# Li ChenUniversity of the District of Columbia | UDC · Department of Electrical and Computer Engineering

Li Chen

Doctor of Philosophy

## About

101

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Introduction

**Skills and Expertise**

## Publications

Publications (101)

Estimating bone density using mathematical models has direct applications in medical image processing. This paper presents a new measure for bone mineral density analysis based on the dual energy X-ray absorptiometry (DEXA) images. We are proposing an innovative procedure to calculate a scalar value that indicates the connectivity of bone mineral c...

In an recent paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (II)}, we discussed two algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. The first algorithm was the continuation of the work initiated in {\it L. Chen, Algorithms for Deform...

In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. However, the algorithms could not guarantee to be applicable to every case. This paper will be the con...

As an emergent field of inquiry, Data Science serves both the information technology world and the applied sciences. Data Science is a known term that tends to be synonymous with the term Big-Data; however, Data Science is the application of solutions found through mathematical and computational research while Big-Data Science describes problems co...

Classical data processing uses pattern recognition methods such as classification for categorizing data. Such a method may involve a learning process. Modern data science also uses topological methods to find the structural features of data sets. In fact, topological methods should be the first step before the classification method is applied in mo...

What is Data Science? Data contains science. It is much different from the angle of classical mathematics that uses mathematical models to fit the data. Today, we are supposed to find rules and properties in the data set, even among different data sets. In this chapter, we will explain data science and its relationship to BigData, cloud computing a...

In this paper, we propose a geometric framework, called Vector Bundle Learning (VBL) for feature extraction. In our framework, a vector bundle is considered as the intrinsic structure to extract features from high dimensional data. By defining a manifold to model the structure of sample set, features sampled from each fibre of a vector bundle can b...

Machine learning usually requires getting a training and testing set of samples. The training set is used to obtain the model, and then, the testing set is used to verify the model. In general, a machine learning method requires an iterated process for reaching a goal. Machine learning is one of the research areas in artificial intelligence. Machin...

A basic problem in data science is to classify a massive data set into different categories. In addition, when more new data samples are collected or come from online sources, we want to know how to put the new data sample into an appropriate category. This problem contains such information as classification, learning, and reconstruction. As we dis...

BigData has been a buzzword since a few years ago. However, what is exactly the corresponding (theoretical) computer model? What can be done and cannot be done with such a model? These are questions demanding answers. Recently, a model was proposed to address this issue by simulating a restricted version of the PRAM model. In this paper, we propose...

Images and videos are really BigData, even before the concept of BigData was initiated or created. For many companies, image related data occupies 80 % of their storage. Therefore, data processing related to images is an essential topic in data science. The tasks concerning images and videos are mainly object search, recognition, and tracking. Curr...

Mathematical imaging aims to develop new mathematical tools for the emerging field of image and data sciences. Automatic segmentation in the variational framework is a challenging as well as a demanding task for a number of imaging tasks. Achieving robustness and reliability is a major problem. The two-phase piecewise-constant model of the Mumford–...

Data science utilizes all mathematics and computer sciences. In this chapter, we give a brief review of the most fundamental concepts in data science: graph search algorithms, statistical methods especially principal component analysis (PCA), algorithms and data structures, and data mining and pattern recognition. This chapter will provide an overv...

In this exploration paper, we design algorithms for deforming and contracting
a simply connected discrete closed manifold to a discrete sphere. Such a
contraction is a kind of shrinking or reducing process. In our algorithms, we
need to assume an ambient space for the discrete manifold, and this ambient
space also a simply connected discrete space...

In this paper we give a discrete proof of the general Jordan-Schoenflies
Theorem. The classical Jordan-Schoenflies Theorem states that a simple closed
curve in the two-dimensional sphere $S^2$ separates the space into two
connected components where each component is homeomorphic to an open disk. The
common boundary of these two components is this c...

This book describes current problems in data science and Big Data. Key topics are data classification, Graph Cut, the Laplacian Matrix, Google Page Rank, efficient algorithms, hardness of problems, different types of big data, geometric data structures, topological data processing, and various learning methods. For unsolved problems such as incompl...

