A contour interpolation algorithm based on Fréchet distance

We describe two (1+epsilon)-approximation algorithms for computing the Frechet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Frechet distance delta. (1) A 2^{O((n + ( (Area(R)+Area(S))/(epsilon.delta)^2 )^2 )} time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant). (2) An O(D/(epsilon.delta)^2) n + 2^{O(D^4/(epsilon^4.delta^2))} time algorithm if R and S are polyhedral terrains over [0,1]^2 with slope at most D. Although, the Frechet distance between curves has been studied extensively, very little is known for surfaces. Our results are the first algorithms (both for surfaces and terrains) that are guaranteed to terminate in finite time. Our latter result, in particular, implies a linear time algorithm for terrains of constant maximum slope and constant Frechet distance.

A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the {O}(n^2) dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to O(n^2/log^{2}n) in several ways and using a variety of ingenious tricks. This line of research, also known as the art of shaving log factors, lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time O(n^2/log^3n)?\nPerhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on boolean formulas (Formula-SAT) faster than exhaustive search. They show that an O(n^2/log^{1000} n) algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear.\nIn this paper, we push this approach to its limit and, in particular, prove that a well-known barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fréchet distance problem from Computational Geometry, we show that an O(n^2/log^{7+epsilon}{n}) runtime would imply new Formula-SAT algorithms.\nOur main result is a reduction from SAT on formulas of size s over n variables to LCS on sequences of length N=2^{n/2} * s^{1+o(1)}. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with N=2^{n/2} * s^c, for some c >= 100.

This study aims to develop an algorithm for interpolating NURBS surfaces on machine tools with limited interpolation time. NURBS surfaces are usually applied to represent free-form surfaces because of the flexibility in modeling processes. However, the copious and complicated operations limit NURBS surfaces in machining applications. To interpolate NURBS surfaces in limited interpolation time, an interpolation algorithm with a new computation structure is developed in this paper. Comparing with the existing approaches, the proposed approach provides good interpolation results with little interpolation time for interpolating NURBS surfaces in actual machining processes.

Stripe noise is one of the most important factors, which influence the qualities of MODIS 1B images. In this paper, the main causes and the characteristics of the stripe noises in MODIS 1B images are deeply analyzed. To remove the stripe noises, three kinds of algorithms, the improved Fourier Transform, Wavelet Transform and the Interpolation Algorithm, are developed based on the above analysis. On academic level, the abilities to remove the stripe noises and their limitations are compared. Analyzed from the processed results, the three kinds of algorithms can more or less remove the stripe noises in MODIS 1B images, but have great differences among them: the improved Fourier Transform has more influences on non-noises region than the other two and the Interpolation Algorithm has the less negative influences on MODIS 1B images in removing the stripe noises. Based on the comparation of the mean value, standard deviation and the edge information of the images processed by above three kinds of algorithms, the conclusion that in removing the MODIS stripe noises, the improved Interpolation Algorithm is the best, and the Wavelet Transform is better than the Fourier Transform can be made.

This thesis focuses on finding ways to differentiate the gray matter (GM) and white matter (WM) in magnetic resonance (MR) images of the human spinal cord (SC). The aim of this project is to quantify tissue loss in these compartments to study their implications on the progression of multiple sclerosis (MS). To this end, we propose segmentation algorithms that we evaluated on MR images of healthy volunteers. Segmentation of GM and WM in MR images can be done manually by human experts, but manual segmentation is tedious and prone to intra- and inter-rater variability. Therefore, a deterministic automation of this task is necessary. On axial 2D images acquired with a recently proposed MR sequence, called AMIRA, we experiment with various automatic segmentation algorithms. We first use variational model-based segmentation approaches combined with appearance models and later directly apply supervised deep learning to train segmentation networks. Evaluation of the proposed methods shows accurate and precise results, which are on par with manual segmentations. We test the developed deep learning approach on images of conventional MR sequences in the context of a GM segmentation challenge, resulting in superior performance compared to the other competing methods. To further assess the quality of the AMIRA sequence, we apply an already published GM segmentation algorithm to our data, yielding higher accuracy than the same algorithm achieves on images of conventional MR sequences. On a different topic, but related to segmentation, we develop a high-order slice interpolation method to address the large slice distances of images acquired with the AMIRA protocol at different vertebral levels, enabling us to resample our data to intermediate slice positions. From the methodical point of view, this work provides an introduction to computer vision, a mathematically focused perspective on variational segmentation approaches and supervised deep learning, as well as a brief overview of the underlying project's anatomical and medical background.

