Scientific Visualization mainly employs two ways to visualise the content in a 3D data set; they are: isosurface and volume rendering. We propose that the visualisation methods normally used in Scientific Visualisation can be very useful for the visualisation of information content in a 3D data set.

In this paper we present a method to visualise a somewhat abstract concept which is the spatial behaviour of the data points in a 3D data set. We classify the behaviour of groups of points in a 3D data set as constant, linear and non-linear. Then, the constant, linear and non-linear characteristic behaviour of the points in the 3D data set is visualised using direct volume rendering.

Wavelets have been widely used in signal processing and more recently in computer graphics and visualisation [1,2]. Here we propose that the three-dimensional visualisation of the wavelet coefficients of a 3D data set could provide useful information about certain spatial behaviour of the data points. More often, the collected points from an experiment or from a simulation have an unknown behaviour and it is important to get the information about this behaviour in a localised and quick way. Wavelets, as contrasted with the Fourier transform, can give localised information about a data set of points.

To the author's knowledge, the visualisation of the detail coefficients of the wavelet has been done only in one- and two-dimensional co-ordinates. We propose that the visualisation of the wavelet coefficients in three-dimensional co-ordinates can give different and useful insight about the behaviour of 3D data points. In fact, the space-scale information provided by the wavelet transform could be associated with each point in the 3D space of the data set, when the appropriate output given by the wavelet transform is used.

We are interested to visualise, in a localised form, the behaviour of a group of points in a 3D data set classified as constant, linear and non-linear. This classification allows the user to get a quick grasp of the complexity of the data that is being visualised and to get a good understanding of where, in space, there may be some interesting behaviour of the data.

In order to associate the magnitude of the wavelet coefficients with the constant, linear and non-linear behaviour of the data points a special property of the wavelet is used. In fact, wavelets have what is called Vanishing Moments [2] expressed by the following equation:

                n=0,.....,M-1

Where is the wavelet function and M are the vanishing moments of the wavelet bases. This equation means that the projection coefficients of the wavelet transform will be zero, or under a specified threshold, for functions that approximate a polynomial of order M-1. In other words, the magnitude of the wavelet coefficients will be near zero any time that the behaviour of the data points approximates to a polynomial of order M-1. Thus we can recognise the behaviour of the data points in a 3D data set by looking at the coefficients magnitude associated with each data point in the data set. Therefore, different wavelet bases with different vanishing moments must be chosen. In effect, to recognise constant zones, the Haar basis is used while, to separate linear from non-linear data points behaviour, the linear B-spline basis is used . The Haar basis (with one vanishing moment) yields zero coefficients only for the constant behaviour of the data points and the B-spline basis (with two vanishing moments) yields zero coefficients for both the constant and linear behaviour of the data points. Using these properties and logic concepts, it is easy to separate the behaviour of the data points in a 3D data set into constant, linear and non-linear.

In this work, the wavelet transform is applied to each co-ordinate axis separately and then a three-dimensional image for each axis is obtained. This way of doing the three-dimensional wavelet transform allows the user to get some insight into the directional behaviour of the data set.

When the projection coefficients of the three-dimensional wavelet transform at the synthesis side is obtained, a new 3D data set with the coefficients magnitude is created. Associated with each point in the new data set there is a look-up table which indicates if the coefficient magnitude corresponds to constant, linear or non-linear behaviour. Thus, using the new data set and the look-up table, the visualisation is performed with the known technique called Direct Volume Rendering [3]. A different colour is given to constant, linear and non-linear associated coefficient. Also, different levels of intensity, for one colour, is given to the pixel depending on the magnitude of the wavelet coefficients.

The images obtained with the volume rendering allow the user to visualise the behaviour of groups of points in a global way, since the volume rendering gives a semi-transparent image where the user can look inside the volume of the data set. The user can easily spot zones of interest in the data set just by looking at the image and he can also infer the shape of an object that could be within the volume. Visualising a rather subjective concept such as the linear and non-linear behaviour of a set of points in a 3D data set, could give useful insight about the content of that data set.

References

[1] Eric j. Stollnitz, Tony D. Derose and David H. Salesin. Wavelets for Computer Graphics. Morgan Kaufmann Publishers, Inc. 1996.

[2] Gilbert Strang and Truong Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press. 1996.

[3] A. E. Kaufman,ed. Volume Visualization. Los Alamitos, Calif.: IEEE CS press, 1991.

Short Paper Presentations

Information Visualization: General

Multiple representations of navigable worlds by Laurence Nigay and Frédéric Vernier

Visualisation of the Information Provided by the Wavelet Coefficients of a 3D Data Set by Ricardo Sánchez and Marcelo Carvajal

Visualization and Processing of Document

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