Nowadays, differential geometry is not only still one of the most profound research areas of mathematics after having had great influence in physics for more than a century, but it has also recently begun to play a very important role in computer graphics and image processing. The Poincare conjecture was believed to be proven by Perelman in 2004. H...

In Chap. 2, we introduced some algorithms for graphs. In this chapter, we specifically design algorithms for digital object recognition and tracking. These algorithms are mainly for digital surfaces and manifolds. There are two types of questions to solve in this chapter: (1) Given a set of data M, decide or recognize whether the data represents a...

This chapter presents a general-purpose definition of discrete curves, surfaces, and manifolds. This definition only refers to a simple graph, \(G=(V,E)\) and its topological structure. Similar to digital manifolds defined in Chap. 5, the ideas presented in this chapter still use recursive definitions for discrete curves, surfaces, solid objects, a...

The best way to describe discrete objects is to use graphs. A graph consists of vertices and edges. The vertex usually represents a part of an object, an whole object, or the location of an object; the edge represents a relationship between two vertices. A graph can be defined as \(G=(V,E)\) where V is a set of vertices and E is a set of edges, eac...

This chapter contains three parts. First, we discuss data reconstruction including curve and surface fitting. Second, we cover principal component analysis, one of the most important geometric data analysis methods. Third, we present mathematical transformations for data analysis. This chapter is highly related to concurrent data sciences from theo...

In this chapter, we present advanced topics in digital geometry and topology including digital curvatures and their applications. First we give a brief overview of current development of computational topology that overlaps digital topology. Second, we introduce digital Gaussian curvatures and prove the digital form of the Gauss-Bonnet theorem. The...

This chapter introduces Euclidean spaces, topological spaces, and their relationships to discrete spaces. We first introduce the concept of metrics, the distance measure of Euclidean spaces. Then, we introduce general continuous spaces—topological space. At the end, we discuss the relationship between continuous spaces and discrete spaces.
In conti...

Topology is the study of the equivalence between two general spaces under continuous mappings. Two spaces are called homemorphic if there is an invertible continuous function between them. Triangles or simplexes are used in topological analysis of a space since we want to decompose a complex space into some simpler shapes to better understanding th...

The digital surface is one of the main topics of this book. We know that digital curves are digital paths and discrete surfaces can be described as triangulations. Therefore, is a digital surface a simple digitization of a continuous surface? The answer is no. This is because the basic 2D cell of digital surface in direct adjacency is a unit square...

In this chapter, we introduce basic 2D digital geometry. The main topic in 2D geometry is curves. A 2D digital curve is a simple path in \(\Sigma_{2}\). A simple closed digital curve is usually the boundary of a connected component. We first discuss how we precisely define a curve in a graph and Euclidean space, then we discuss how we represent dig...

This chapter deals with the relationships among objects in discrete, digital, and continuous spaces. If Chap. 7 can be viewed as covering the theoretical aspect of discrete spaces and its objects, then this chapter can be viewed as covering the practical methods to make actual moves from one space to another. Likewise, Chap. 9 then connects classic...

Digital geometry is a relatively new research area. It is difficult to show the characteristics of digital geometry as a well-developed theory. On the other hand, discrete geometry used to focus on combinatorial methods such as simplicial decomposition, counting, and tillings. However, it is now also much interested in differential geometry methods...

In ancient times, the need for measuring land resulted in the development of geometry, much like the need for counting yielded arithmetic. The easiest example is to measure the distance between two points as we discussed in Chap. 3. In this chapter, we cover basic geometric measurements including curve length, surface area, and solid volumes in cla...

In this chapter, we focus on cutting edge problems in geometric data processing. These problems have common properties and usually can be summarized as generally as: Given a set of n data points \(x_1,...,x_n\) in m-dimensional space,R
m
, how do we find the geometric structures of the sets or how do we use the geometric properties in real data pro...

Digital Functions and Data Reconstruction: Digital-Discrete Methods provides a solid foundation to the theory of digital functions and its applications to image data analysis, digital object deformation, and data reconstruction. This new method has a unique feature in that it is mainly built on discrete mathematics with connections to classical met...