The dynamic influence on each axis will significantly influence the machining stability and product precision while high speed machining (HSM) is significant but always be ignored in most interpolators.The proposed axis-based look-ahead NURBS interpolator (ALANI) not only guarantees the contour precision of workpiece,but also confines the component acceleration according to the capability of the servo system on each axis,restricting the component jerk to avoid immoderate vibration and shocks.Based on the kinematics analysis,an algorithm of interpolation strategy modification was designed,which can efficiently modify the feedrate while encountering a bad point during the look-ahead calculation.The simulation shows that the ALANI algorithm has better characteristic compared to other interpolators.

An algorithm of image reconstruction is presented, which is based on one-variable Thiele's interpolating continued fraction. In this case, the RGB values of a image pixel is taken to be a vector, we use the gray value instead of the vector, neighbor pixels are taken to be interpolation knots. Based on the theory of interpolation by Thiele's interpolating continued fraction, we offer an algorithm to calculate an isolation point. The experimental results show that the algorithm is more effective in computation and more efficient than those by conventional methods in implementing image reconstruction.

This thesis focuses on the rigid-body registration of a three-dimensional model of an object to a set of its two-dimensional projections. The main contribution is the development of two registration algorithms that use a continuous model of the volume based on splines, either in the space domain or in the frequency domain. This allows for a well-defined gradient of the dissimilarity measure, which is a necessary condition for efficient and accurate registration. The first part of the thesis contains a review of the literature on volume-to- image registration. Then, we discuss data interpolation in the space domain and in the frequency domain. The basic concepts of our registration strategy are given in the second part of the thesis. We present a novel one-step approach for fast ray casting to simulate space-based volume projections. We also discuss the use of the central-slice theorem to simulate frequency-based volume projections. Then, we consider the question of the registration robustness. To improve the robustness of the space-based approach, we apply a multiresolution optimization strategy where spline-based data pyramids are processed in coarse-to-fine fashion, which improves speed as well. To improve the robustness of the frequency-based registration, we apply a coarse-to-fine strategy that involves weights in the frequency domain. In the third part, we apply our space-based algorithm to computer-assisted orthopedic surgery while adapting it to the perspective projection model. We show that the registration accuracy achieved using the orthopedic data is consistent with the current standards. Then, we apply our frequency-based registration to three-dimensional electron-microscopy application. We show that our algorithm can be used to obtain a refined solution with respect to currently available algorithms. The novelty of our approach is in dealing with a continuous space of geometric parameters, contrary to the standard methods which deal with quantized parameters. We conclude that our continuous parameter space leads to better registration accuracy. Last, we compare the performance of the frequency-based algorithm with that of the space-based algorithm in the context of electron microscopy. With these data, we observe that frequency-based registration algorithm outperforms the space-based one, which we attribute to the suitability of interpolation in the frequency domain when dealing with strictly space-limited data.

We present a hybrid approach to generate plausible motions for high-DOF human-like articulated figures in constrained environment with multiple obstacles. The method combines motion planning and motion blending techniques to efficiently produce natural and collision-free human motion for different tasks.

