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An iterative constrained least squares method for continuous piecewise linear approximation

COMPUTERS & CHEMICAL ENGINEERING 2022年 168卷

作者： Kim, Ji Hee Choi, Naeun Heo, Seongmin Dankook Univ Dept Chem Engn Yongin 16890 South Korea

In this work, a novel iterative least squares method is proposed to approximate nonlinear functions with continuous piecewise linear (CPWL) functions, where the continuity is guaranteed by constrained least squares. A typical least-squares-based method for CPWL fitting consists of two major steps: to perform least squares for a fixed set of breakpoints (i.e. for a specific partition of function domain), and to update breakpoints to reduce the overall fitting error. The proposed method aims to improve the existing CPWL method, the one with canonical representation, by modifying both major steps. Instead of performing ordinary least squares with canonical representation, partitioned least squares and constrained least squares are successively performed to reduce the computational complexity. For the update of breakpoints, an iterative procedure is proposed, where gradient descent with momentum method is employed to improve the convergence characteristics. The advantages of proposed method are illustrated using illustrative examples.

关键词： piecewise linear approximation Constrained least squares Iterative least squares Gradient descent Momentum

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The Art of piecewise linear approximation in MMSE Estimator for Most Accurate and Fast Frequency Extraction in DIFM Receivers

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT 2021年 70卷 1页

作者： Keshani, Shamim Masoumi, Nasser Univ Tehran Sch Elect & Comp Engn Circuits & Syst & Test Lab CST Lab Dept ElectColl Engn Tehran *** Iran Univ Tehran Sch Elect & Comp Engn Dept Elect Coll Engn Tehran *** Iran

Detection of wideband passive radar signals, accomplished by instantaneous frequency measurement (IFM) systems, is crucial for gaining a profound knowledge of environmental circumstances. The need for fast pulse detection has hindered the investigation of frequency extraction with higher accuracy in IFM receivers. A noteworthy research in recent years shows that fast and reliable frequency extraction in high bit-count (HBC) IFMs while attaining a better frequency accuracy [called intelligent, reliable, and design-independent (InReDI)] is feasible. In addition, some investigated process-intensive algorithms like the minimum mean squared error (MMSE) estimator promise excellent frequency accuracy. In this work, we present a new method of MMSE algorithm modified by a piecewise linear approximation [named linear approximation of MMSE (LAMMSE)]. By utilizing the InReDI methodology and the novel LAMMSE algorithm, we propose a new architecture for a fast, accurate, and reliable frequency extraction (FAREx) in HBC digital IFMs (DIFM) receivers. The proposed FAREx architecture resolves the compromise challenge between frequency accuracy and extraction speed that has been long lingered in DIFM receivers. We have implemented the new architecture using Xilinx fieldprogrammable gate array (FPGA) on a fabricated 15 bit, 2-4 GHz, and 70 dB dynamic range IFM receiver. The measurement results show an average frequency rms error of 1.43 MHz for a CW signal, 2.49 MHz average rms error for a 50 ns pulsed input signal, and 2.7 MP/s detection speed, representing a state-ofthe-art receiver and a breakthrough compared to the latest works.

关键词： Algorithm architecture digital processing fast accurate and reliable frequency extraction (FAREx) instantaneous frequency measurement (IFM) receiver minimum mean squared error (MMSE) estimator piecewise linear approximation

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fastSW: Efficient piecewise linear approximation of Quaternion-Based Orientation Sensor Signals for Motion Capturing with Wearable IMUs

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SENSORS 2021年第15期21卷 5180-5180页

作者： Gruetzmacher, Florian Kempfle, Jochen Van Laerhoven, Kristof Haubelt, Christian Univ Rostock Inst Appl Microelect & Comp Engn D-18051 Rostock Germany Univ Siegen Dept Elect Engn & Comp Sci D-57076 Siegen Germany

In the past decade, inertial measurement sensors have found their way into many wearable devices where they are used in a broad range of applications, including fitness tracking, step counting, navigation, activity recognition, or motion capturing. One of their key features that is widely used in motion capturing applications is their capability of estimating the orientation of the device and, thus, the orientation of the limb it is attached to. However, tracking a human's motion at reasonable sampling rates comes with the drawback that a substantial amount of data needs to be transmitted between devices or to an end point where all device data is fused into the overall body pose. The communication typically happens wirelessly, which severely drains battery capacity and limits the use time. In this paper, we introduce fastSW, a novel piecewise linear approximation technique that efficiently reduces the amount of data required to be transmitted between devices. It takes advantage of the fact that, during motion, not all limbs are being moved at the same time or at the same speed, and only those devices need to transmit data that actually are being moved or that exceed a certain approximation error threshold. Our technique is efficient in computation time and memory utilization on embedded platforms, with a maximum of 210 instructions on an ARM Cortex-M4 microcontroller. Furthermore, in contrast to similar techniques, our algorithm does not affect the device orientation estimates to deviate from a unit quaternion. In our experiments on a publicly available dataset, our technique is able to compress the data to 10% of its original size, while achieving an average angular deviation of approximately 2 degrees and a maximum angular deviation below 9 degrees.

