Volume rendering via ray casting

Volume rendering via ray casting of head, chest and feet.

Methods for visualization of three dimensional data are summarized as Volume Rendering (VR). It is a process for visualizing sampled functions of three spatial dimensions by computing 2-D projections of a colored, semitransparent volume. VR is a major application area in medical imaging where data is mostly obtained by Computed Tomography (CT) or Magnetic Resonance Imaging (MRI). There are a number of different methods to perform volume rendering, a common method is Ray Casting (RC) which is a direct volume rendering method; a 3D view results from performing the algorithm on a data set. In this procedure, which transforms data into a 3D projection, rays are traced from the view point into the viewing volume. 

The figure illustrates the principle approach of RC. It shows a volume which has a density D(x,y,z) and is  penetrated by a ray R.

From the light source S an illumination I (x,y,z) can be estimated at each point (x,y,z) along the ray. The intensity gritted along the ray to the eye depends on this value, a reflection function P and the local density D (x,y,z). The dependence on density expresses the fact that a few bright particles will scatter less light in the eye direction than a number of dimmer particles. The density function along the ray can be written as:

 D(x(t),y(t),z(t))=D(t)

The illumination equation from the source as:

I(x(t),y(t),z(t))=I(t)

Furthermore the illumination scattered along R from a point distance t along the ray can be written as: 

I(t)D(t)P(cos∅)

∅ is the angle between R and L, thus the light vector, from the point of interest. To determine I(t) the attenuation of radiation from the light sources and the shadow when passing through the volume to the point of interest must be calculated. This is the same as the computation of how the light scattered at point (x,y,z) is affected in its way along R to the eye. In most algorithms, this calculation is ignored and I(x,y,z) is set to be uniform throughout the volume. In medical imaging, it would be impossible to see into areas surrounded by bone if the bone were considered dense enough to shadow light. On the other hand, in applications where internal shadows are desired, this integral has to be computed. The attenuation due to the density function along a ray can be calculated as: 

exp⁡(-τ∫D(s)ds) |t1,t2 

τ is a constant that transforms density to attenuation. The intensity of the light reaching the eye along direction R due to all the elements along the ray is defined as:

B=∫(exp⁡(-∫D(s)ds))I(t)D(t)P(cos∅)|t1,t2;|t1,t2

The Ray Casting algorithm is usually implemented with a Transfer Function (TF). This is either a linear or parabolic equation which defines the coherence between opacity and color for each density value. With the TF colored an transparent effects can be created on the data set. Many medical image viewer provide an option to define a TF.