Spinal degenerations can lead to segmental instability that is regarded as a major cause of back pain and is often an important factor in deciding on surgical fusion or prosthesis implant. The spinal kinematics analysis can provide useful information for diagnosis of instability and for the assessment of therapy and surgical treatment or for performance evaluation of disc prostheses. Digitized videofluoroscopy permits to analyze spinal motion during the full patient’s movement, with an acceptable low X-ray dose. By recognizing the vertebrae position on successive fluoroscopic images through manual selection or automated algorithms the relative kinematics between pairs of adjacent vertebrae (i.e. intervertebral kinematics) can be easily estimated. The application of fluoroscopy in the study of spinal kinematics is, however, limited because large errors can occur in the measurements. This thesis presents a comprehensive study of an innovative technique designed to provide a more accurate estimation of intervertebral kinematics. The recognition of vertebrae along the fluoroscopic sequence is implemented using an automated template-matching algorithm and involving a strong enhancement of the outline of vertebrae by resorting to derivative operators. Particular attention is devoted to fluoroscopic noise suppression and to edge-preserving filter design. Spline interpolation of the kinematic data extracted by videofluoroscopy is applied in order to obtain a more complete, continuous description of spinal kinematics and, more specifically, of instantaneous center of rotation. In the introductory part of the thesis (Chapter I and II) the motivation of the study and a survey of spinal measurement techniques are given. The feasibility of videofluoroscopic analysis of spinal motion is extensively discussed. In Chapter III common kinematic parameters (such as range of motion, center of rotation, etc.) utilized for describing intervertebral spinal behaviour are presented, providing particular emphasis on the difficulty to determine a “boundary” between normal and abnormal measures of segmental kinematics for the definition of spinal instability. An extensive review of recent proposals in analysis of segmental motion is reported. Manual recognition of anatomical landmarks in videofluoroscopy can be very problematic. It is also well-known that derivative operators, commonly used for automatic recognition, are highly sensitive to noise. Chapter IV attempts to address this issue: fluoroscopic noise model, also in presence of non-linear gray-level transformations for image enhancement, is presented; various denoising algorithms specifically designed for signal-dependent noise and AWGN are examined and a performance comparison among them is carried out. In Chapter V the proposed algorithm for automated vertebrae recognition is described and its performance is experimentally analyzed on fluoroscopic images of a calibration model. A comparison with a manual selection procedure and other automated algorithms on real lumbar fluoroscopic sequences is presented. In Chapter VI a continuous-time description of intervertebral motion by cubic smoothing spline interpolation is presented and the evaluation of instantaneous center of rotation of spinal motion segments by videofluoroscopy is discussed.

Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.

داشتن درک درستی از چگونگی تغییرات زمانی و مکانی متغیر های هواشناسی در مطالعات محیطی از اهمیت بسزایی برخوردار است. اخیرا از مدل‌های منطقه ای پیش بینی عددی هواشناسی در تحقیقات بسیاری استفاده می شود. با این وجود برای بررسی چگونگی تغییرات زمانی و مکانی نتایج حاصل از مدل ها تحقیقات زیادی صورت نگرفته است. بر این اساس و با توجه به اهمیت این موضوع یک رویکرد زمانی - مکانی در این مقاله بررسی می شود. در این رویکرد نتایج پیش‌بینی مدل میان مقیاس 4MM5 با مجموعه داده های مشاهده ای، برای میانگین ماهانه و سالانه دما و بارش در دوره سال‌های 1990-2010 در منطقه شمال شرق ایران مورد تحلیل و بررسی قرار گرفت. در این تحقیق ارزیابی کمی شباهت مکانی و زمانی بین مجموعه داده‌ها و مدل آب و هوایی، و نیز شناسایی مناطق جغرافیایی و فصولی که در مدل سازی‌ها مشکل‌ساز می‌باشند انجام شده است. این کار با استفاده از نقشه‌های تهیه شده بر مبنای الگوریتم تشابه5 جبری، که یک روش موثر برای تحلیل مجموعه داده‌های مکانی مختلف، نقشه‌ها و مدل‌ها می‌باشد، انجام شد. مشخص شد که شباهت داده‌های واقعی با نتایج مدل برای متغیر دما بیش‌تر از بارندگی است. نتایج حاصل از مدل برای بارش و دما در فصل‌های گرم سال در مقایسه با داده‌های واقعی از لحاظ پراکندگی مقادیر تشابه، تناقض داشت. هم‌چنین در فصل‌های سرد سال پیش‌بینی دقیق‌تری به وسیله مدل ارائه شده است. علاوه بر این چند منطقه که در آنجا خطای پیش‌بینی بیش‌تر بود به وسیله الگوریتم نقشه‌پردازی تشابه شناسایی شدند. این تحلیل نشان داد که در محدوده‌های کوچکی از جنوب و ناحیه مرکزی منطقه مورد مطالعه بین مدل و داده‌های واقعی تشابه کم‌تری وجود دارد.

Publication in the conference proceedings of EUSIPCO, Bucharest, Romania, 2012

SOPC becomes popular nowadays, which combines software and hardware in a single chip. In this paper, a multi-axis motion controller is implemented based on a SOPC system with a NIOSII processor in a single FPGA chip. The motion controller includes DDA module, spindle module and NURBS calculation module realized in Verilog HDL. DDA algorithm is used to drive the motors while spindle module is to drive the spindle. Because of NURBS recursive nature, NURBS calculation module is used to accelerate the calculation which takes the advantage of parallel structure. The NIOSII processor realizes the feedrate scheduling, in which constant feedrate is adopted. The second-order interpolation algorithm is used to control the feedrate.

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.