关键词： piecewise linear approximation segmentation motion capturing IMU wearable sensors orientation quaternion

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Segmentation of gray-scale images using piecewise linear approximation

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SYSTEMS AND COMPUTERS IN JAPAN 1996年第1期27卷 69-76页

作者： Yamasaki, I Ohshima, T Hasegawa, M Furukawa, T FUJI PHOTO FILM CO LTD KANAGAWA 258JAPAN SHINSHU UNIV FAC TEXT SCI & TECHNOLDEPT KANSEI ENGNMATSUMOTONAGANOJAPAN

This paper deals with clock truncation coding for gray-scale images. An image region is segmented into unequal-sized square blocks according to local luminance variations and luminance values in the block are approximated by triangular plane patches constructed from the luminance values at the four vertices of the block. Computational complexities of the unequal-sized block segmentations are discussed. Segmentations are classified into three types: top-down splitting;bottom-up merging;and generalized split-and-merge method. It is derived from the theoretical model that initial block size in the split-and-merge method for a given image of size 256 x 256 is 17 x 17. Segmentation by accumulated square error is preferred to human visual sensitivity. Experimental results using a standard image have clarified that the segmentation by the accumulated square error improves the quality of approximated image to 1 to 2 dB compared with segmentation by the mean square error. Moreover, it is confirmed that the approximation by triangular plane patches is superior to JPEG for the compression of the images that have much flat areas.

关键词： block truncation coding piecewise linear approximation triangular plane patches split-and-merge method accumulated square error

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Dynamic step selection algorithm for piecewise linear approximation of complex control trajectories

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OCEAN ENGINEERING 2023年第1期280卷

作者： Tan, Liguo Li, Liyi Su, Haoxiang Novikova, S. V. Zhang, Xinbin Mingaliyev, Z. Z. Harbin Inst Technol Harbin Peoples R China Southwest Univ Sci & Technol Mianyang Peoples R China Kazan Natl Res Tech Univ Named AN Tupolev Kazan Russia Natl Res Mordovia State Univ Saransk Russia

The paper proposes a method for replacing a complex function used as a target for automatic control of un-manned vehicles with a simplified piecewise linear function, which requires much less computing resources. A numerical method has been developed for adaptive selection of a variable step for approximating a nonlinear one-dimensional function, the analytical expression of which is not given, by a piecewise linear function. The article shows that selection of approximation step (grid) is an important task for minimizing the calculations required number, if the computing device on board the unmanned vehicle must be miniature. The developed algorithm includes the calculation of successive intervals lengths that eventually cover the entire domain of the function with a predetermined approximation accuracy. The determination coefficient is used as a measure of accuracy. Numerical experiments are presented for calculating the linear-piecewise function parameters for the approximating a target trajectory problem with complex configuration for unmanned aerial underwater vehicle automatic control. The proposed method efficiency is compared with methods with a constant step, as well as with methods for calculating the dynamic step in various approaches. The advantage of the developed method from the point of view of computational costs is proved with the same accuracy, the limitations of the appli-cability of the method are determined.

关键词： Automatic control Target trajectory Unmanned vehicle piecewise linear approximation Variable step approximation grid step Numerical methods

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Adaptive error bounded piecewise linear approximation for time-series representation

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2023年第PartB期126卷

作者： Zhou, Zhou Baratchi, Mitra Si, Gangquan Hoos, Holger H. Huang, Gang Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China Leiden Univ Leiden Inst Adv Comp Sci Leiden Netherlands Xi An Jiao Tong Univ Sch Elect Engn State Key Lab Elect Insulat & Power Equipment Xian 710049 Peoples R China Rhein Westfal TH Aachen Chair AI Methodol Aachen Germany Univ British Columbia Vancouver BC Canada Zhejiang Univ Coll Elect Engn Hangzhou 310027 Zhejiang Peoples R China

