Fast Wavelet Based Volume Rendering by Accumulation of Transparent Texture Maps

Abstract

In the following paper, a new method for fast and accurate volume intensity and color integration is elaborated, which employs wavelet decompositions and texture mapping. At this point, it comprises and unifies the advantages of recently introduced Fourier domain volume rendering techniques and wavelet based volume rendering. Specifically, the method computes analytic solutions of the ray intensity integral through a single wavelet by slicing its Fourier transform and by backprojecting it into the spatial domain. The resulting slices can be considered as RGB textures where R, G, and B account for the decomposed volume color function. Due to the similarity of the basis functions, the computation of the texture map has to be figured out only once for each 3D mother wavelet. Hence, the final volume rendering procedure turns out to be a superposition of self-similar, transparent and colored textures, which is supported by modern hardware accumulation buffers. Linear shading and attenuation can be introduced by modifications of the wavelet's Fourier transform. ___________________________________________________________________________ 2The main advantages of this method are the provision of accurate solutions and quantification of error bounds, the absence of any expensive prefiltering and the independence of the computational costs from the image resolution. Furthermore, any required discretization, such as the resolution of the basis textures is defined within the computational framework of the wavelet transform. The method is not restricted to a specific type of wavelet unless is provides an analytic Fourier description, such as any B-spline wavelets do.

Fast visualization of volume emissions using conservative subdivision

“…• The coefficients are projected onto the image plane before a two-dimensional inverse transform that produces the image (coefficient splatting, used by Lippert and Gross [16]). • The order of the spatial accuracy of the algorithm is determined by the order of the subdivision algorithm used.…”

“…Several wavelet-based algorithms for volume rendering have been published [3,[5][6][7]16,19], but the wavelet bases used have not been suited for the integration needed when projecting the intensities onto the image plane, requiring expensive transformations to point values on a fine grid, projections of basis function footprints, or numerical quadrature. Lippert and Gross [16] presents a wavelet-based algorithm that uses splatting of wavelet coefficients, but the basis is such that the necessary computations for each splatted coefficient makes the projection expensive. Also, the representation is built from a fine grid, so the work is proportional to N 3 .…”

Frequency domain volume rendering by the wavelet X-ray transform

“…However, since reconstruction also can be performed in frequency space, it is not necessary to carry out these inverse FFT's. Instead, a reconstruction at a desired level is first computed in the Fourier domain by (10) and similarly for the dual filters. Also, is the matrix with elements , with defined analogously.…”

“…Wavelet splatting (WS) [3], [10], [11] modifies the basic splatting algorithm in two ways: i) it uses wavelets as reconstruction filters, and ii) it provides a mechanism to visualize data at different levels of detail. First, the algorithm performs a 3-D wavelet decomposition of the volume data.…”

“…This method supports occlusion, shading, and perspective projection, but suffers from color bleeding or "popping" [8] and aliasing [9]. Previously, wavelet splatting has been proposed by Lippert and Gross [10] as an extension of the splatting method, see also [3], [11]. Wavelet splatting modifies the splatting algorithm by using wavelets as reconstruction filters, so that data can be visualized at different levels of detail.…”