Visualizing Geometric Uncertainty of Surface Interpolants

Introduction

Central to the work of scientists, engineers and designers is the task of constructing models of data sets obtained by instruments or created by users. However, in most situations, there is no clear choice of one model over another. Therefore, scientists, engineers and designers are very keenly interested in comparing the results from di erent models, and analyzing their relative advantages and disadvantages.

Data interpolation is one of the most important examples of this task. Franke compared several data interpolation techniques and evaluated the interpolants based on several characteristics such as accuracy, sensitivity to parameters and visual aspects Fra82]. Mann et al. also compared several interpolants for triangulated scattered data in R 3 and evaluated the interpolants based on the shaded images of the interpolants or their Gaussian curvature plots MLL + 92]. E ective display of geometric information associated with surface interpolants has become an important tool in evaluating and comparing the quality of surface interpolants in computer graphics, computer aided geometric design and scienti c visualization.

We introduce the term geometric uncertainty as a measure of interpolation error, level of con dence or quality of an interpolant. Geometric uncertainty can be estimated as a scalar or a vector-valued function that depends upon geometric characteristics of interpolants associated with the underlying data. These characteristics include position, normals, isophotes, principal curvatures and directions, and mean and Gaussian curvatures. Other measures of geometric uncertainty will be discussed in Section 2.1.

Visualizing geometric uncertainty is a very valuable aid in evaluating the e ectiveness of an interpolation scheme. Side-by-side display of interpolants or some geometric property of interpolants such as Gaussian curvature is perhaps the most popular technique for comparing interpolants. Other popular techniques include pseudo-coloring, di erencing, overlay and animation. Although these techniques have been found to be successful to some extent, no one technique is exible or powerful enough to provide the wide range of information that a user typically seeks. Moreover, most of the past methods provide no control to the user for probing the quality or geometry of the interpolants.

In this work we present new techniques for visualizing geometric uncertainty of surface interpolants. There are two major strengths of the system that we have designed. First, we have used a wide range of visualization techniques that combine the advantages of traditional techniques with new glyph-based techniques that capture the geometric information through shape, size and color of glyphs. We have also incorporated texture-based visualization techniques that include bump mapping, displacement mapping and spot mapping. Second, our system provides an interactive control to the user for probing geometric information of surface interpolants in many useful and convenient ways. Examples include displaying only a subregion of interest that satisfy certain constraints, or mapping geometric information of interest to visual objects such as glyphs, or manipulating characteristics such as width or display resolution of glyphs in order to create graphics that are convenient to view.

In order to demonstrate the e ectiveness of our techniques, we have implemented several interpolation schemes that include multiquadrics, inverse multiquadrics and thin plate splines. We have also implemented bilinear interpolation and C 2 bicubic B-spline interpolation schemes that work only on gridded data. The geometric uncertainty information of these interpolants has been computed and visualized. Experimentation with the visualization techniques brings out a wealth of information about the interpolants in a convenient and e ective manner.

The rest of the paper is organized as follows. Section 2 describes previous work on de ning and visualizing uncertainty in general and geometric uncertainty in particular. Section 3 presents new techniques for visualizing geometric uncertainty. Section 3 also describes interactive features of the system that allow the user to probe the interpolants e ectively and conveniently. Section 4 presents the implementation and discusses the results of our experimentation. Section 5 presents two experiments on visualizing the geometric uncertainty of surface interpolants. Section 6 concludes with nal remarks and future work.

2. Background

In this section we describe the previous work on de ning and visualizing uncertainty with an emphasis on geometric uncertainty.

Uncertainty

Uncertainty is a term that has been used to describe several di erent features of scienti c data including error, accuracy, con dence level and quality of data. Error can be de ned as the discrepancy between a given value and its true value GBW94]. Inaccuracy is the di erence between the given value and its modeled or simulated value GBW94]. Con dence level is the level of con dence that can be associated with data and can be computed based on statistical methods or evaluation by scienti c judgement TK93]. Data quality is a very broad term that encompasses many concepts including data validity and data lineage BBC91,Moe88].