According to a general definition of discrete curves, surfaces, and
manifolds. This paper focuses on the Jordan curve theorem in 2D discrete
spaces. The Jordan curve theorem says that a (simply) closed curve separates a
simply connected surface into two components. Based on the definition of
discrete surfaces, we give three reasonable definitions o...

Based on previous results of digital topology, this paper focuses on
algorithms of topological invariants of objects in 2D and 3D Digital Spaces. We
specifically interest in solving hole counting of 2D objects and genus of
closed surface in 3D. We first prove a new formula for hole counting in 2D. The
number of of holes is $h=1 + (|C_4|-|C_2|)/4$ w...

Numerically solving partial differential equations (PDE) has made a significant impact on science and engineering since the last century. The finite difference method and finite elements method are two main methods for numerical PDEs. This chapter presents a new method that uses gradually varied functions to solve partial differential equations, sp...

As a practical method,digital-discrete data reconstruction uses both discrete and continuous methods for data interpolation and approximation. Today,a significant development in discrete mathematics is digital technology. Digital methods contain a flavor of graphical presentation and artificial intelligence. It is very interesting to explore the re...

Gradually varied functions or smooth gradually varied functions were developed for the data reconstruction of randomly arranged data points, usually referred to as scattered points or cloud points in modern information technology. Gradually varied functions have shown advantages when dealing with real world problems. However, the method is still ne...

In this chapter, we generalize the concept of gradually varied functions to gradually varied mappings. In Chap. 3, we mainly discussed the function from a discrete space to {1, 2, ⋯ , n} or {A 1, A 2, ⋯ , A n }. This chapter focuses on gradually varied functions from a discrete space to another discrete space. For instance, a digital surface is def...

This chapter is the continuation of Chap. 7. It focuses on various applications and their data representations of the digital-discrete method. This chapter is divided into three parts: (1) An introduction to real problems and then a discussion of the data structure for the reconstruction problems. Then we focus on the implementation details. This p...

In this chapter, we introduce the concepts of digital functions and their basic properties. We start with digital “continuous” functions by Rosenfeld, then on to gradually varied functions proposed by Chen. We also present the necessary and sufficient condition of existence for gradually varied interpolations. For practical uses of these concepts,...

The most popular problem in data reconstruction is fitting a curve or a surface. Data fitting has two meanings: interpolation and approximation. Interpolation means fitting the data exactly on the sample values while approximation means a fitted function can be set near the sample values. In this chapter, to introduce data reconstruction, we do a c...

In this chapter, we present a systematic digital-discrete method for obtaining continuous functions with smoothness of a certain order (C n ) from randomly arranged data points. The new method is based on gradually varied functions and the classical finite difference method. This method is independent from existing popular methods such as the cubic...

This chapter overviews the basic concepts of functions and relations including continuous functions and their differentiations in Euclidean space. We also introduce functions in discrete spaces, specifically graphs and grid spaces. This chapter contains three parts: continuous functions in Euclidean space, graphs and discrete spaces, and advanced t...

In science, changing one curve α into another curve β continuously is called deformation. To describe this action, we usually use a sequence of curves in a sketch: the beginning curve C 0 is the original curve α and the final curve C 1 indicates the targeting curve β. Therefore, deformation can be defined as a function f α(t) = C t , where t ∈ [0,...

The goal of smooth function reconstruction on a 2D or 3D manifold is to obtain a smooth function on surfaces or higher dimensional manifolds. It is a common problem in computer graphics and computational mathematics, especially in civil engineering including structural analysis of solid objects. In this chapter, we introduce a new method using harm...

The number of holes in a connected component in 2D images is a basic
invariant. In this note, a simple formula was proven using our previous results
in digital topology (Chen 2004, Chen and Rong (2010). The new is: $h =1+
(|C_4|-|C_2|)/4$, where h is the number of holes, and $C_i$ indicate the set of
corner points having $i$ direct adjacent points...

Finite difference method and finite element method are popular methods for
solving groundwater flow equations. This paper presents a new method that uses
gradually varied functions to solve such equation. In this paper, we have
established a mathematical model based on gradually varied functions for
groundwater data volume reconstruction. These fun...

