Multiresolution maximum intensity volume rendering by morphological adjunction pyramids

Abstract

We describe a multiresolution extension to maximum intensity projection (MIP) volume rendering, allowing progressive refinement and perfect reconstruction. The method makes use of morphological adjunction pyramids. The pyramidal analysis and synthesis operators are composed of morphological 3-D erosion and dilation, combined with dyadic downsampling for analysis and dyadic upsampling for synthesis. In this case the MIP operator can be interchanged with the synthesis operator. This fact is the key to an efficient multiresolution MIP algorithm, because it allows the computation of the maxima along the line of sight on a coarse level, before applying a two-dimensional synthesis operator to perform reconstruction of the projection image to a finer level. For interpolation and resampling of volume data, which is required to deal with arbitrary view directions, morphological sampling is used, an interpolation method well adapted to the nonlinear character of MIP. The structure of the resulting multiresolution algorithm is very similar to wavelet splatting, the main differences being that (i) linear summation of voxel values is replaced by maximum computation, and (ii) linear wavelet filters are replaced by (nonlinear) morphological filters.

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

“…For that purpose, morphological sampling can be used, an interpolation method well adapted to the nonlinear character of MIP. As shown in [11], the result of this analysis is that after voxel projection a final morphological closing of the projection images has to be applied.…”

“…Then the MIP operation can be performed on a coarse level, where the size of the data is reduced, before performing a 2-D EXPAND operation to a finer level, thus leading to a computationally efficient algorithm. For the pyramids defined above, such commutativity of MIP and pyramid synthesis holds because both the upsampling operation and the dilation δ A commute with the maximum operation [11,12].…”

“…Then the MIP operator can be applied to the volume data, and the resulting 2-D image is finally expanded j 1 times by a 2-D EXPAND operator which has the same form as (7), that is, 2-D upsampling followed by a 2-D dilation, but with a structuring elementà which is the MIP of A [11,12]. This algorithm is applicable for all detail pyramids where the synthesis operator is a dilation.…”

“…However, when using (8) we can refine directly in the image plane, as follows from (10). This case has been treated in [11,12]. The pseudo-code of this algorithm is given in Algorithm 3.2.…”

“…In a previous paper [12] (see [11] for an expanded version) we proposed a pyramid scheme for MIP volume rendering with progressive refinement based on morphological pyramids [3,6]. Such pyramids, which involve nonlinear spatial filtering by morphological operators [5,14], systematically split the volume data into approximation and detail signals.…”