Volume rendering

Fourier volume rendering

Volume rendering is traditionally done in Euclidean space. Here we present a method that works from data represented in the frequency domain. It employs the Fourier projection slice theorem, used in various recontruction applications, most popular being Computed Tomography. Here we basically invert the process of CT.

A conventional uniformly sampled volume aquired with MRI, CT, PET, etc is transformed into the frequency domain as a preprocess. This representation can now be sampled along a two dimensional slice of the 3D fourier data. When inverse fourier transformed, this slice yields the spatial projection of the original dataset, in a direction perpendicular to the resampling slice in the frequency domain. The net result is a factor of 100 - 1000 in computational complexity over a spatial approach. This is due to the fact that traditional volume rendering requires sampling in 3 dimensions, whereas here only sample a two dimensional slab in frequency space.

3D photography

A central problem in computer vision is the estimation of 3D structure from multiple images of a scene. Although methods have been developed for specific classes of scenes, a general, robust solution to this problem has proved elusive. We are pursuing a new approach pioneered by Steve Sietz called voxel carving that uses an underlying volumetric representation. If the location and orientation of all cameras are know, one can project each voxel into all source images and determine the statistical color consistency between those footprints as evidence of the voxel containing a surface.

Our algorithms concentrate on 3 main contributions. Firstly, we can ensure information from all cameras available can be used. Secondly, our treatment of visibility can use layered depth images which provide computational advantages.

Lastly, our reconstructions can take place over an semi-infinite domain as opposed to a small, predefined region of space. The image shown is a rendering of a recovered model from a novel viewpoint.

3D ultrasound

Two dimensional image slices acquired with ultrasound can be combined to reconstruct a three dimensional imaged volume. We use an Ascention magnetic position tracker to determine the spatial location and orientation of each slice acquired in real time. This data is resampled into a uniformly sampled array using filters that are fast and smooth. Particularly effective are the segmentation techniques provided by the ultrasound machine itself. Acoustic Quantification and similar methods can differentiate blood from tissue at frame rates. Voxels labeled with AQ are tesselated into a polygonal representation and rendered interactively using 3D graphics hardware. It takes a few seconds to patient, a few seconds to resample and then renderings are available in real time from arbitrary perspectives.

At right is a rendering of several veins in a liver, imaged with a HP Sonos 1500.

Volume sampling

A step in almost every volume rendering algorithm is the interpolation of volumetric sample values. A common example is during the ray casting of preclassified volume data. Here sample points are defined along the rays and color and opacity values are sampled in the volume, typically using trilinear interpolation.

The common method is to sample color and opacity values independently at each ray point. This method is flawed and leads to sampling artifacts. The correct approach is to interpolate the color values preweighted by their opacities (in a rendering of human vertebrae, above left). If this is not done colors assigned to empty regions of space (zero opacity) can wind up bleeding onto surfaces. The red color (above right) was assigned only for zero opacity voxels and should not appear in any final rendering. Note that it does appear when separately interpolating color and opacity.

Artery tracking

Atherosclerotic vascular disease is the most common cause of death and disability in the western world. Magnetic Resonance is on the verge of having the resolution required to image the coronary arteries and detect lesions and plaque buildup. Here we have developed a method that assists in the visualization of arterial datasets in general, regardless of imaging modality. Given a seed point on the artery, we can track the path of the artery in 3D and use this path to extract a ruled spline surface through the arterial medial axis. Intensity values from the raw imaging data can be mapped onto the resampling surface providing a visualization of the cross-section of the artery.

This cross section can be rotated interactively and is stereoscopically rendered and displayed from arbitrary viewpoints in conjunction with the original slice data. The challenging aspect of this method is the robust tracking of artery.

This is accomplished with a parametric finite impulse response filter. This center surround filter has a number of sample points arranged in two concentric rings. Under parametric control the filter can be arbitrarily oriented and sized and responds maximally when the interior sample points lie inside the artery and exterior sample points lie outside the artery. Since the cost of evaluating the filter is independent of size or orientation, the parameter space can be automatically searched in real time to find the path of the artery.