Geometric uncertainty, likewise, is a scalar or a vector-valued function that captures error, accuracy, quality or con dence level of the geometry of a surface. The geometric characteristics of interest typically include several pieces of geometric information that are based on positional, rst, second and sometimes even third derivative information. The rst derivative information of interest at a point on the surface includes tangent plane information, normals and isophotes. Given a normalÑ(p) at the point p on a surface and a directionL of the light source, the isophote surface IL(p) can be de ned as IL(p) =Ñ(p) L , where denotes the dot product. There is a continuum of isophote surfaces depending upon the direction of the light source. Contours of isophote surfaces have been used to interrogate surface geometry HHS + 92]. Most of the geometric measures that capture second derivative information are based on minimum and maximum principal curvatures 1 and 2 and the associated principal directionsẽ 1 and e 2 respectively. We refer the reader to any standard textbook on di erential geometry for details dC76]. Important geometric measures for surfaces are Gaussian curvature K = 1 2 and mean curvature H = 1 2 ( 1 + 2 ). Both Gaussian and mean curvatures are geometric invariants that capture the local geometry of the surface. The quantity 2 1 + 2 2 measures the strain energy of exure and torsion in a thin rectangular elastic plate with small de ection, and is typically used as a standard fairness criterion for surfaces in engineering HS91]. Third derivative information is captured by the sum of the variations of the principal curvatures along the principal directions, that is, ( d 1 dẽ 1 ) 2 + ( d 2 dẽ 2 ) 2 , which has also been used as a fairness metric MS94]. Other more sophisticated criteria have also been adopted HB93, MS94]. In addition, re ection lines, orthotomics and focal surfaces have also been proposed for surface interrogation HHS + 92]. In principal, any of the above measures or weighted combination of these measures or di erences between these measures can be used as an estimate of geometric uncertainty. The exact choice depends upon the application at hand.

Visualizing Uncertainty

Popular techniques for visually comparing surface interpolants are side-by-side comparisons, di erence comparison and pseudo-coloring. Franke compared visual aspects of several interpolants by drawing wireframe perspective plots side-by-side Fra82]. Isophotes have been compared by drawing the contours of isophote surfaces side-by-side HHS + 92, PHD91]. Examples of side-by-side comparison also occur in comparing 2D images after wavelet compression DJL92] and comparing 2D images of 3D volumetric data after hierarchical volume rendering and compression WG94]. Di erence comparison is a technique where the di erence between two images, surfaces or volumes is computed point-by-point and the di erence image, surface or volume is rendered. Examples of this occur in comparing images by Tvedt Tve91 Other techniques for visual comparisons include transparency, overlay and animation. Use of transparency for comparing surface interpolants is presented in PFN94]. Related concepts of blends (including techniques based on percentage classi cation of materials), fuzziness, fog or blurs have been proposed in FLN90,BBC91]. The idea of overlaying two curves or surfaces and connecting the respective points by straight lines has also been used LSG94]. Animation has been used to visualize fuzzy data Ger92].

Although glyphs or textures have not been used for comparing or visualizing surface interpolants, they are quite common in data displays. Glyphs are symbols that represent data through visual properties such as size, shape, color, position and orientation. They have also been called probes, geometrical primitives, stars, boxes and icons PG88]. Glyphs have been used to represent univariate data Tuk84, Tuf83b, Tuf83a, Cle85]. Di erent types of glyphs such as stars, Cherno faces, boxes, pro les, Kleiner-Hartigan tress and Andrew's plots have been used to represent multivariate data CBB91]. Glyphs for representing vector and tensor elds are shown in dLvW93]. Texture mapping has been used for generating photo-realistic images Hec86] and scienti c visualization vW91]. Displacement mapping and bump mapping are also standard techniques in computer graphics FvDFH90].

In addition to the techniques mentioned above, most of the work in visualization of uncertainty has been in the eld of Geographic Information Systems, for which we refer the reader to HG93] or WSF + 95]. We also mention that several techniques have been proposed for visualizing surfaces over surfaces and multi-valued volumetric visualization FL90, FL91, Nie87, NFHL91], but none of them seems to have addressed the question of visually comparing surfaces or visualizing geometric uncertainty. Finally, visual comparison of sequences also have been studied HW91].

