A Framework for the Design of Accurate and Efficient Interpolation and Derivative Filters for Volume Rendering

In this work, we describe a new method for analyzing, classifying, and evaluating filters, which can be applied to interpolation filters as well as to arbitrary derivative filters of any order. Our analysis is based on the Taylor series expansion of the convolution sum. As a result of our analysis, we show the need and also derive the method for the normaliza tion of derivative filter weights. Under certain minimal restrictions of the underlying function, we are able to compute tight absolute error bounds of the reconstruction process. As an example, we demonstrate the utilization of our methods to the analysis of the class of cubic BC-spline filters. As our technique is not restricted to interpolation filters, we are able to show that the Catmull-Rom spline filter and its derivative are the most accurate reconstruction and derivative filter, respectively, among the class of BC-spline filters. Through our analysis of these filters we also discover a new derivative filter which features a better spatial accuracy than any derivative BC-spline filter, and is optimal within our framework. Our analysis shows that a good choice of a derivative filter has a much higher impact than the interpolation filter. We conclude by demonstrating the use of these optimal filters for accurate interpolation and gradient estimation in volume rendering.

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In other work, we describe existing methods for normal estimation in volume rendering and classify them into four classes. After showing that three of these classes are numerically equivalent, we present a framework to compare them with the fourth scheme. We conclude that gradient estimation can be done in one convolution instead of two without losing numerical accuracy. More detail is provided here:

Our latest work is motivated by the observation that for visualization algorithms it is more natural to specify a filter in terms of the smoothness of the resulting reconstructed function and the spatial reconstruction error. To account for this, we present a methodology for designing filters based on spatial smoothness and accuracy criteria. We first state our design criteria and then provide an example of a filter design exercise. We also use the filters so designed for volume rendering of sampled data sets and a synthetic test function. We demonstrate that our results compare favorably with existing methods. Here are the URLs for this work:

This research effort is still on-going, and the M^3 Y^2 team, Torsten Moeller, Raghu Machiraju, Klaus Mueller, Roni Yagel and Yair Kurzion, works relentlessly and restlessly to improve the quality of your datasets.


View onto an MRI brain dataset


Zoomed in at high magnification:
discontinuous filter, 1EF
(grid is very noticable)
discontinuous filter, 3EF
(grid is less noticable)
C0 continuous filter, 1EF 
C0 continuous, 3EF 
(smooth, but more detail)

Thus, it is important to use a filter that is both smooth and has a high order of error function (EF).

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