Mathematica on 3D Display from X3D Technologies

Ingo Relke

X3D Technologies GmbH

Abstract

We believe that in the near future every display and TV set will be 3D or 2D-3D switchable. Already on the market there are a lot of different software products that use the rendition and calculation of 3D objects. One especially interesting program for the visualization of 3D display is the computer algebra system Mathematica. Therefore, it will be interesting and useful to adjust the existing software programs, such as Mathematica for the 3D display. Currently, these programs use the 3D information for the rendition, but they represent the rendered 3D objects on a typical 2D display. This process not only loses the impression from the rendered 3D object, but it also loses part of visualized information. 
 
The autostereoscopic display from our company contains the ordinary flat TFT LCD, PDA, or even plasma display, in front of which the optical filter is situated with the special structure developed by the X3D Technologies GmbH. In comparison to nature, where one can observe the continuum set of the perspectives, we present on the display the discrete set of eight perspectives. However if the angle between perspectives is small, then the discrete set may be considered as a quasi-continuum set, and the obtained impression is very near to the reality.

It should be noted that our technology is able to provide a high-quality stereo image for the 3D display of any size from 6 inches to 50 inches and even more. 

The use of the multichannel system, where the number of the presented display channels is much more than two, leads to the appearance of greater quality in the demonstrated stereo image. In particular, it means that 3D images may be looked at by many observers simultaneously and without any view aids such as helmets, glasses, and so on. At the Wolfram Technology Conference 2004 we are going to present some models of our glass-free multi-user 3D displays demonstrated with Mathematica 3D objects.

In general form, every stereo image is presented as a matrix defined by

stereo[i_, 
        j_] := (∑_(k = 1)^8) DiscreteDelta[
            k - S[i, j]]*TwoD[k][[i, j]]

Here the matrixes stereo[i,j] and TwoD[k][[i,j]] correspond to the stereo and the two-dimensions image respectively. The matrix S[i,j] contains all the information about the distribution of the 2D images in one stereo image. In Mathematica notation, this matrix may be presented as

S[i_, 
          j_] := (∑_(m = 1))(DiscreteDelta[j, 
                  i - m] (PN + i - (PN/9) m - 
                    PN*Quotient[PN + i - (PN/9) m, PN, 1]) + 
              DiscreteDelta[i, 
                  1 + j - m] (i + (PN/9) ( m - 1) - 
                    PN*Quotient[i + (PN/9) ( m - 1), PN, 1]))

where the parameter PN describes the total number of used perspectives. 

It should be noted that we use Mathematica both for image creation and checking the parameters of the 3D display. In my opinion the success of our company in the creation of the 3D display and development of the corresponding methods for the image processing is due to the use of the Mathematica on all stages of calculations and renditions.

Using the power and universality of Mathematica, we present in our paper and demonstrations the different approaches for evaluating the 3D images for our 3D display.

It should be noted that for the visualization of these stereo images one needs to use 3D displays from X3D Technologies. In the visualization process on our 3D display, we present for the viewer only two from the set of the definite perspectives. Thus, each eye of the observer may watch one definite perspective and only in the human brain will the volumetric perception of the observed scene appear.