Arccosine Function Research Articles

A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

A Hybrid Forecasting Model for Non-Stationary Time Series

In time series analysis, an important problem is how to extract the information hidden in the non-stationary and noise data and combine it into a model for forecasting. In this paper, the authors propose a TEI@I based hybrid forecasting model. A novel feed forward neural network is developed based on the improved particle swarm optimization with adaptive genetic operator (IPSO-FNN) for forecasting. In the proposed IPSO, inertia weight is dynamically adjusted according to the feedback from particles’ best memories, and acceleration coefficients are controlled by a declining arccosine and an increasing arccosine function. Subsequently, a crossover rate which only depends on generation and an adaptive mutation rate based on individual fitness are designed. The parameters of FNN are optimized by binary and decimal particle swarm optimization. Further, the forecast results of IPSO-FNN are adjusted with the knowledge from text mining and an expert system. The empirical results on the container throughput forecast of Tianjin Port show that forecasts with the proposed method are much better than some other methods.

Direct phase extraction from interferograms with random phase shifts

A new method that extracts phase directly from randomly phase-shifted interferograms is proposed. In this method, the background intensity and the modulation amplitude are determined by a temporal method and used as the system parameters and then the phase is extracted from one single interferogram by arc cosine function with range of (0, π). By partitioning the extracted phase distribution into watershed regions and determining the sign of phase in each region, the extracted phase is recovered to its principle phase with range of (-π, π). Numerical simulation and experiment are implemented to verify the effectiveness of this method. The method has applications in both static and dynamic phase measurement.

Absolute Fringe Order Determination in Digital Photoelasticity

An improvement of the already proposed phase unwrapping algorithm for directly processing the triangular-typed wrapped phase map of the isochromatic parameter, photoelastically generated by an arccosine function based on the well-known technique of phase-shifting, is presented. In phase unwrapping algorithm, at any pixel, a three-dimensional plane and its normal vector are generated using three fractional (relative) fringe order values. Its generated plane is then adjusted using the orthogonal projection associated with the reference plane. Such point is unwrapped on the basis of regularization. The quality guide map is used to guide phase unwrapping and to mask out the conflictive regions such that they are lastly processed. The circular disk under compression demonstrated the performance of the improved algorithm. Results showed the accuracy improvement in such conflictive regions with reasonable agreement to theory.

Improved the-law-of-cosines-based fast search method for vector quantization by updating angular information

Vector quantization (VQ) is a well-known signal compression method. In the framework of VQ, fast search method is one of the key issues because it is the time bottleneck for practical VQ applications. By introducing the-law-of-cosines and directly using the angular information to reject a candidate codeword, the previous work [Chung, K.L., Lai, J.Y., 2004. Pattern Recognition Lett. 25 (14), 1613–1617] has proposed a very efficient fast search method for VQ encoding. However, there still exist two problems in this work as (1) a complicated arccosine function (i.e., COS −1) is used and (2) the reference vector for a given input vector is fixedly selected as the initial best-matched codeword in terms of the minimum difference between L 2 norm and no updating to the reference vector is conducted during the whole search process. This paper aims at improving these two problems further by (1) avoiding using the COS −1 function completely and (2) updating the reference vector whenever it is possible so as to be able to set up a better reference vector for the following search process. Because a better reference vector always helps to reject a candidate codeword, it in principle can speed up the search process. Experimental results confirmed that the proposed method outperforms the previous work [Chung, K.L., Lai, J.Y., 2004. Pattern Recognition Lett. 25 (14), 1613–1617] obviously.

Phase extraction from a single fringe pattern based on guidance of an extreme map

A method for automatic phase extraction from a single fringe pattern based on the guidance of an extreme map is introduced. The method uses an adaptive weighted filter to reduce noise and enhance contrast and to locate the fringe extremes. Wrapped phase values are calculated by use of an arccosine function obtained from the extreme map. With this method, wrapped phase values can be efficiently demodulated from a single fringe pattern without the need for assigning fringe order or interpolating fractional fringe order. The validity of the method is demonstrated by use of closed-fringe patterns generated by digital speckle interferometry.

Correction to “Hybrid Frequency-Polarization Shift-Keying Modulation for Optical Transmission”

In a previous paper [1], the cosine function in (23) and (26) should be replaced by the inverse cosine function (arccosine function).

Nonlinear temperature response of lake ice breakup

A uniquely comprehensive set of four decades of ice breakup data from 196 Swedish lakes covering 13° of latitude (55.7°N to 68.4°N) shows the relationship between the timing of lake ice breakup and air temperature to be an arc cosine function. The nonlinearity inherent in this relationship results in marked differences in the response of the timing of lake ice breakup to changes in air temperature between colder and warmer geographical regions, and between colder and warmer time periods. The spatial and temporal patterns are mutually consistent, suggesting that climate change impacts on the timing of lake ice breakup will vary along a temperature gradient. This has potentially important ramifications for the employment of lake ice phenologies as climate indicators and for the future behavior of lacustrine ecosystems.

Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting.

The use of an optical fringe projection method with two-step phase shifting for three-dimensional (3-D) shape measurement of small objects is described. In this method, sinusoidal linear fringes are projected onto an object's surface by a programmable liquid-crystal display (LCD) projector and a long-working-distance microscope (LWDM). The image of the fringe pattern is captured by another LWDM and a CCD camera and processed by a phase-shifting technique. Usually a minimum of three phase-shifted fringe patterns is necessary for extraction of the object shape. In this method, a new algorithm based on a two-step phase-shifting technique produces the 3-D object shape. Unlike in the conventional method, phase unwrapping is performed directly by use of an arccosine function without the need for a wrapped phase map. Hence, shape measurement can be speeded up greatly with this approach. A small coin is evaluated to demonstrate the validity of the proposed measurement method, and the experimental results are compared with those of the four-step phase-shifting method and the conventional mechanical stylus method.