We now present an overview of our system for visualizing geometric uncertainty of surface interpolants and the key factors that in uenced the design of the system. First, although traditional visualization techniques such as pseudo-coloring or di erencing have been successful to some extent, no one technique is exible or powerful enough to provide the wide range of information that a user typically seeks. Therefore, our system creates a wide range of visualization possibilities that incorporate the complementary advantages of di erent visualization techniques. Second, in our visualizations, we have attempted to incorporate the important principles of data-ink maximization Tuf83b] and maximum impact Tuk84] by providing a clutter-free presentation and focusing on the substance of the presentation. More importantly, we are guided by the principle of maximum utility to the user. Therefore, the user is provided with an interactive query-driven toolbox that allows the facility to control many parameters such as geometric uncertainty parameters, subregion selection, scaling, lighting, zooming, translation, rotation, color ramps to create their own views. Moreover, in our visualizations, we have included many retinal or visual variables such as shape, size, and color based on Bertin's classi cation Ber83]. We now discuss both these features in greater detail.

Visualization Techniques

In order to capture diverse geometric information together, we have created visualizations based on geometry glyphs. Geometry glyphs are visual objects that convey geometry through its visual properties such as size, shape, color and position. The user can choose between many di erent shapes that include boxes, spheres and ellipsoids. Shapes, sizes and colors can be mapped to user-preferred geometric parameters. These choices provide a wide range of possible glyphs. We now describe speci c examples of some glyphs that we have found useful. 2) is a vector-glyph that displays the triangular region between two vectors at the same point. We have used triangular glyphs to display the geometric uncertainty of normals and principal curvature directions at a point. We have also created a volume-lling glyph ( Figure 3.1) that encloses the volume between two surfaces by spheres whose radii are proportional to the di erence between two surfaces.

In order to create visualizations that are clutter-free and easy to perceive, we have used texture mapping for capturing geometric uncertainty information. Three di erent techniques .2 and 4.3) for visualizing any one geometric feature of interest. Our contribution here is to allow the user to choose from a wide variety of geometric uncertainty parameters, described in Section 2.1.

Combinations of these techniques provide a richer and more useful class of techniques. For example, overlay surfaces can be combined with displacement glyphs (Figure 3.5), di erence surface can be pseudo-colored (Figure 3.6), or cross-hair and triangular glyphs can be used with transparency (Figure 4.2). By combining these techniques judiciously, we have created a wide range of new possibilities for probing the geometry of surfaces. Advantages of these visualization techniques are presented in Section 4.

Interactive Features

This visualization system provides the user with query-driven interactive control of several features in order to create graphics that are useful and convenient to view.

Visual Parameter Selection: With every visualization technique, there are several visual parameters that can be controlled by the user. In glyph-based techniques the user can choose the display resolution as well as the size, shape and color of the glyphs. In texture-based techniques, the user can choose the randomness factor. In transparency or pseudo-coloring, the amount of transparency or the choice of the color ramp is up to the user. In addition, there are several visual parameters that are not tied to any particular visualization technique. For example, the user can position the lights, choose the intensity and colors of the light and choose material properties of the surface such as the coe cients of re ectivity for ambient, di use and spectral light. The user also has the ability to view a wireframe representation or a shaded representation. This exibility can be used for three di erent purposes: Figure 3.4: Spot texture mapping with triangular strips indicating uncertainty in normals above a certain threshold for MQ and TPS interpolants 1. To create views that are easy to navigate and understand: This objective is achieved by mapping visual parameters according to convenience of viewing. For example, the display resolution can be chosen for a dense (Figure 4.1) or a sparse presentation (Figure 4.2). Size of the glyphs have been scaled in Figure 4.3 because the original glyphs were too small to view indicating that the absolute di erences between the two interpolants are very small. A green-red ramp is chosen in Figures 3.2 and 3.6 over a standard grey ramp, because it indicates not only the magnitude of the di erences between the two surfaces by the brightness, but also the sign of the di erences by the color. The amount of transparency has been manipulated in Figures 4.2 and 4.3 to display a transparent surface where the di erences are small and relatively opaque where the di erences are large. The randomness factor in displacement mapping has been chosen in Figure 3.3 to present a certain level of contrast that is meant to represent the level of con dence in the interpolant. Regions of low level of con dence appear uncertain due to its rough texture. 2. To overload an image with additional cues: Visual parameters are mapped to the same geometric information in order to reinforce the data with di erent visualization techniques. Figure 3.5 displays an isophote of the multiquadric interpolant in green and the corresponding isophote of the thin plate spline interpolant in red. The di erences between the two isophotes are then lled in by displacement glyphs. Both the mappings { overlay and the displacement glyphs { encode the same information about the position of the isophotes. However displacement glyphs provide additional cues. As another example, Figure 3.2 displays a surface that has been pseudo-colored according to the di erence between the two interpolants in addition to the glyphs that encode the same information through their heights. Both the mappings { the pseudo-color and the glyphs { provide the same information but reinforce each other in a strong way to provide a much better understanding of both relative and absolute values. 3. To create a single graphic that brings together diverse geometric information together:

