A method of representing points in a
space of given dimensions by coordinates from an origin.
Coordinate Transformations
A coordinate transformation is a
conversion from one system to another, to describe the same space. With every
bijection from the space to itself two coordinate transformations can be
associated:
1).such that the new coordinates of
the image of each point are the same as the old coordinates of the original
point (the formulas for the mapping are the inverse of those for the coordinate
transformation).
2).such that the old coordinates of
the image of each point are the same as the new coordinates of the original
point (the formulas for the mapping are the same as those for the coordinate
transformation).
For example, in 1D, if the mapping
is a translation of 3 to the right, the first moves the origin from 0 to 3, so
that the coordinate of each point becomes 3 less, while the second moves the
origin from 0 to -3, so that the coordinate of each point becomes 3 more.
2D Coordinate Systems
§ Cartesian coordinate system
§ Polar coordinate system
§ Parabolic coordinate system
§ Bipolar coordinates
§ Hyperbolic coordinates
§ Elliptic coordinates
3D Coordinate Systems
§ Cartesian coordinate system
§ Cylindrical coordinate system
§ Spherical coordinate system
§ Parabolic coordinate system
§ Parabolic cylindrical coordinates
§ Paraboloidal coordinates
§ Oblate spheroidal coordinates
§ Prolate spheroidal coordinates
§ Ellipsoidal coordinates
Resolution !
Refers to the sharpness
and clarity of an image. The term is most often used to describe monitors,
printers, and bit-mapped graphic images. In the case of dot-matrix and laser
printers, the resolution indicates the number of dots per inch. For example, a
300-dpi (dots per inch) printer is one that is capable of printing 300 distinct
dots in a line 1 inch long. This means it can print 90,000 dots per square
inch.
For graphics monitors,
the screen resolution signifies the number of dots (pixels) on the entire
screen. For example, a 640-by-480 pixel screen is capable of displaying 640
distinct dots on each of 480 lines, or about 300,000 pixels. This translates
into different dpi measurements depending on the size of the screen. For
example, a 15-inch VGA monitor (640x480) displays about 50 dots per inch.
Printers, monitors,
scanners, and other I/O devices are often classified as high resolution, medium
resolution, or low resolution. The actual resolution ranges for each of these
grades is constantly shifting as the technology improves.
The basic building
block of any graphic image is a pixel, a contraction of "Picture
Element", arranged in precise rows and columns. The number of rows and
columns of pixels is referred to as the "resolution" of the image and
is usually expressed by the number of horizontal pixels (rows) multiplied by
the number of vertical pixels (columns), for example: 800*600, 1024*768,
1152*864. Note that these resolutions are sized at a 4:3 ratio. The reason for
this is that monitors are manufactured with a 4:3 aspect ratio, therefore for a
pixel to appear square it must appear at the same 4:3 width to height aspect
ratio.
However, the resolution
alone defines nothing except the physical size of the image. Each of these
resolutions also has an associated "bit depth", which defines the
number of colors that can be displayed. The bit depth refers to how many bits
of data are associated with each pixel and are available to store a colour
value. A 24 bit image, commonly referred to as "true colour", allows
for the storage of 24 bits of data per pixel. To clarify the measurement, there
are of course 8 bits in every byte. This raises the question of how colors are
defined.
Homogeneous Coordinates Syatem !
One of the many
purposes of using homogeneous coordinates is to capture the concept of
infinity. In the Euclidean coordinate system, infinity is something that does
not exist. Mathematicians have discovered that many geometric concepts and
computations can be greatly simplified if the concept of infinity is used. This
will become very clear when we move to curves and surfaces design. Without the
use of homogeneous coordinates system, it would be difficult to design certain
classes of very useful curves and surfaces in computer graphics and
computer-aided design.
Let us consider two
real numbers, a and w, and compute the value of a/w. Let us hold the value of a
fixed and vary the value of w. As w getting smaller, the value of a/w is
getting larger. If w approaches zero, a/w approaches to infinity! Thus, to
capture the concept of infinity, we use two numbers a and w to represent a
value v, v=a/w. If w is not zero, the value is exactly a/w. Otherwise, we
identify the infinite value with (a,0). Therefore, the concept of infinity can
be represented with a number pair like (a, w) or as a quotient a/w.