The smooth function reconstruction needs to use derivatives. In 2010, we used
the gradually varied derivatives to successfully constructed smooth surfaces
for real data. We also briefly explained why the gradually varied derivatives
are needed. In the this paper, we present more reasons to enlighten of forcing
derivatives to be continuous is necess...

In computer graphics, smooth data reconstruction on 2D or 3D manifolds
usually refers to subdivision problems. Such a method is only valid based on
dense sample points. The manifold usually needs to be triangulated into meshes
(or patches) and each node on the mesh will have an initial value. While the
mesh is refined the algorithm will provide a s...

A new algorithm can derive one or more minimal surfaces from an initial arbitrary surface with a fixed boundary. A discrete surface is a mesh represented as a set of vertices in 3D. The method extends earlier work, which used linear elastic springs to simulate a spider web animated in 3D.

A systematic digital-discrete method for obtaining continuous functions with smoothness to a certain order (C^(n)) from sample data is designed. This method is based on gradually varied functions and the classical finite difference method. This new method has been applied to real groundwater data and the results have validated the method. This meth...

This paper concerns with computation of topological invariants such as genus and the Betti numbers. We design a linear time algorithm that determines such invariants for digital spaces in 3D. These computations could have applications in medical imaging as they can be used to identify patterns in 3D image.Our method is based on cubical images with...

A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth surface in the real world on a very small scale. However, we do have a concept of smooth surfaces in practice si...

This paper presents some applications using recently developed algorithms for smooth-continuous data reconstruction based on the digital-discrete method. The classical discrete method for data reconstruction is based on domain decomposition according to guiding (or sample) points. Then the Spline method (for polynomial) or finite elements method (f...

The most common problem in data reconstruction is to fit a function based on the observations of some sample (guiding) points. This paper provides a methodological point of view of digital-discrete surface reconstruction. We explain our method along with why it is a truly universal and nonlinear method unlike most popular methods, which are linear...

This paper presents some applications of using recently developed algorithms for smooth-continuous data reconstruction based on the digital-discrete method. The classical discrete method for data reconstruction is based on domain decomposition according to guiding (or sample) points. Then uses Splines (for polynomial) or finite elements method (for...

Groundwater flow in Washington DC greatly influences the surface water quality in urban areas. The current methods of flow estimation, based on Darcy's Law and the groundwater flow equation, can be described by the diffusion equation (the transient flow) and the Laplace equation (the steady-state flow). The Laplace equation is a simplification of t...

This paper deals with computing topological invariants such as connected components, boundary surface genus, and homology groups. For each input data set, we have designed or implemented algorithms to calculate connected components, boundary surfaces and their genus, and homology groups. Due to the fact that genus calculation dominates the entire t...

It is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G≅G1×G2×⋯×Gt, where each Gi is a cyclic group of order pj for some prime p and integer j≥1. If ai generates the cyclic group of Gi, i=1,2,…,t, then the elements a1,a2,…,at are called a basis of G. We show a randomized algorithm such that give...

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we get some interesting properties related to gradual variation? In this note, we try to prove that many harmoni...

In this paper, we propose using curvatures in digital space for 3D object analysis and recognition. Since direct adjacency has only six types of digital surface points in local configurations, it is easy to determine and classify the discrete curvatures for every point on the boundary of a 3D object. Unlike the boundary simplicial decomposition (tr...

In this paper, we design linear time algorithms to recognize and determine topological invariants such as the genus and homology groups in 3D. These properties can be used to identify patterns in 3D image recognition. This has tremendous amount of applications in 3D medical image analysis. Our method is based on cubical images with direct adjacency...

The key step of generating the well-known Hsiao code is to construct a {0,1}-check-matrix in which each column contains the same odd-number of 1's and each row contains the same number of 1's or differs at most by one for the number of 1's. We also require that no two columns are identical in the matrix. The author solved this problem in 1986 by in...

Image segmentation is to separate an image into distinct homogeneous regions belonging to different objects. It is an essential step in image analysis and computer vision. This paper compares some segmentation technologies and attempts to find an automated way to better determine the parameters for image segmentation, especially the connectivity va...