In order to achieve this objective, visual parameters such as glyph parameters, texture parameters, amount of transparency or the color ramp are mapped to di erent geometric uncertainty parameters. Figure 4.2 displays the multiquadric interpolant, where di erences between the multiquadric and the thin plate spline interpolant are highlighted using transparency technique, di erences in normals are shown by triangular strips and cross-hair glyphs have been utilized to display the di erences in mean and Gaussian curvatures. This graphic combines the positional, the rst derivative and the second derivative uncertainty information in a single graphic. Query-Based: This refers to the ability of the user to highlight or display only a part of the entire graphic that satis es certain constraints or queries. These queries are tied to the geometric properties of the surface. An example of such a query is to display only those glyphs that represent large di erences between normals (Figure 3.4) or represent di erences between Region Selection: This refers to the ability of the user to select certain subregions of interest. For example, the viewer can choose to view only the region around a hill or a saddle point.

Our system provides the facility to the user for viewing only that part of graphics that are associated with a curve or a point. The user can select these subregions either by clicking with a mouse or by providing the location of the point or the equation of the curve. This feature is useful for probing the surface at a given point, surrounding regions or along boundary curves. Glyphs along the curves can be animated with animated probes. In this case a glyph such as an ellipsoidal ball or a box moves along a curve on one surface and expands or shrinks according to the di erence between two surfaces along that curve. The user can control the speed of the probe. Alternatively, the glyphs along the curves can be swept along a desired curve and retained for subsequent viewing in swept probes (Figure 3.2).

The system allows standard geometric and viewing transformations such as translation, scaling, rotation and zooming. We also have a 3D-trackball that allows user to pick a direction of the light source interactively in order to create an isophote surface.

Implementation and Analysis

We now describe the interpolation schemes and data sets used in the experimentation of our visualization system. We then discuss the results of our experiments.

Interpolants

We have implemented several interpolation techniques, that are quite popular in computer graphics, computer aided geometric design and scienti c visualization applications. These interpolants are C 0 piecewise linear interpolant (based on a triangulation of the data), bilinear interpolant, and C 2 bicubic B-spline interpolant for gridded data. For the bicubic B-spline interpolants, we have used the generalization of not-a-knot boundary condition Wol90] for constructing tensor-product interpolants. We have also implemented Hardy's multiquadrics, inverse multiquadrics, and thin plate splines. The motivation for choosing these radial interpolants is that these three radial interpolants are the only ones (besides one more radial interpolant) that received an`A' rating in visual category in Franke's survey Fra82].

Examples and Data Sets

We have experimented with Franke's six analytic test functions Fra82], which include a wide variety of shapes including hills, valleys, cli s, saddles and a part of a sphere. The equations for these functions are available in Nie87]. We have set the value of the free parameter for multiquadrics and inverse multiquadrics interpolants for Franke's test functions to be the one reported by Foley et al. Fol94], which is nearly optimal for a slightly di erent distribution of data. For each of these functions, the interpolants can be constructed by sampling the analytic functions for di erent data distributions Nie87]. We have also experimented with some meteorological and oceanographic data obtained by instruments. Due to limited space, in this paper all the gures correspond to interpolants constructed by sampling Franke's rst analytic function (that contains two hills, a valley and a saddle), on a 10 10 grid. Geometric uncertainty in these gures is computed as the di erence between the geometric quantity of the two interpolants. For example, in Figure 4.2 the height of the cross-hairs depict the di erence between the mean curvatures (in red) and the Gaussian curvatures (in green) of the two interpolants.

