VISUALIZATION OF MULTI DIMENSIONAL FUNCTIONS
OTHER APPROACHES
Several tools have been developed to visualize the nature of multi dimensional functions. Some of the most interesting ones are as follows
WORLD WITHIN WORLDS
In this approach every dimension is embedded within another dimension. Hence springs the name "World within Worlds". For example if we consider a three dimensional function a possible way in which the visualization appears is to have the z coordinate space as the outer world, then within this world lies the y coordinate world and within it lies the x coordinate world. Thus for any particular value of z and y the x world can be viewed and/or manipulated to understand the nature of the function.
The main advantages of this method are that it avoids cluttering of the screen with data from several dimensions. However this method has a number of disadvantages such as absence of an overview and occlusion of several dimensions. I feel that this method is not very effective to visualize a large number of dimensions.
More information from http://www.cs.columbia.edu/graphics/projects/AutoVisual/AutoVisual.html
HYPER SLICE
This is a very interesting approach. Their main approach is to see only two dimensions at a time because the human eye cannot perceive more than four dimensions at any point of time. Hence the slice the space and show a plane at a time (i.e. all dimensions except n-2 dimensions are fixed) and a number of these slices are arranged on the screen in a grid like fashion to understand the nature of the function. Thus slices obtained when different dimensions are kept constant can be seen ultimately. Each of these individual slices can be manipulated by the user.
The advantages are that the solution scales up pretty well and the relationship between two variables can be deciphered very easily. The disadvantage of this method are that there is repetition of slices which occupies some unnecessary screen space. However currently this seems to be a pretty good solution.
More information from http://citeseer.nj.nec.com/13223.html
MULTIVARIATE DATA
This approach was conceived by Mihalisin. In this visualization the functional values for all possible combinations of input are shown. For e.g. consider a three dimensional function where each dimension can assume values from 1 to 10. The visualization would show the value of the function for all possible combination of values for x, y and z.
The main advantages are that a good overview is obtained, a lot of data can be seen at a time and this method is not lossy as there is no sampling of data involved. However the disadvantages are that the solution does not scale up to higher dimensions.
More information from http://seamonkey.ed.asu.edu/~alex/pub/multi-vis/multi-vis.html
OTHER POINTERS
1. General information on the topic of multidimensional visualizations
http://seamonkey.ed.asu.edu/~alex/pub/multi-vis/multi-vis.html
Updated on Sept 2nd 2001