VISUALIZATION OF 3D GRAY-TONE IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 215

The projection of a perpendicular plane to an original slice and a projection
to a slant plane are reduced to the projection of a 2D image to a line. This is
basically the same as the calculation of a section. The same type of method
(and in particular the arithmetic method) is utilized in the reconstruction of
a 2D section image from a CT projection.
In any of the mentioned methods, a high quality image is obtained only
when an angle between a projection plane and a slice is relatively small. The
calculation of projections and sections is one of the fundamental problems
common to the display (or visualization) of 3D images and the reconstruction
from projections. Special purpose hardware has been developed for obtaining
sections and projections in a vertical direction. A method of rendering pre-
sented here is actually a good model showing what a 3D object and a 3D
scene look like when we see them from a specific position in the real world.
In a sense visualization is similar to painting a 3D scene or taking pictures.
Considering this we often use words like a “viewpoint” and a “view direction”
that are originally used for human image generating.
7.4 The concept of visualization based on ray casting -
(1) projection of a point
The idea and methods presented here are basically the same as those in 3D
computer graphics. We will explain them briefly considering the convenience
of applying them to the visualization of a 3D gray-tone image.
For starters, let us imagine the 3D world containing a 3D image (object)
to be rendered and an image plane (screen) for rendering. No other things
exist in this world. We call this imaginary ( virtual ) 3D space and denote it by
S I . A light source and a ray in S v are assumed only to be convenient tools
for this visualization. They usually do not have any specific relation to those
in the real world. This is one of the major differences from the generation of
high-quality realistic images in photorealistic rendering in computer graphics.
In the subsequent part of this section we explain the case of a continuous
image only for the sake of simplicity. It should be noted that digitization is
performed in many steps like in the sampling on a ray as described before.
7.4.1 Displaying a point
Assume the coordinate system ( x, y, z ) in a 3D imaginary space S I that is
called the world coordinate system and the 2D coordinate system ( X, Y )ona
display plane H p which we call the image plane coordinate system .
First of all we need to determine where an arbitrary point P = ( x, y, z )
in space S I should be mapped onto a display plane H p ,thatis,whichpoint
on a plane H p is the image of mapping. We call this mapping of a 3D point
to a 2D plane ray casting . Positions of the plane H p and a view point P e are