerful control mechanism for determining how the interpolation should appear, while still assuring against intersection and guaranteeing termination of the algorithm. Our algorithm also allows additional control by accommodating a set of algebraic constraints that can be weakly enforced throughout the interpolation sequence.

The Average Errors for the Grünwald Interpolation in the Wiener Space

We determine the weakly asymptotically orders for the average errors of the Grünwald interpolation sequences based on the Tchebycheff nodes in the Wiener space. By these results we know that for the -norm approximation, the -average error of some Grünwald interpolation sequences is weakly equivalent to the -average errors of the best polynomial approximation sequence.

Motion-Compensated Frame Interpolation for Intra-Mode Blocks

Motion-compensated frame interpolation (MCFI) is widely used to smoothly display low frame rate video sequences by synthesizing and inserting new frames between existing frames. The temporal shift interpolation technique (TSIT) is popular for frame interpolation of video sequences that are encoded by a block-based video coding standard such as MPEG-4 or H.264/AVC. TSIT assumes the existence of a motion vector (MV) and may not result in high-quality interpolation for intra-mode blocks that do not have MVs. This paper proposes a new frame interpolation algorithm mainly designed for intra-mode blocks. In order to improve the accuracy of pixel interpolation, the new algorithm proposes sub-pixel interpolation and the reuse of MVs for their refinement. In addition, the new algorithm employs two different interpolation modes for inter-mode blocks and intra-mode blocks, respectively. The use of the two modes reduces ghost artifacts but potentially increases blocking effects between the blocks interpolated by different modes. To reduce blocking effects, the proposed algorithm searches the boundary of an object and interpolates all blocks in the object in the same mode. Simulation results show that the proposed algorithm improves PSNR by an average of 0.71 dB compared with the TSIT with MV refinement and also significantly improves the subjective quality of pictures by reducing ghost artifacts.

Interpolation of infinite sequences by entire functions

The study was originally motivated by a theoretical question raised by Warming and Hyett in a famous publication on the Modified Equation Approach [R.F. Warming, B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput Phys. 14 (1974) 159–179]. Their classical accuracy analysis of a finite-difference method applied to a time-dependent problem implicitly relies on the assumption that a function interpolating the numerical values can be expanded, over an indefinite domain, in Taylor's series of the independent variables x and t. Here, we establish constructively that the problem of interpolation of an arbitrary infinite sequence of real numbers by an entire function of x (and possibly t) admits uncountably many solutions. In the case of a single variable, if the values are bounded, the interpolant can be made bounded, and all its derivatives bounded. Besides, the construction is generalized to the interpolation of the values of the function and its derivatives up to an arbitrarily prescribed order ( Hermitian interpolation). The proposed interpolant depends on a free parameter λ, and its behavior as λ varies is illustrated by a numerical example.

Interpolation of multidimensional stationary sequences

Interpolation of multidimensional stationary sequences

Complemented invariant subspaces and interpolation sequences

It is shown that the invariant subspace of the Bergman space A p A^p of the unit disc, generated by a finite union of Hardy interpolation sequences, is complemented in A p A^p .

Interpolation of Sequences

We present a generalization of the following result of Y. Benyamini. There is a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that for each (/$x_n)_{n \in \mathbb{Z}}\in [0,1]^\mathbb{Z}$, there is $t \in \mathbb{R}$ such that $x_n=f(t+n)$ for all $n\in \mathbb{Z}$.

Reconstruction of nonuniformly sampled images in spline spaces

This paper presents a novel approach to the reconstruction of images from nonuniformly spaced samples. This problem is often encountered in digital image processing applications. Nonrecursive video coding with motion compensation, spatiotemporal interpolation of video sequences, and generation of new views in multicamera systems are three possible applications. We propose a new reconstruction algorithm based on a spline model for images. We use regularization, since this is an ill-posed inverse problem. We minimize a cost function composed of two terms: one related to the approximation error and the other related to the smoothness of the modeling function. All the processing is carried out in the space of spline coefficients; this space is discrete, although the problem itself is of a continuous nature. The coefficients of regularization and approximation filters are computed exactly by using the explicit expressions of B-spline functions in the time domain. The regularization is carried out locally, while the computation of the regularization factor accounts for the structure of the nonuniform sampling grid. The linear system of equations obtained is solved iteratively. Our results show a very good performance in motion-compensated interpolation applications.

Extended Cesàro operators on Bergman spaces

We define an extended Cesàro operator T g with holomorphic symbol g in the unit ball B of C n . For a large class of weights w we characterize those g for which T g is bounded (or compact) from Bergman space L p a, w ( B) to L q a, w ( B), 0< p, q<∞. In addition, we obtain some results about equivalent norms, the norm of point evaluation functionals, and the interpolation sequences on L p a, w ( B).

Image Interpolation Using Enhanced Multiresolution Critical-Point Filters

There are increasing demands for image interpolation in various fields such as virtual reality, computer animation, and video transmission. The critical-point filters (CPF) we have proposed previously enable completely automatic matching of two images. In our previous method, however, it has taken a long time to compute the optimal parameter values by iterative searches. This paper proposes an improved algorithm called the enhanced critical-point filters (ECPF) where the parameters are stably computed using together the inverse mapping from the destination to the source. It takes only a second to match images whose size is 64 × 64. The algorithm is also improved in its precision by directly handling color images while the previous algorithm has taken only the intensity value into account. We apply ECPF to keyframe interpolation of video sequences. We also apply ECPF to interpolating two different views of an object to generate any intermediate views. In this case, there are many constraints that can be used to determine the mapping between the images. We propose a method to use such constraints to improve the accuracy of the mappings. As the results, image-based pseudo-3D models are easily created from a set of views without any special devices.

