GEOTHER 1.1: Handling and Proving Geometric Theorems Automatically
GEOTHER provides an environment for handling and proving theorems in geometry automatically. In this environment, geometric theorems are represented by means of predicate specifications. Several functions are implemented that allow one to translate the specification of a geometric theorem into an English or Chinese statement, into algebraic expressions, and into a logic formula automatically. Geometric diagrams can also be drawn automatically from the predicate specification, and the drawn diagrams may be modified and animated with a mouse click and dragging. Five algebraic provers based on Wu’s method of characteristic sets, the Grobner basis method, and other triangularization techniques are available for proving such theorems in elementary (and differential) geometry. Geometric meanings of the produced algebraic nondegeneracy conditions can be interpreted automatically, in most cases. PostScript and HTML files can be generated, also automatically, to document the manipulation and machine proof of the theorem. This paper presents these capabilities of GEOTHER, addresses some implementation issues, reports on the performance of GEOTHER’s algebraic provers, and discusses a few challenging problems.

title: Automated Generation of Diagrams with Maple and Java
abstract: This note shows how to draw diagrams automatically from the predicate specification of a given set of geometric relations among a set of points in the plane. It is done first in Maple by translating the geometric relations into polynomial equations, decomposing the obtained system of polynomials into irreducible representative triangular sets, and finding an adequate numerical solution from each triangular set. A Java class coding the solution and the polynomials in each triangular set is generated, compiled, and then executed with the main Java programs to draw a diagram. The whole process combining symbolic elimination in Maple with numerical computation, graphic drawing, and letter labeling in Java is fully automatic. The drawn diagrams may be animated and fine-tuned by mouse click and dragging and saved as PostScript files. We present the drawing method, discuss some techniques of implementation, and give several sample diagrams drawn by our program.

title: Reasoning about Surfaces Using Differential Zero and Ideal Decomposition
abstract: This paper presents methods for zero and ideal decomposition of partial differential polynomial systems and the application of these methods and their implementations to deal with problems from the local theory of surfaces. We show how to prove known geometric theorems and to derive unknown relations automatically. In particular, an algebraic relation between the first and the second fundamental coefficients in a very compact form has been derived, which is more general and has smaller degree than a relation discovered previously by Z. Li. Moreover, we provide symmetric expressions for Li's relation and clarify his statement. Some examples of theorem proving and computational difficulties encountered in our experiments are also discussed.

title: On Wu's Method for Proving Constructive Geometric Theorems
abstract: In 1977 Wu Wen-tsiin discovered an efficient method for mechanical theorem proving. This method has been applied to prove and discover theorems in various kinds of geometries. Based on an extension of one of Wu's algorithmic procedures, the author implemented a prover CPS for proving so-called constructive geometric theorems, as well as a general theorem prover PS. Using this prover we proved more than one hundred non-trivial theorems and discovered also a number of interesting ones. This paper is a summarization of our early work, as a abbreviation of the chapter 2 from my Chinese doctoral dissertation.

title: GEOTHER: A Geometry Theorem Prover

title: A simple method for implicitizing rational curves and surfaces
abstract: This paper presents a simple method for converting rational parametric equations of curves and surfaces into implicit equations. The method proceeds via writing out the implicit polynomial F of estimated degree with indeterminate coefficients u i , substituting the rational expressions for the given parametric curve or surface into F to yield a rational expression g / h in the parameter s (or s and t ), equating the coefficients of g in terms of s (and t ) to 0 to generate a sparse, partially triangular system of linear equations in u i with constant coefficients, and finally solving the linear system for u i . If a nontrivial solution is found, then an implicit polynomial is obtained; otherwise, one repeats the same process, increasing the degree of F . Our experiments show that this simple method is efficient. It performs particularly well in the presence of base points and may detect the dependency of parameters incidentally.

