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อ. สนั่น ศรีสุข Computer Graphics and Multimedia (EECP0462) อ. สนั่น ศรีสุข Bachelor of Engineering (MUT) Master of Engineering (MUT) http://www.mut.ac.th/~sanun

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อ. สนั่น ศรีสุข Computer Graphics & Image Processing Reference Books: 1. Donald Hearn. and M. Pauline Baker., “Computer Graphics C Version 2nd Edition.” 2. Gerhard X. Ritter. and Joseph N. Wilson., “Computer Vision Algorithms in Image Algebra.” 3. G. X. Ritter., “Image Algebra.” available via anonymous ftp from ftp://ftp.cis.ufl.edu/pub/src/ia/documents 4. J. R. Parker., “Algorithms for Image Processing and Computer Vision.”

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อ. สนั่น ศรีสุข Score Midterm30 % Final50 % Homework5 % Project15 % Total 100 %

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อ. สนั่น ศรีสุข Course Descriptions Part I (Computer Graphics) •Week1: Mathematics for Computer Graphics and Image Processing •Week2: Line Drawing Algorithms •Week3: Circle and Ellipse Generating Algorithms •Week4: Basic Transformations •Week5: Clipping Operations •Week6: Filling Algorithms •Week7: Three-Dimensional Concepts and Transformations

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อ. สนั่น ศรีสุข Course Descriptions Part II (Image Processing) •Week8: Gray-Level Segmentaion •Week9: Thinning and Skeletonizing •Week10: Edge-Detection Techniques •Week11: Image Matching (Hausdorff Distance) •Week12: Basic Neural Network I •Week13: Basic Neural Network II •Week14: Basic Neural Network III

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อ. สนั่น ศรีสุข Two-Dimensional Cartesian Reference Frames Coordinate origin at the lower left screen corner Coordinate origin at the upper left screen corner

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อ. สนั่น ศรีสุข Polar coordinate reference Relationship between polar and Cartesian coordinates

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อ. สนั่น ศรีสุข Polar coordinates in the xy plane Polar to Cartesian coordinates Cartesian to Polar coordinates

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อ. สนั่น ศรีสุข Points and Vectors •Vector V in the xy plane of a Cartesian reference Vector V is: Vector magnitude using the Pythagorean theorem is:

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อ. สนั่น ศรีสุข Elements of Point set Topology The concept of set is basic to all of mathematics and mathematical applications. We think of a set as something made up by all the object that satisfy some given condition, such as the set of integers, the set of pages in book. The objects making up the set are called the elements, or member, of the set and may themselves be sets. A set X is comprised of elements, for example, the equation. X = {1, 2, 3, 4} means that a set X made up of the four elements 1, 2, 3 and 4. A set may be not by any particular order, Thus X might be. X = {1, 4, 2, 0}

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อ. สนั่น ศรีสุข The elements of a set X may have duplicates. For example. X = {1, 2, 3, 3, 4, 5, 4, 1}={1, 2, 3, 4, 5}. Each elements must distinct each other. If a set is a large finite set or an infinite set, we can describe it by listing a property necessary for membership. For example, the equation. Y = {y | y is a positive, even integer} reads “Y equals the set of all y such that y is a positive, even integer,” that is, Y consists of the integers 2, 4, 6, and so on. If X is a finite set, we let |X| = number of elements in X If x is in the set X, we write reads “x is an element of X,” and if x is not in X, we write reads “x is not an element of X.”

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อ. สนั่น ศรีสุข For example, if X={x | x is a positive integer}, Y={-1, -3, -5}. if x=2, then, but. The set with no element is called the empty set and is denoted by. Thus The sets X and Y are equal and we write X=Y if X and Y have the same elements. To put it another way, X=Y if whenever, then and whenever, then. In image algebra we write reads “for all x such that x is an element of X if and only if x is an element of Y.”, read “X is a subset of Y,” signifies that each element of X is an element of Y, that is, We call X a proper subset of Y whenever and. The set whose elements are all the subsets of a given set X is called the power set of X and is denoted by.

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อ. สนั่น ศรีสุข The following statements are evident:

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อ. สนั่น ศรีสุข The algebra of Sets When defining operations on and between sets it is customary to view the sets under consideration as subsets of some larger set U, called a universal set or the universe of discourse. Example: Consider the equation. If R is the universal set, then X={-1, 3/2}. Let X and Y be given sets. The union of X and Y, written, is defined as the set whose elements are either in X or in Y (or in both X and Y). Thus,

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อ. สนั่น ศรีสุข The intersection of X and Y, written is defined as the set of all elements that belong to both X and Y. Thus, For example, X={0, 1, 2, 3}, Y={-2, -1, 0, 1, 2}. Two sets X and Y are called disjoint if they have no elements in common, that is, if obviously, If then the complement of X (with respect to U) is denoted by and is defined as The difference of two sets is denoted by X\Y. and defined as

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อ. สนั่น ศรีสุข Some of the more important laws governing operations with sets. Here X, Y, and Z are subsets of some given universal set U. Because of associativity, we can designate Similarly, a union (or intersection) of four sets, say by associativity, the distribution of parentheses is irrelevant, and by commutativity, the order of terms plays no role. By induction, the same remarks apply to the union (or intersection) of any finite number of sets.

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อ. สนั่น ศรีสุข The statements and are all equivalent.

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อ. สนั่น ศรีสุข Identity Laws Idempotent Laws Complement Laws Associative Laws (Laws of Operations with Sets)

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อ. สนั่น ศรีสุข Commutative Laws Distributive Laws Demorgan’s Laws (Laws of Operations with Sets)

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อ. สนั่น ศรีสุข Distance Function (or Distance Measures) In many applications, it is necessary to find the distance between two pixels or two components of an image. Unfortunately, there is no unique method of defining distance in digital images. One can define distance in many different ways. For all pixels p, q, and r, any distance metric must satisfy all of the following three properties: Let,

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อ. สนั่น ศรีสุข Several distance functions have been used in digital geometry. Some of the more common distance functions are: Euclidean City-block Chessboard

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อ. สนั่น ศรีสุข Euclidean distance City-block distance Chessboard distance

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