.help volumes

[OUT OF DATE (Jan 89); this is an original pre-design document]

.ce
Conceptual Model for 2D Projections of Rotating 3D Images

Consider the problem of visualizing a volume containing emissive and
absorptive material.  If we had genuine 3D display tools, we could imagine
something like a translucent 3D image display that we could hold in our
hand, peer into, and rotate around at will to study the spatial distribution
of materials inside the volume.

Lacking a 3D display device, we can resort to 2D projections of the interior
of the volume.  In order to render absorptive material, we need a light
source behind the volume; this light gets attenuated by absorption as it
passes through the volume toward the projection plane.  In the general case
light emitted within the volume contributes positively to the projected
light intensity, but it also gets attenuated by absorbing material between
it and the projection plane.  At this point the projection plane has a range
of intensities representing the combined absorption and emission of material
along columns through the volume.  But looking at a single 2D projection,
there would be no way to determine how deep in the original volume a particular
emitting or absorbing region lay.  One way around this is to cause the volume
to rotate, making a series of 2D projections.  Playing the projections back
as a movie gives the appearance of seeing inside a translucent rotating
volume.

Modelling the full physics of light transmission, absorption, refraction,
etc. with arbitrary projective geometries would be quite computationally
intensive and could rival many supercomputer simulations.  However, it is
possible to constrain the model such that an effective display can be
generated allowing the viewer to grasp the essential nature of the spatial
relationships among the volume data in reasonable computational time.  This
is called volume visualization, which can include a range of display
techniques approximating the actual physics to varying extents.  There is
some debate whether visualization problems can best be attacked by simplified
direct physical models or by different models, such as ones that might better
enhance the \fBperception\fR of depth.  We will stick with direct physical
models here, though simplified for computer performance reasons.

For computational purposes we will constrain the projection to be orthogonal,
i.e. the light source is at infinity, so the projection rays are all parallel.
With the light source at infinity behind the volume (a datacube), we need not
model reflection at all.  We will also ignore refraction (and certainly
diffraction effects).

We can now determine a pixel intensity on the output plane by starting
at the rear of the column of voxels (volume elements) that project from
the datacube onto that pixel.  At each successive voxel along that column
we will attenuate the light we started with by absorption, and add to it
any light added by emission.  If we consider emission (voxel intensity)
alone, the projection would just be the sum of the contributing intensities.
Absorption alone would simply decrease the remaining transmitted light
proportionally to the opacity of each of the voxels along the column.
Since we are combining the effects of absorption and emission, the ratio
of the intensity of the original incident light to that of the interior
voxels is important, so we will need a beginning intensity.

The opacities have a physical meaning in the model.  However, we are more
interested here in visualizing the volume interior than in treating it as
a pure physical model, so we add an opacity transformation function.  This
allows us to generate different views of the volume interior without having
to modify all the raw opacity values in the datacube.  For maximum flexibility
we would like to be able to modify the opacity function interactively, e.g.
with a mouse, but this would amount to computing the projections in real
time and is not likely at present.

.nf

Let:	i     = projected intensity before considering the current
	        voxel
	i'    = intensity of light after passing through the current
	        voxel
	I0    = initial light intensity (background iillumination
		before encountering the volume)
	Vo    = opacity at the current voxel, range 0:1 with
	        0=transparent, 1=opaque
	Vi    = intensity at the current voxel
	f(Vo) = function of the opacity at the current voxel,
		normalized to the range 0:1
	g(Vi) = function of the voxel's intensity, normalized
		to the range 0:1

Then:	i' = i * (1 - f(Vo)) + g(Vi)
		[initial i = Imax, then iterate over all voxels in path]

.fi

We want to choose the opacity and intensity transformation functions in such
a way that we can easily control the appearance of the final projection.
In particular, we want to be able to adjust both the opacity and intensity
functions to best reveal the interior details of the volume during a
rotation sequence.  For example, we might want to eliminate absorption
"noise" so that we can see through it to details of more interest, so we
need a lower opacity cutoff.  Likewise, we would want an upper opacity
cutoff above which all voxels would appear opaque.  We will need the same
control over intensity.

.nf
Let:	o1   = lower voxel opacity cutoff
	o2   = upper voxel opacity cutoff
	t1   = lower transmitted intensity cutoff
	t2   = upper transmitted intensity cutoff
	i1   = lower voxel intensity cutoff
	i2   = upper voxel intensity cutoff
	Imax = output intensity maximum for int transform function
.fi

Now all we need is the form of the opacity and intensity functions between
their input cutoffs.  A linear model would seem to be useful, perhaps with
logarithmic and exponential options later to enhance the lower or upper
end of the range.  f(Vo) is constrained to run between 0 and 1, because
after being subtracted from 1.0 it is the intensity attenuation factor
for the current voxel.

.nf

		Opacity Transformation Function f(Vo):

	        { Vo < o1       : 	    0.0		    }
	        {					    }
	        { o1 <= Vo < o2 : (t2 - (Vo - o1)(t2 - t1)) }
	        {		  (	---------	  ) }
	f(Vo) = {		  (	(o2 - o1)	  ) }
	        {		  ------------------------- }
	        {			    I0	    }
	        {					    }
	        { o2 <= Vo      : 	    1.0		    }


backg. int.  I0-|
		|
	     t2-|------			Transmitted Intensity
		|        `		as function of opacity
	        |           `		(ignoring independent
i * (1 - f(Vo))-|..............`	voxel intensity contri-
                |               . `	bution)
    		|               .    `
		|               .       `
	     t1-|		.	  |
		|		.	  |
		+____________________________________________
		       |        |         |
		       o1	Vo 	  o2
			   Voxel Opacity

       ------------------------------------------------------------

		Intensity Transformation Function g(Vi):

		{ Vi < i1       :    0.0
		{
		{ i1 <= Vi < i2 : (Vi - i1) * Imax
		{		  ---------
	g(Vi) = {		  (i2 - i1)
		{
		{ i2 <= Vi	:    Imax
		{
		{
		{


	        |
		|
	   Imax-|                       ---------------------
		|                    /
	  g(Vi)-|................./
      		|              /  .
		|           /     .
		|        /        .
	    0.0	+___________________________________________
		     |            |     |
		     i1	          Vi    i2
			Voxel Intensity
.fi