3D Rendering with Iso-surfaces

I am back on iso-surface rendering. After reading a lot of stuff on the subject I decided to experiment using points to render the surfaces. This is very easy to implement: you go through the whole volume, detect where an iso-surface lies (take a ‘cube’ composed of 8 voxels and see if all of them have values higher or lower than the iso-surface value; if they do, there is no iso-surface there, otherwise mark the center of the cube as an iso-surface point), and draw an OpenGL point for every detected position. Points can be drawn using glDrawArrays for very fast results. You also need to enlarge the size of the points, so that they do not leave holes between them, otherwise the ‘surface’ looks like a point cloud. Do that with glPointSize. Round points can be drawn with glEnable(GL_POINT_SMOOTH).
The results were not very nice. The image looked very ‘blocky’, as you can see here (click on image to enlarge).

To improve this I thought I had to resort to constructing a true surface (of triangles). The best-known algorithm for doing that is the ‘marching cubes’, but it is patented (!). I found some alternatives, but I was concerned of the speed factor, besides the fact that they all seemed rather complicated to implement. Then I read ‘Interactive Ray Tracing for Isosurface Rendering’ and ‘Iso-splatting: A Point-based Alternative to Isosurface Visualization’ (search Google and you will find the pdf files). From these papers I realized that the problem was not the points per se, or the supposedly reduced resolution of the data, but the position of the points. I was placing each point in the center of the ‘cube’ of eight voxels, but the iso-surface could be passing through the cube at any position. In this way, the points were all aligned to each other, producing the ‘blocky’ effect. The solution was to move the points towards the iso-surface. The papers, mentioned above, describe some sophisticated ways to do that (using Newton-Raphson or solving a cubic equation), but, not being mathematically so proficient, I thought of a simpler method: I start from the center of the cube and iteratively move towards the iso-surface as follows: I use the gradient vector inside the cube to specify the direction I will move along. The gradient points from lower to higher values. Therefore, I start from the center of the cube and calculate the value there. If it is smaller than the iso-surface value (i.e. I am inside the surface), I move along the gradient by an amount equal to 1/4 the side of the cube, otherwise I move in the opposite direction. Then I calculate the value at the new position and repeat the process, but each time I halve the distance that I move. 3 or 4 such steps are enough to give the result shown in the second image. The solution is approximate but the result is great and the extra code incurs a very small time penalty.

Notice that, no matter how many times you iterate, you will never end up outside the ‘cube’. Conversely, if the iso-surface is very close to one of the corners, you will never reach it, but that does not seem to affect the result much.