Discussion

We now discuss the results of our experimentation with visualizing geometric uncertainty.

The key observation is that a static visualization system is highly constrained to be of much value in a practical situation. The key to a successful system is providing exibility in creating visualizations by possible combinations of (i) visualization techniques, (ii) geometric uncertainty parameters, and (iii) visual parameters. This exibility was heavily utilized in creating examples of visualizations presented in this paper and in conducting the experiments for probing the quality of surface interpolants described in Section 5. Examples and advantages of exibility in choosing visual parameters are described in Section 3.2. Here we focus on analyzing the advantages and disadvantages of di erent techniques for visualizing geometric uncertainty.

Glyphs: We have found both the displacement glyphs and volume lling glyphs to be one of the most useful and precise techniques for comparing surfaces visually. Displacement glyphs give a very good idea of absolute di erences between surfaces. They also provide the information as to where these di erences are located as well as the relative positions of the two surfaces. Volume lling glyphs are very useful in providing a good sense of the error by lling the total volume enclosed between the two surfaces. Even if the absolute di erences are rather small, this method can be made very e ective by scaling the glyphs, by choosing di erent glyph shapes, by adjusting the spacing between glyphs and by zooming into the areas of interest. For example, spheres are better than boxes for small di erences but worse for large di erences because they tend to bulge out.

Texture Mapping: Displacement mapping, bump mapping and spot mapping provide relatively easy to view information about the regions where the two surfaces disagree. Although these methods seem to do a crude job of providing precise quantitative information, they are very e ective both as additional cues and in having a clutter-free presentation even after adding more information about an additional geometric feature.

Transparency: Transparency uses much less data-ink to portray the same information and is very helpful in providing clutter-free presentation. This technique is also useful due to its seethrough mechanism. However, this method does not provide a precise idea of absolute di erences between the two quantities.

Di erence Surface: This method is very e ective in assessing the absolute di erence between two quantities. By scaling, this method can also bring out regions of high relative di erences. The location of these di erences can also be grasped very easily relative to the domain, but not with respect to the range.

Overlays: Overlays provide satisfactory information about the relative placement of two surfaces or the two geometric quantities. However they are rather di cult to view due to intersections between two surfaces. Pseudo-color: Pseudo-coloring technique is e ective in bringing out the regions of high relative di erences. However it is di cult to gain good understanding of the absolute value of the di erences using this method.

Animation: We found it rather di cult to get much useful information from a simple animation between two surfaces. However when combined with animated probes that expand in proportion to di erences between surfaces along prescribed curves over which they move, they become an e ective method for detailed information in regions of interest.

Side-by-side comparison: This method is e ective in revealing large structural di erences only when they exist. However the eye cannot detect many subtle and even intermediate scale di erences particularly when the di erences are shifts of similar features.

Systematic usage of the variations and combinations of these techniques yields a wealth of information, that is not available when restricting oneself to only one variation or technique. We mention only a few examples. Overlaid surfaces along with displacement glyphs ( Figure 3.5) provide the user with a much better understanding (in an interactive mode) of both the relative positions of two surfaces as well as the magnitude of the di erences between the two surfaces, and overcomes the di culties encountered by other popular techniques acting alone such as side-by-side comparison, di erence surfaces and pseudo-coloring. Glyphs with pseudocolor provide both absolute and relative di erence information relative to the features of the surfaces (Figure 3.2). Di erence surface with transparency provides both relative and absolute di erence information on the domain (Figure 4.3).

Applications

We now describe two experiments for probing the surface geometry of interpolants using visualization techniques developed in this work.

Experiment 1

This experiment describes comparisons of multiquadric (MQ) interpolant with the thin plate spline (TPS) interpolant for the data set mentioned in the previous section. Both these interpolants were assigned an`A' rating in visual aspects by Franke Fra82]. We wanted to probe the geometry of these interpolants in order to make ner distinctions between these two interpolants.