Error-bounded piecewise linear approximation (1.0-PLA) has been proven effective in addressing challenges in data management and analytics. It works by approximating the original time series with linear segments such that the approximation error on each data point does not exceed a pre-defined threshold. To achieve this goal, most prior works on 1.0-PLA use a fixed error threshold during the entire approximation process, assuming a stable time series. However, in many real-world applications, the distribution of values on the time series undergoes substantial changes in different temporal stages. This introduces a need to adaptively tune the error threshold based on the properties of the time series while considering the trade-off between representation error and storage resources. In this work, we propose a general framework for constructing 1.0- PLA with adaptive error bounds (AEPLA). It works by dividing the original time series into a set of intervals and assigns adaptive error bounds to these sub-sequences based on their fluctuation levels. Next, these subsequences are approximated using a user-defined 1.0-PLA method. We implement this framework by employing three different types of 1.0-PLA methods and evaluate the performance of our approach on an extensive set of real-world time-series datasets from industrial and scientific domains. Our experiments show that constructing 1.0-PLA using the AEPLA framework can provide better a trade-off between representation error and storage resources and achieves lower time and space costs than applying the original 1.0-PLA methods.

关键词： Time series Data representation piecewise linear approximation Adaptive error bounds

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Time and Memory Efficient Online piecewise linear approximation of Sensor Signals

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SENSORS 2018年第6期18卷 1672-1672页

作者： Gruetzmacher, Florian Beichler, Benjamin Hein, Albert Kirste, Thomas Haubelt, Christian Univ Rostock Inst Appl Microelect & Comp Engn D-18051 Rostock Germany Univ Rostock Inst Comp Sci D-18051 Rostock Germany

piecewise linear approximation of sensor signals is a well-known technique in the fields of Data Mining and Activity Recognition. In this context, several algorithms have been developed, some of them with the purpose to be performed on resource constrained microcontroller architectures of wireless sensor nodes. While microcontrollers are usually constrained in computational power and memory resources, all state-of-the-art piecewise linear approximation techniques either need to buffer sensor data or have an execution time depending on the segment's length. In the paper at hand, we propose a novel piecewise linear approximation algorithm, with a constant computational complexity as well as a constant memory complexity. Our proposed algorithm's worst-case execution time is one to three orders of magnitude smaller and its average execution time is three to seventy times smaller compared to the state-of-the-art piecewise linear approximation (PLA) algorithms in our experiments. In our evaluations, we show that our algorithm is time and memory efficient without sacrificing the approximation quality compared to other state-of-the-art piecewise linear approximation techniques, while providing a maximum error guarantee per segment, a small parameter space of only one parameter, and a maximum latency of one sample period plus its worst-case execution time.

关键词： piecewise linear approximation segmentation sensor data abstraction sensor signal compression CPLR wireless sensor networks embedded sensor processing

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An algorithm for the optimum piecewise linear approximation of digitized curves

An algorithm for the optimum piecewise linear approximation ...

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International Conference on Pattern Recognition

作者： J.-C. Perez E. Vidal Dpto. de Sistemas Informáticos y Computación (DSIC) Universidad Politécnica de Valencia Valencia Spain

A dynamic programming algorithm for piecewise linear approximation of a digitized curve is presented. This approximation yields a global minimization of the total error between the approximating straight-line segments and the original points. Many error measures can be used with this algorithm although a lower computational cost can be achieved if error measures that can be computed incrementally are used. Two incremental formulations of classical error measures, along with the results of a sample experiment, are also presented.< >

关键词： approximation algorithms piecewise linear approximation Dynamic programming Signal processing algorithms Heuristic algorithms piecewise linear techniques Computational efficiency Pattern analysis Signal analysis Pattern classification

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piecewise linear ORTHOGONAL approximation

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2010年第3期48卷 840-856页

作者： App, Andreas Reif, Ulrich Univ Stuttgart Fachbereich Math D-70569 Stuttgart Germany Tech Univ Darmstadt Working Grp Geometry & Approximat D-64289 Darmstadt Germany

We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in R-d are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield sup... 详细信息

关键词： piecewise linear approximation Sobolev space orthogonality

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Hilbert transform by divide-and-conquer piecewise linear approximation

Hilbert transform by divide-and-conquer piecewise linear app...

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European Conference on Circuit Theory and Design, ECCTD

作者： Lacrimioara Grama Corneliu Rusu Faculty of Electronics Telecommunications and Information Technology Technical University of Cluj-Napoca Cluj-Napoca Romania

For many years it has been known that having a piecewise linear fitting of a function, one can obtain a good approximation of its Hilbert transform. There are few ways to determine the slopes of a broken line approximation, but the most popular approaches need to specify the breakpoints. In this paper we determine the breakpoints using a divide-and-conquer approach, then the Hilbert transform can be computed in two different ways. Simulations results provided here show that the piecewise linear approximation of function and the resulting Hilbert transform approximation are relatively accurate, and the complexity of implementation is not large.

关键词： piecewise linear approximation Transforms linear approximation Simulation TV Gratings

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