Interpolation Sequences for the Class of Functions of Finite -Type Analytic in the Unit Disk

We establish conditions for the existence of a solution of the interpolation problem f(λn) = bn in the class of functions f analytic in the unit disk and such that $$\left( {\exists \;c_1 > 0} \right)\;\left( {\forall z,\;|\;z\;| < 1} \right):\;\;\left| {f\left( z \right)} \right|\;\; \leqslant \;\;\;\exp \left( {c_1 \eta \left( {\frac{{c_1 }}{{1 - \left| z \right|}}} \right)} \right).$$ Here, η : [1; +∞) → (0; +∞) is an increasing function convex with respect to ln t on the interval [1; +∞) and such that ln t = o(η(t)), t → ∞.

A forward-trajectory global semi-Lagrangian transport scheme

A forward-trajectory semi-Lagrangian scheme for advection on the surface of the sphere is proposed. The advection scheme utilizes the forward (downstream) trajectory originating at Eulerian grid points and cascade interpolation, a sequence of 1D interpolations, to transfer data from the downstream Lagrangian points to the Eulerian points. A new and more accurate algorithm determines pole values. The resulting forward-trajectory semi-Lagrangian scheme can easily incorporate high-order trajectory integration methods. This avoids the standard iterative process in a typical backward-trajectory scheme. Two third-order accurate schemes and a second-order accurate scheme are presented. A mass-conservative version of the forward-trajectory semi-Lagrangian scheme is also derived within the cascade interpolation framework. Mass from a Lagrangian cell is transferred to the corresponding Eulerian cell with two 1D remappings through an intermediate cell system. Mass in the polar region is redistributed by way of an efficient local approximation. The resulting scheme is globally conservative, but restricted to meridional Courant number, C θ ⩽1.

Finite unions of interpolation sequences

A unified and relatively simple proof is given for some well-known results involving finite unions of uniformly separated sequences.

Multiple interpolation and extremal functions in the Bergman spaces

Multiple interpolation sequences for the Bergman space are characterized. In addition, relationships between interpolation sequences and extremal functions are explored.

A genetic algorithm approach to image sequence interpolation

Image sequence interpolation, or to obtain an up-sampled image sequence equivalently from a corresponding low-resolution image sequence, is an ill-posed inverse problem. In this study, three processing steps, namely, regularization, discretization and optimization, are used to convert the image sequence interpolation problem into a solvable optimization problem. In regularization, a fitness function combining a set of spatial and temporal performance measures for rating the quality of the interpolated (up-sampled) images is defined, which is used to convert the original ill-posed interpolation problem into a well-posed optimization problem. Discretization transforms the well-posed problem into a discrete one so that it can be solved numerically. Genetic algorithms (GAs) are used to optimize the solution in the discrete solution space using three basic operations, namely, reproduction, crossover and mutation. In the proposed approach, instead of only the spatial information within the current image frame employed in most existing methods, both the spatial and temporal information within the image sequence can be employed. Based on the experimental results obtained in this study, the interpolation results by the proposed approach are always better than those from the three existing approaches used for comparison. This shows the feasibility of the proposed approach.

On Interpolation Sequences of One Class of Functions, Analytic in the Half-Plane

A new criterion of solvability of the interpolation problem f(λn)=bn in the class of functions f, analytic in the right half-plane \(\mathbb{C}_ + \) and such that there exists c1∈(0;+∞) such that |f(z)|≤c1exp(η(c1|z|)) for all z ∈\(\mathbb{C}_ + \), where η is a positive increasing continuous differentiable function on [0;+∞), for which η(t)→+∞ as t→+∞ and there exists c2∈(0;+∞) such that $$\frac{{\eta (t)\ln \eta (t)}}{{t\eta '(t)}} \leqslant c_2 $$ for all t≥ 1 is described.

On Interpolation Sequences of One Class of Functions Analytic in the Unit Disk

We establish a criterion for the existence of a solution of the interpolation problem f(λ n ) = b n in the class of functions f analytic in the unit disk and satisfying the relation $$\left( {\exists {\tau }_{1} \in \left( {0;1} \right)} \right)\;\left( {\exists c_1 >0} \right)\;\left( {\forall z,\left| z \right| < 1} \right):\;\left| {f\left( z \right)} \right| \leqslant \exp \left( {c_1 \gamma ^{{\tau }_{1} } \left( {\frac{{c_1 }}{{1 - \left| z \right|}}} \right)} \right),$$ where γ: [1; +∞) → (0; +∞) is an increasing function such that the function lnγ(t) is convex with respect to lnt on the interval [1; +∞) and lnt = o(lnγ(t)), t → ∞.

On seip's description of sampling sequences for bergman spaces

In his 1993 paper (K. SeipInvent. Math. 113 (1993) 21–39), Kristian Seip characterizes sampling and interpolation sequences for the space A −n and gives an outline of the proof of the corresponding theorem for the Bergman space A 2. Using his techniques, we provide a complete proof of the result concerning sampling sequences not only for A 2, but also for A ρ, 1 ≤ p < ∞.

Weak conditions for interpolation in holomorphic spaces

An analogue of the notion of uniformly separated sequences, expressed in terms of extremal functions, yields a necessary and sufficient condition for interpolation in Lp spaces of holomorphic functions of Paley-Wiener-type when $0 < p \leq 1$, of Fock-type when $0 < p \leq 2$, and of Bergman-type when $0 < p < \infty$. Moreover, if a uniformly discrete sequence has a certain uniform non-uniqueness property with respect to any such Lp space ($0 < p < \infty$), then it is an interpolation sequence for that space. The proofs of these results are based on an approximation theorem for subharmonic functions, Beurling's results concerning compactwise limits of sequences, and the description of interpolation sequences in terms of Beurling-type densities. Details are carried out only for Fock spaces, which represent the most difficult case.