Dyadic Splines

Dyadic splines are a simple and efficient function representation that supports multiresolution design and analysis. These splines are defined as limits of a process that alternately doubles and perturbs a sequence of points, using B-spline subdivision to smoothly perform the doubling. An interval-query algorithm is presented that efficiently and flexibly evaluates a limit function for points and intervals. Methods are given for fitting these functions to input data, and for minimizing the energy and redundancy of the representation. Several methods are given for designing dyadic splines by controlling the perturbations of the limit process. Several applications are explored, including shape design, synthesis of terrain and other natural forms, and compression

Streaming Aerial Video Textures

We present a streaming compression algorithm for huge time-varying aerial imagery. New airborne optical sensors are capable of collecting billion-pixel images at multiple frames per second. These images must be transmitted through a low-bandwidth pipe requiring aggressive compression techniques. We achieve such compression by treating foreground portions of the imagery separately from background portions. Foreground information consists of moving objects, which form a tiny fraction of the total pixels. Background areas are compressed effectively over time using streaming wavelet analysis to compute a compact video texture map that represents several frames of raw input images. This map can be rendered efficiently using an algorithm amenable to GPU implementation. The core algorithmic contributions of this work are methods for fast, low-memory streaming wavelet compression and efficient display of wavelet video textures resulting from such compression

Multiresolution Techniques for Interactive Texture-Based Rendering of Arbitrarily Oriented Cutting Planes

We present a multiresolution technique for interactive texture based rendering of arbitrarily oriented cutting planes for very large data sets. This method uses an adaptive scheme that renders the data along a cutting plane at different resolutions: higher resolution near the point-of-interest and lower resolution away from the point-of-interest. The algorithm is based on the segmentation of texture space into an octree, where the leaves of the tree define the original data and the internal nodes define lower-resolution versions. Rendering is done adaptively by selecting high-resolution versions. Rendering is done adaptively by selecting high-resolution cells close to a center of attention and low-resolution cells away from it. We limit the artifacts introduced by this method by blending between different levels of resolution to produce a smooth image. This techinique can be used to produce a viewpoint-dependent renderings

Hierarchical Large-scale Volume Representation with 3rd-root-of-2 Subdivision and Trivariate B-spline Wavelets

Multiresolution methods provide a means for representing data at multiple levels of detail. They are typically based on a hierarchical data organization scheme and update rules needed for data value computation. We use a data organization that is based on what we call $\sqrt[n]{2}$ subdivision. The main advantage of $\sqrt[n]{2}$ subdivision, compared to quadtree (n=2) or octree (n=3) organizations, is that the number of vertices is only doubled in each subdivision step instead of multiplied by a factor of four or eight, respectively. To update data values we use n-variate B-spline wavelets, which yield better approximations for each level of detail. We develop a lifting scheme for n=2 and n=3 based on the $\sqrt[n]{2}$-subdivision scheme. We obtain narrow masks that provide a basis for out-of-core techniques as well as view-dependent visualization and adaptive, localized refinement

A Selective Refinement Approach for Computing the Distance Functions of Curves

Abstract. We present an adaptive signed distance transform algorithm for curves in the plane. A hierarchy of bounding boxes is required for the input curves. We demonstrate the algorithm on the isocontours of a turbulence simulation. The al-gorithm provides guaranteed error bounds with a selective refinement approach. The domain over which the signed distance function is desired is adaptively triangulated and piecewise discontinuous linear approximations are constructed within each triangle. The resulting transform performs work only where requested and does not rely on a preset sampling rate or other constraints.

Dyadic Splines

Dyadic splines are a simple and efficient function representation that supports multiresolution design and analysis. These splines are defined as limits of a process that alternately doubles and perturbs a sequence of points, using B-spline subdivision to smoothly perform the doubling. An interval-query algorithm is presented that efficiently and flexibly evaluates a limit function for points and intervals. Methods are given for fitting these functions to input data, and for minimizing the energy and redundancy of the representation. Several methods are given for designing dyadic splines by controlling the perturbations of the limit process. Several applications are explored, including shape design, synthesis of terrain and other natural forms, and compression

Boundry Determination for Trivariate Solids

The trivariate tensor-product B-spline solid is a direct extension of the B-spline patch and has been shown to be useful in the creation and visualization of free-form geometric solids. Visualizing these solid objects requires the determination of the boundary surface of the solid, which is a combination of parametric and implicit surfaces. This paper presents a method that determines the implicit boundary surface by examination of the Jacobian determinant of the defining B-spline function. Using an approximation to this determinant, the domain space is adaptively subdivided until a mesh can be determined such that the boundary surface is close to linear in the cells of the mesh. A variation of the marching cubes algorithm is then used to draw the surface. Interval approximation techniques are used to approximate the Jacobian determinant and to approximate the Jacobian determinant gradient for use in the adaptive subdivision methods. This technique can be used to create free-form solid objects, useful in geometric modeling applications

Progressive Precision Surface Design

We introduce a novel wavelet decomposition algorithm that makes a number of powerful new surface design operations practical. Wavelets, and hierarchical representations generally, have held promise to facilitate a variety of design tasks in a unified way by approximating results very precisely, thus avoiding a proliferation of undergirding mathematical representations. However, traditional wavelet decomposition is defined from fine to coarse resolution, thus limiting its efficiency for highly precise surface manipulation when attempting to create new non-local editing methods. Our key contribution is the progressive wavelet decomposition algorithm, a generalpurpose coarse-to-fine method for hierarchical fitting, based in this paper on an underlying multiresolution representation called dyadic splines. The algorithm requests input via a generic interval query mechanism, allowing a wide variety of non-local operations to be quickly implemented. The algorithm performs work proportionate to the tiny compressed output size, rather than to some arbitrarily high resolution that would otherwise be required, thus increasing performance by several orders of magnitude. We describe several design operations that are made tractable because of the progressive decomposition. Free-form pasting is a generalization of the traditional control-mesh edit, but for which the shape of the change is completely general and where the shape can be placed using a free-form deformation within the surface domain. Smoothing and roughening operations are enhanced so that an arbitrary loop in the domain specifies the area of effect. Finally, the sculpting effect of moving a tool shape along a path is simulated