Image segmentation is to separate an image into distinct homogeneous regions belonging to different objects. It is an essential step in image analysis and computer vision. This paper compares some segmentation technologies and attempts to find an automated way to better determine the parameters for image segmentation, especially the connectivity va...

Detecting spatial outliers can help identify significant anomalies in spatial data sequences. In the field of meteorological data processing, spatial outliers are frequently associated with natural disasters such as tornadoes and hurricanes. Previous studies on spatial outliers mainly focused on identifying single location points over a static data...

This paper presents some novel theoretical results as well as practical algorithms and computational procedures on fuzzy relation equations (FRE). These results refine and improve what has already been reported in a significant manner. In the previous paper, the authors have already proved that the problem of solving the system of fuzzy relation eq...

In this paper, we design fast algorithms for segmenting/classifying 2D images or 2D spatial data. The data is stored in quadtree and Rtree formats, and it may be extracted from spatial databases. The topological and graph-theoretic properties will be used to speed up the segmentation process. The key feature of this paper is to perform a segmentati...

A reexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reexiv e graphs and U V (H), then a vertex map f : U ! V (G) is called nonexpansive if for every two vertices x; y 2 U, the distance between f(x) and f(y) in G is at most that between x and y in H. A reexiv e graph G is said to have the extension...

This note presents a new measure for bone mineral density analysis based on DEXA images. We are proposing an innovative procedure to calculate a scalar value that indicates the connectivity of bone mineral components. This method may provide a totally new measure for bone density study using DEXA scan images that can only calculate T and Z values a...

This paper presents three new algorithms for λ-connected segmentation and fitting. It deals with a discrete system in which the elements are connected. The connectivity (known as the degree of connectedness) has the property of gradual variation. The first algorithm proposed is a direct segmentation method for quadtree represented images. The algor...

Rough sets was first studied by Pawlak to describe the approximation of a set X by using its lower bound L(X) and upper bound U(X). λ-connectedness was originally proposed as a technique to search layers in 2D or 3D digital seismic data. This note introduces
λ-connected components to represent lower and upper approximations for rough sets. Accordin...

In this article, we develop a new method and an algorithm to solve a system of fuzzy relation equations. We first introduce
a solution-base-matrix and then give a tractable mathematical logic representation of all minimal solutions. Next, we design
a new universal algorithm to get them. Two simplification rules are found to simplify the solution-b...

Rough sets was first studied by Pawlak to describe the approximation of a set X by using its lower bound L(X) and upper bound U(X). ¿-connectedness was originally proposed as a technique to search layers in 2D or 3D digital seismic data. This note introduces ¿-connected components to represent lower and upper approximations for rough sets. Accordin...

This paper proposes a technique that combines a fuzzy search
method called λ-connected search and rough sets, in data mining.
λ-connected searching was originally proposed to search seismic
layers in seismic data processing. Although λ-connected searching
is designed for digital spaces, or numerical data analysis, it can be
used for any domain, as...

Introduces a systematic approach, which we call a
λ-connectedness method, that can be applied to several real image
processing problems, such as segmentation/classification, searching and
data reconstruction. In this paper, we integrate previous research work
into a network/graph-based system to build a unified framework for these
processes. This t...

Digital surfaces deal with properties of surfaces in digital spaces. In the early 80's, researchers began to establish definitions for digital surfaces. Unlike surfaces in continuous spaces, digital surfaces have different characteristics. A general and intuitive definition for digital surfaces is still an open problem. This paper presents a proof...

This paper describes the foundations for a class of fuzzy neural
networks. Such a network is a composite or two-stage network consisting
of a fuzzy network stage and a neural network stage. It exhibits the
ability to classify complex feature set vectors with a configuration
that is simpler than that needed by a standard neural network, Unlike a
sta...

Morgenthaler and Rosenfeld gave the first formal definition of digital surfaces. Kong and Roscoe embedded digital spaces to continuous spaces to show the intuitive meaning of Morgenthaler-Rosenfeld's surfaces. Reed and Rosenfeld discussed the recognition algorithms for digital surface. Chen and Zhang considered an intuitive and simple definition of...

