Frequency domain volume rendering by the wavelet X-ray transform

Abstract

We describe a wavelet based X-ray rendering method in the frequency domain with a smaller time complexity than wavelet splatting. Standard Fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed. We review the implementation of the fast wavelet transform in the frequency domain. The wavelet X-ray transform is derived, and the corresponding Fourier-wavelet volume rendering algorithm (FWVR) is introduced, FWVR uses Haar or B-spline wavelets and linear or cubic spline interpolation. Various combinations are tested and compared with wavelet splatting (WS). We use medical MR and CT scan data, as well as a 3-D analytical phantom to assess the accuracy, time complexity, and memory cost of both FWVR and WS. The differences between both methods are enumerated.

Multiresolution maximum intensity volume rendering by morphological adjunction pyramids

“…As a consequence, maximum projection of the coarser signals can be computed first, followed by a 2-D pyramid synthesis operation and a 2-D closing, resulting in a computationally efficient algorithm. The algorithm is very similar to wavelet splatting [11,24], the main differences being that (i) linear summation of voxel values is replaced by maximum computation, and (ii) linear wavelet filters are replaced by morphological filters (dilation and erosion). If the volume data are of integer type (bytes or shorts, depending on the dynamic range), then in contrast to the case of wavelet-based volume rendering, no floating point computations are required, but all operations are carried out as minimum and maximum calculations on integers.…”

“…An extensively studied class of multiresolution models in X-ray volume rendering is based on wavelets [7,16,26]. Recent developments include wavelet splatting [11,12], which extends splatting [27] by using wavelets as reconstruction filters, and Fourierwavelet volume rendering [20,24], which is a generalization of standard Fourier volume rendering [14], and uses a frequency domain implementation of the wavelet transform.…”

Mathematical Morphology in Computer Graphics, Scientific Visualization and Visual Exploration

“…For maximum intensity projection (MIP), the transform is nonlinear, so the standard linear multiresolution models based on wavelets (see, e.g., [50]) are not applicable. Instead, the framework of morphological pyramids as developed by Goutsias and Heijmans [14] can be used as the basis for developing multiresolution algorithms for MIP; see [36] for a survey.…”

Morphological Pyramids in Multiresolution MIP Rendering of Large Volume Data: Survey and New Results

“…So the projections increase pointwise as one goes down the pyramid. This algorithm is very similar to that of wavelet splatting [7,8,16]. The main differences are that (i) linear summation of voxel values is replaced by maximum computation, and (ii) linear wavelet filters are replaced by morphological filters.…”