title: Wavelet transform-based frequency tuning ILC
abstract: A discrete wavelet transform-based cutoff frequency tuning method is proposed and experimental investigation is reported. In the method, discrete wavelet packet algorithm, as a time-frequency analysis tool, is employed to decompose the tracking error into different frequency regions so that the maximal error component can be identified at any time step. At each time step, the passband of the filter is from zero to the upper limit of frequency region where the maximal error component resides. Hence, the filter is a function of time as well as index of cycle. The experimental results show that this method can suppress higher frequency error components at proper time steps. While at the time steps where the major tracking error falls into lower frequency range, the cutoff frequency of the filter is set lower to reduce the influence of noises and uncertainties. This way, learning transient and long-term stability can be improved.

title: Scene analysis by integrating primitive segmentation and associative memory
abstract: Scene analysis is a major aspect of perception and continues to challenge machine perception. This paper addresses the scene-analysis problem by integrating a primitive segmentation stage with a model of associative memory. The model is a multistage system that consists of an initial primitive segmentation stage, a multimodule associative memory, and a short-term memory (STM) layer. Primitive segmentation is performed by a locally excitatory globally inhibitory oscillator network (LEGION), which segments the input scene into multiple parts that correspond to groups of synchronous oscillations. Each segment triggers memory recall and multiple recalled patterns then interact with one another in the STM layer. The STM layer projects to the LEGION network, giving rise to memory-based grouping and segmentation. The system achieves scene analysis entirely in phase space, which provides a unifying mechanism for both bottom-up analysis and top-down analysis. The model is evaluated with a systematic set of three-dimensional (3-D) line drawing objects, which are arranged in an arbitrary fashion to compose input scenes that allow object occlusion. Memory-based organization is responsible for a significant improvement in performance. A number of issues are discussed, including input-anchored alignment, top-down organization, and the role of STM in producing context sensitivity of memory recall.

title: Making Predicate Abstraction Efficient: How to Eliminate Redundant Predicates
abstract: In this paper we consider techniques to identify and remove redundant predicates during predicate abstraction. We give three criteria for identifying redundancy. A predicate is redundant if any of the following three holds (i) the predicate is equivalent to a propositional function of other predicates. (ii) removing the predicate preserves safety properties satisfied by the abstract model (iii) removing it preserves bisimulation equivalence. We also show how to efficiently remove the redundant predicates once they are identified. Experimental results are included to demonstrate the effectiveness of our methods.

title: Oriented statistical nonlinear smoothing filter
abstract: This paper presents a nonlinear smoothing method which is based on an orientation-sensitive probability measure. By incorporating geometrical constraints through the coupling structure, we obtain a robust nonlinear smoothing algorithm. Even when noise is substantial the proposed smoothing algorithm can still preserve salient boundaries. Compared with anisotropic diffusive approaches, the proposed nonlinear algorithm not only performs better in preserving boundaries but also has a non-uniform stable state, whereby reliable results are available within a fixed number of iterations independent of images. A system using the proposed method and LEGION network has been developed and applied in noisy image segmentation and hydrographic feature extraction from digital ortho-photo quadrangles. Experimental results using synthetic and real images are provided.

title: Theoretical Foundation for Nonlinear Edge-Preserving Regularized Learning Image Restoration
abstract: Image restoration is an important issue in image processing, which helps recovering of degraded images caused by various factors in different circumstances. Neural networks models have been successfully applied in handling image restoration problems and some progress and promising results have been reported in the past. A popular neural networks model for image restoration is the Hopfield network due to its ability on dealing with optimization problems. A degraded image may have multiple corresponding solutions, i.e., the restored images, and the obtained solution can be sub-optimal which is related to a local minimum point in the weight space of the Hopfield network. This paper gives an algebraic characterization on images and shows that images with lower complexity can be restored uniquely from their degraded images and edge information. The obtained result in this paper establishes a mathematical basis for employing the mapping neural networks to realize a learning based image restoration scheme. The linear image restoration model is firstly generalized to a nonlinear one to broaden the scope of application. Secondly, we view the image restoration task as a set of approximation problems in a high dimensional space, and a mapping relationship between the degraded images with edge information and the source images is then built using feed-forward neural networks and the well trained neural network can be used to restore the degraded images in realtime. Computer simulations demonstrate the effectiveness of the learning image restoration techniques proposed in this paper.