The Geometry of Perspective Projection
Dea Rahma Sari
Gunadarma University, Indonesia
Abstract
Images
are 2D projections of real-world scenes. Computer graphics is a process to
create an image based on the description of the object and the background is
contained in the image. To create an image with the best quality it takes
a good image processing includes the process of withdrawal of information or a
description of the object or the introduction of objects contained in the
image. In the real world it is well known the Cartesian coordinate system
of two dimension (2D) and three-dimension (3D). Point O is as the center
coordinate system. This is different from a computer screen that displayed
only acknowledges the two-dimensional coordinate system by
coordinate. Therefore this article aims at the study to generate the image
effect of perspective projection for computer graphics
Keyword : Perspective Projection, camera models, projection
1.
Preliminary
Computer Graphics is one area of
computer science that is interesting in terms of the rapid development and
adoption and utilization. Computer graphic visually present information
that can give one or more visual information. Graphic also be interpreted
as a combination of images, symbols, symbols, letters, numbers, word, pictures
and sketch are used as a category to provide concepts and ideas of users in the
delivery of information.
Human beings should have the eye as a
means to see the world. Camera is computerâ € ™ s eye, camera is
structurally the same as the eye. The camera is equipped with a lens and
aperture. While the human eye is equipped with cornea and pupil.
2.
Research Methods
2.1
Data Collecting Methods
2.1.1
Literature Methods
This
method is used by reading books of literature or reference that related to
computer graphic and learning about reports and other books that related for
research.
2.1.2
Data Collecting from Internet or Browsing
This
method is used by searching data and information such as text, picture and
program source code that related to research by using computer network.
2.1.3
Observation
A
directly observation method on study about computer graphic. Besides in doing
observation, it is doing document records that related to research subject.
2.2
User Requirements Analysis
The college students needs is a system that can
support students to understand transformation matrix material. This thing can
be done if the system that is built contains the materials that they need. A
complete, interesting and user friendly system are very important factors that
is needed for students. So that teaching and learning process can be
appropriate with SAP.
3. Camera Models
3.1
Thin
Lens Model
Most modern cameras use a lens to focus light into
the view plane (sensory surface). It makes a person can capture enough light in a
sufficiently short period of time that the objects do not move appreciably, and
the image is bright enough to show significant detail over a wide range of
intensities and contrasts.
Lens models can be quite complex, especially for
compound lens found in most cameras. We consider perhaps the simplist case,
known widely as the thin lens model. In the thin lens model, rays of light
emitted from a point travel along paths through the lens, convering at a point
behind the lens. The key quantity governing this behaviour is called the focal
length of the lens. The focal length, | f |, can be defined as distance behind
the lens to which rays from an infinitely distant source converge in focus.
3.2
Pinhole
Camera Model
A pinhole camera is an idealization of the thin lens
as aperture shrinks to zero. Pinhole camera is the simplest imaging device
which, captures accurately the geometry of perspective projection. Rays of
light enters the camera through an infinitesimally small aperture.
Light from a point
travels along a single straight path through a pinhole onto the view plane. Pinhole
Camera at the intersection with the light rays forming the image of the object
image plane.Such a mapping from three dimensions onto two dimensions is called perspective
projection.
We use a right-handed
coordinate system for the camera, with the x-axis as the horizontal direction
and the y-axis as the vertical direction. This
means that the optical axis (gaze direction) is the negative z-axis.
The image you’d get corresponds to drawing a ray
from the eye position and intersecting it with the window. This is equivalent
to the pinhole camera model, except that the view plane is in front of the eye
instead of behind it, and the image appears rightside-up, rather than upside
down. (The eye point here replaces the pinhole). To see this, consider tracing
rays from scene points through a view plane behind the eye point and one in
front of it:
3.3 Camera Projections
The
view of the three dimension object oriented synthetic camera use because it is
quite flexible. In this case the
image can be taken from any angle at varying distances and with varying camera
direction. Perspective using synthetic camera system has three main
components, namely the field of view in which the window is placed, the
coordinate system is referred to as the coordinate system point of view and the
eye in the system. Eye view some objects through the window in the field of view. Results
of view that will be drawn on the computer screen with a two-dimensional
system.
