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อ. สนั่��นั่ศรี�สข
Computer Graphics and Multimedia (EECP0462)อ.สนั่��นั่ ศรี�สข
Bachelor of Engineering (MUT)
Master of Engineering (MUT)
http://www.mut.ac.th/~sanun
อ. สนั่��นั่ศรี�สข
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.”
อ. สนั่��นั่ศรี�สข
Score
Midterm 30 %
Final 50 %
Homework 5 %
Project 15 %
Total 100 %
อ. สนั่��นั่ศรี�สข
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
อ. สนั่��นั่ศรี�สข
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
อ. สนั่��นั่ศรี�สข
Two-Dimensional Cartesian Reference
Frames
xx i
y i
yxx i
y i
y
Coordinate origin at the lower left screen corner
Coordinate origin at the upper left screen corner
อ. สนั่��นั่ศรี�สข
Polar coordinate reference
r
θ
xx i
y i
y
r
θ
P
Relationship between polar and Cartesian coordinates
อ. สนั่��นั่ศรี�สข
Polar coordinates in the xy plane
θx
yr
P
cosrx
Polar to Cartesian coordinates
sinry
Cartesian to Polar coordinates 22 yxr
x
y1tan
อ. สนั่��นั่ศรี�สข
Points and Vectors• Vector V in the xy plane of a Cartesian
reference
Vector V is:
xx 2
y 2
y
|v |P2
x 1
y 1 P1
12 PPV ),( 1212 yyxx
),( yx VVVector magnitude using the Pythagorean theorem is:
22yx VVV
x
y
V
V1tan
อ. สนั่��นั่ศรี�สข
Elements of Point set TopologyThe 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}
อ. สนั่��นั่ศรี�สข
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.”
XxXx
อ. สนั่��นั่ศรี�สข
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 .
Xx
Y x
{}.
Xx YxYxXx
.| YxXxx
YX
.| YxXxx
YX YX X2
อ. สนั่��นั่ศรี�สข
The following statements are evident:.set every for XXX
. then , and if ZXZYYX
. and both ifonly and if XYYXYX
.set every for XX
.2 and 2 XX X
.2}{ and ,2 XX xXxYXY
อ. สนั่��นั่ศรี�สข
The algebra of SetsWhen 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,
0)1)(32)(1( 2 xxx
YX
}.or |{ YzXzzYX
อ. สนั่��นั่ศรี�สข
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
,YX
}. and |{ YzXzzYX
}.3,2,1,0,1,2{ YX
}.2,1,0{YX
.YX disjoint. are and ZZ
,UX X
}.,|{ XxUxxX UYX ,
}.|{\ that Note }.|{\ XxUxXUXYxXxYX
อ. สนั่��นั่ศรี�สข
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.
.by simply )( ZYXZYX
because, as written becan ),()( ZYXWYZXW
. ison intersecti theand , written is ,,..., sets, ofunion Then
111 i
ii
n
in XXXXn
อ. สนั่��นั่ศรี�สข
The statements
and
are all equivalent.
:bygiven is and ,,between relation The
,)( YXI
,)( YXXII
,)( YXYIII
,)( XYIV
.)( YXV
อ. สนั่��นั่ศรี�สข
Identity Laws
Idempotent Laws
Complement Laws
Associative Laws
(Laws of Operations with Sets)
UUX XUX
XX X
XXX XXX
XX )( UU ,
UXX XX
)()( ZYXZYX )()( ZYXZYX
อ. สนั่��นั่ศรี�สข
Commutative Laws
Distributive Laws
Demorgan’s Laws
(Laws of Operations with Sets)
XYYX XYYX
)()()( ZXYXZYX )()()( ZXYXZYX
YXYX )( YXYX )(
อ. สนั่��นั่ศรี�สข
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,
}.,{},,{},,{ 332211 yxryxqyxp
,0),( and 0),( .)( qpqpdqpdI
),,(),( .)( qpdpqdII
).,(),(),( .)( rqdqpdrpdIII
อ. สนั่��นั่ศรี�สข
Several distance functions have been used in digital geometry. Some of the more common distance functions are:
Euclidean
City-block
Chessboard
2
12
122
122
122
12 )()()()(),( yyxxyyxxqpdEuclidean
1212),( yyxxqpdcity
),max(),( 1212 yyxxqpdchess
อ. สนั่��นั่ศรี�สข
Euclidean distance City-block distance
Chessboard distance
3
85258
52125
3210123
52125
85258
3
3
323
32123
3210123
32123
323
3
3333333
3222223
3211123
3210123
3211123
3222223
3333333
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