Let us apply this to
the xy-coordinate plane. If we replace x and y with x/w and y/w, a function
f(x,y)=0 becomes f(x/w,y/w)=0. If function f(x,y) = 0 is a polynomial,
multiplying it with wn will clear all denominators, where n is the degree of
the polynomial.
For example, suppose we
have a line Ax + By + C = 0. Replacing x and y with x/w and y/w yields A(x/w) +
B(y/w) + C = 0. Multiplying by w changes it to
Ax + By + Cw = 0.
Let the given equation
be a second degree polynomial Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0. After
replacing x and y with x/w and y/w and multiplying the result with w2, we have
Ax2 + 2Bxy + Cy2 + 2Dxw
+ 2Eyw + Fw2 = 0
If you look at these
two polynomials carefully, you will see that the degrees of all terms are
equal. In the case of a line, terms x, y and w are of degree one, while in the
second degree polynomial, all terms (i.e., x2, xy, y2, xw, yw and w2) are of
degree two.
Given a polynomial of
degree n, after introducing w, all terms are of degree n. Consequently, these
polynomials are called homogeneous polynomials and the coordinates (x,y,w) the
homogeneous coordinates.
Given a degree n
polynomial in a homogeneous coordinate system, dividing the polynomial with wn
and replacing x/w, y/w with x and y, respectively, will convert the polynomial
back to a conventional one. For example, if the given degree 3 homogeneous
polynomial is the following:
x3 + 3xy2 - 5y2w + 10w3
= 0
the result is
x3 + 3xy2 - 5y2 + 10 =
0
This works for
three-dimension as well. One can replace a point (x, y, z) with (x/w, y/w, z/w)
and multiply the result by w raised to certain power. The resulting polynomial
is a homogeneous one. Converting a degree n homogeneous polynomial in x, y, z
and w back to the conventional form is exactly identical to the two-variable
case.
Note
Converting from a
homogeneous coordinates to a conventional one is unique; but, converting a
conventional coordinates to a homogeneous one is not.
A
Simple Geometric Interpretation
Given a homogeneous
coordinate (x,y,w) of a point in the xy-plane, let us consider (x,y,w) to be a
point in space whose coordinate values are x, y and w for the x-, y- and w-
axes, respectively. The line joining this point and the coordinate origin
intersects the plane w = 1 at a point (x/w, y/w, 1). Please verify this fact
yourself. The following figure illustrates this concept.
This transformation
treats a two-dimensional homogeneous point as a point in three-dimensional
space and projects (from the coordinate origin) this three-dimensional point to
the plane w=1. Therefore, as a homogeneous point moves on a curve defined by
homogeneous polynomial f(x,y,w)=0, its corresponding point moves in
three-dimensional space, which, in turn, is projected to the plane w=1. Of
course, (x/w,y/w) moves on a curve in plane w=1.
The above figure also
shows clearly that while the conversion from the conventional Euclidean
coordinates to homogeneous coordinates is unique, the opposite direction is not
because all points on the line joining the origin and (x,y,w) will be projected
to (x/w,y/w,1).
Scan Conversion ! (Details form book ROY PLASTOCK)
Scan conversion or scan
rate converting is a technique for changing the vertical / horizontal scan
frequency of video signal for different purposes and applications. The device
which performs this conversion is called scan converter. The application of
scan conversion is wide and covers video projectors, cinema equipment , TV and
video capture cards, standard and HDTV televisions, LCD monitors and many
different aspects of picture and video processing.
Scan conversion
process, not only needs to make changes in synchronization (sync) frequencies,
changes in picture information data rate are also mandatory in most cases.
There are two distinct methods for doing the process:
1).Analog Methods (Non
retentive, memory-less or real time method)
This conversion is done
using large numbers of delay cells and is appropriate for analog video.
2).Digital methods
(Retentive or buffered method)
In this methods ,
picture is stored in a line or frame buffer with n1 speed (data rate) and is
read with n2 speed, several picture processing techniques are applicable when
the picture is stored in buffer memory including kinds of interpolation from
simple to smart high order comparisons, motion detection and … to improve the
picture quality and prevent the conversion artifacts.