3.4
Orthographic
Projection
There
are two main method in the projection of the object, ie parallel projection and
perspective projection. Orthographic Projection also known as parallel projection. In
parallel projections, coordinate position along the length of the field is
transformed into parallel lines. It
is the projection of a 3D object onto a plane by a set of parallel rays
orthogonal to the image plane.
3.5
Perspective
Projection
For
Perspective Projection, object position is transformed to the field of view
along lines converging to a point called the reference point projection
(projection reference point) or the center of projection. Form of the perpective projection using
the idea of Similar triangles. We Consider a complementary algebraic
formulation.
Below are images of objects projected results are determined by
calculating the intersection of the lines projected by the field of view
Perspective
projection capable of displaying an image which is more realistic than the
Parallel projection, but can not maintain the relative size of the original. For two objects of
the same size when the distance of the field of view is different, so the size
of the projected objects differently.
Image
Perspective
projection of two objects that have the same size at different distances from
the field of view
To obtain
perspective projection, we project the results of perspective transformation on
to a any of the orthographic projection planes, say, z=0 plane.
The effect of the perspective
transformation is to bring a point at infinity to a finite value in the 3D
space
3.6 Camera Position and Orientation
Like someone who will take a picture of an object,
the first thing to do is find the right position to take a picture of an object
(the camera position) then determined the direction of the camera towards the
object.
The image shows the general processing steps
modeling and conversion of an object in a Cartesian coordinate system to the
old system of coordinates of the display screen. The first step is the
position of an object modeled in the Cartesian coordinate system are converted
into the coordinate system point of view. Furthermore, the projected
operation performed to convert the coordinates of the observation in
perspective to the position coordinates in the projection field, which will
then be drawn into the computer screen.
3.7
Model
Framework
Three-dimensional graphics that will be
discussed in this paper is still in the form of a model framework, which is an
image object only in the form of dots and dashes. In a three-dimensional
graphic modeling framework, to consider two aspects: geometrical and
topological aspects. Aspects of geometry in the form of information about
the location of each point form a three-dimensional object. Information about
the location of each point is written in the form of a list of point (vertex
list). Topology aspect or aspects of the connectivity, is used to show the
list of line (edge list) of three-dimensional objects.
3.8 Determination of Direction Camera
In determining the position and orientation of the camera
takes three components, that is:
-
VRP,
that is r = (rx,ry,rz)
-
VPN,
that is n = (nx,ny,nz)
-
Vector
v
The first selected
interesting views or appropriate to establish a coordinate system point of
view, which is referred to as a reference point of view (VRP). This point is the
center point of the coordinate system point of view. A reference point of
view have been in a position close to the midpoint of an object to be viewed. Furthermore,
determined the direction of the field of view or the so-called normal vector
field of view (n). Determine the
position in Cartesian coordinates, and from this point built direction
vector n relative to a reference point of view. For example n1, then the computer asked
to calculate the vector unit using the equation:
|n1|
|
By |n1| states of the vector n1 .
For example, by specifying n 1 = -r, then n1 is a
vector whose direction towards the center point of the Cartesian coordinate
system. After that specify the upward direction of the view plane or
vectorv. The way to determine v is to use the help of
any vector, ie u p, and project it to the view plane in the direction of the vector n, in order to obtain
the vector u p '. Vector up 'can be determined using the equation:
up’
= up – (up . n) n
with u p, n expressed product of vectors of two points. Unit vector v can be
calculated by using the equation:
After the three-dimensional coordinates of the object framework successfully converted into a coordinate system wvn then the next step is to perform a perspective projection of the object into the XY plane or field Z = 0.
Image
Perspective projection of a point
The image above illustrates the geometry of a perspective projection of point P on the three-dimensional space into the realm Z = z * = 0 at the center of the projection at the point Z, the Z axis projection point P is a point P *. By looking at the picture above, it can be used a formula to calculate the similarity of triangles point x * and y *, there is:
4.
Conclusion
Computer step in developing a
three-dimensional image in some ways similar to the process of shooting in
photography, the camera takes a position at a certain point in a room further
determine the direction of the camera to the object to be photographed. In
the process the three-dimensional image on the display screen required
two-dimensional projection techniques (perspective projection).
5.
References
[1] Leow Wee Kheng, Camera Models and Imaging
[2]
Muhammad Abrori, Teknik Proyeksi dan Cara Pandang Kamera Sintetik sebagai
Metode Pembangkitan Citra 3D, April 2005
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