In digital geometry and topology, there are two popular kinds of digital spaces: point spaces and raster spaces. In point-spaces, a digital object is presented by a set of elements. In raster spaces as defined in this note, a digital object is a subset of a 'relation' on the space. In an Euclidean space, given a set S of points which are called sit...

Real time curve tracking is an important topic in radar image processing and can be applied to automated ionogram scaling and remote target tracking. A 2D gray-scale image only contains line segments and curves, and we want to extract them. For straight line tracking, Hough transform can be applied easily. However, to extract an arbitrary curve, Ho...

Intelligent data fitting is often used in seismic data processing to reconstruct large three-dimensional data volumes based on relatively small amounts of 2D seismic and/or well- log data. This technique is useful for such applications because of the many different layers and lithologies in stratum. However, if such data fitting is performed over t...

This paper presents a general-purpose definition for discrete curves, surfaces, and manifolds. We also focus on their tracking algorithms and implementations. This definition only refers to a simple graph, G equals (V,E), which is a generalized discrete space. The idea presented in this paper is to recursively define discrete curves, surfaces, soli...

The Distributed OS Formalization Generating System (DOSFGS) described in this paper consists of a grammar subsystem DOSGS and a semantics subsystem DOSSS. DOSGS is a form of Context Free Grammar. DOSSS is a semantics system with an operating set. DOSGS uses the semantics subsystem DOSSS to automatically generate a simulation version of a distribute...

In this note, we discuss various kinds of 2D unit cells, or surface-unit cells, made by regular polygons (simplexes) in the plane R X R. Mathematically, the plane can be divided by simplexes or regular polygons (decomposition). If we only allow one kind of surface-unit in the plane, there are only three possible choices: regular triangle (3-regular...

In this paper, we give a theoretical analysis for a generalized fuzzy neural network created in our previous papers. This analysis includes a mathematical proof of the training formulas used by such a network. the fuzzy neural network can accept a set of possibility functions as input as well as a vector of scalar values. This network consists of t...

Thesis (M.S.) -- Utah State University. Dept. of Computer Science, 1995. Includes bibliographical references.

This paper describes the framework of a novel approach to fuzzy
neural networks, In this approach, a fuzzy neural network accepts a set
of possibility functions as well as vectors as input and produces a
vector of membership function values as output. A fuzzy neural network
implemented in this approach consists of three components: a parameter
comp...

This paper describes the application of a fuzzy search technique to geophysical data processing. Here, we discuss three kinds of image object search problems: velocity layer search in computerized tomography in cross wells, surface search and reconstruction in 3D seismic images, and critical frequency curve search in digital ionograms. These three...

A new digital surface called the gradually varied surface is introduced and studied in digital spaces, especially in digital manifolds. In this paper, we have proved a constructive theorem: Let i_(Sigma) m be an indirectly adjacent grid space. Given a subset J of D and a mapping fJ : J yields i_(Sigma) m, if the distance of any two points p and q i...

Rosenfeld proposed the concept of the 2D fuzzy subset and successfully applied it to the problem of image segmentation. However, the 2D fuzzy subset approach could be used only for gray scale image segmentation because it fails to handle higher-dimensional range images such as color images. To deal with higher-dimensional range images, we introduce...

This paper presents a mathematical digital surface definition. This definition is able to deal with boundary points as well as `inner' points. Namely, it is able to distinguish inner points from boundary points. The definition is intuitive and provides a basis for designing fast algorithms for surface decision, boundary search, and surface tracking...

In this paper, we present two theorems: classification theorem and corner point theorem for closed digital surfaces. The classification theorem deals with the categorization of simple surface points and states that there are exactly six different types of simple surface points. On the basis of the classification theorem and Euler formula on planar...

The gradually varied surface was introduced and studied in digital and discrete spaces by Chen. The basic idea of introducing gradually varied surfaces is to employ a purely discrete interpolation algorithm to fit a discrete surface when the desired surface is not required to be "smooth." In this paper, we generalize the concept of gradually varied...

## Projects

Project (1)