The data set has two hills in the back (in still views displayed in this paper), a saddle between the two hills and a valley in the front. We rst compared the two interpolants by looking at the pseudo-color. This visualization indicated that the di erences between the two interpolants are relatively worst at the valley followed by the two hills and near the saddle. This observation was rea rmed by transparency technique. Both these techniques however failed to give an idea of the absolute di erence between the two interpolants. This was easily assessed by looking at the di erence surface and even more e ectively by volume-lling glyphs shown in Figure 3.1. We also observed that the MQ interpolant was a better t than the TPS interpolant by reaching higher (and closer to the true analytic value) at the two hills and by dipping lower (and closer to the true analytic value) at the valley by comparing both the multiquadric and the thin plate spline interpolant with the analytic surface. This observation was again rea rmed by comparing other features such as Gaussian and mean curvatures of the MQ and TPS interpolants with the same features of the analytic surface.

To visualize the uncertainty in normals, we used triangular strips that displayed the differences between the normals of the two interpolants. Figure 3.4 shows the displacement of the two interpolants as a spotted texture while only the di erences between normals within a certain range are shown in this gure. Other than the large di erences in normals at the hills and valleys (which are suppressed in this gure), the normals are deviant in the atter regions (on the right of the valley for example). This observation was recon rmed by visualizing a series of corresponding isophotes of two surface interpolants. Figure 3.5 shows isophotes of the multiquadric and thin plate spline interpolants overlaid over each other. The di erences in the isophotes shown as displacement glyphs are thresholded and are more pronounced in atter regions.

We then compared the Gaussian curvature information of two interpolants. First, the two hills and a valley correspond to three hills in the Gaussian curvature plots for the multiquadric interpolant. More signi cantly, the saddle point, which remains relatively unattractive to the eye in displays of interpolants, becomes an important feature as a valley in the Gaussian curvature plot of the multiquadric interpolant. Performance of the Gaussian curvature plot of the thin plate spline was observed to be rather poor. Figure 3.6 shows the di erence between the Gaussian curvature of the two interpolants. Pseudo-coloring is mapped to the di erence between the two interpolants. The interesting observation here is the phenomena of a steep hill adjacent to a steep valley at all the three hills of the Gaussian curvature. This phenomenon is visible in the front in the image shown in 3.6 near the valley. Two more occurrences of this phenomena are near the two hills at the back. Figure 4.1 shows the mean curvature of the MQ interpolant. Here the saddle point appears as a ridge due to approximate cancellation of the two principal curvatures. The dense crosshairs in this gure clearly bring out those red regions where di erences between mean curvatures dominate as compared to green areas where di erences between Gaussian curvatures dominate. Finally, Figure 4.2 compares the positional, rst and second derivative information in a single graphic, that is clutter-free and easy to navigate to obtain additional and precise information.

Experiment 2

This experiment compares multiquadric and inverse multiquadric interpolant for the same data set. Both these interpolants do an excellent job of tting this data set. In fact the di erence surface is essentially at and the eye can hardly capture any di erence. Pseudo-coloring and transparency emphasize relative di erence and fail to provide much meaningful information. We wanted to investigate if we can discover anything further about these two interpolants using visualization techniques developed in this work. We experimented with isophotes and normals to bring out the most signi cant di erences between the two interpolants. The di erences between normals were again mostly insigni cant for the eye to detect. The absolute di erences between the corresponding isophotes also remained small for almost all directions including the one shown in Figure 4.3. Figure 4.3 shows the transparent surface that displays the di erence surface between corresponding isophotes of the two interpolants for a chosen direction of light. The regions of high relative di erences in isophote surfaces are brighter.

We then investigated the di erences between Gaussian curvature and mean curvatures of the two interpolants. These di erences were also rather small. In order to bring out the regions of high relative di erences, we mapped the curvature di erence information and scaled the glyphs manyfold. Then by thresholding the low di erences in curvatures, small green patches (where the two hills and the saddle point at the back and the valley in the front are located) in Figure 4.3 brings out that the two interpolants di er relatively more in Gaussian curvature at these features while the di erences in mean curvature are more signi cant in atter regions on both sides of the valley. By comparing the Gaussian and mean curvature of the interpolants with those of the analytic surface, it became clear that the multiquadric interpolant does a slightly better job than the inverse multiquadric interpolant in the valley region.