Surface Voxels Research Articles

Deformable models for segmentation of CLSM tissue images and its application in FISH signal analysis.

In this paper we present an application of deformable models for the segmentation of volumetric tissue images. The three‐dimensional images are obtained using confocal microscope. The segmented images have been used for the quantitative analysis of the Fluorescence In Situ Hybridization (FISH) signals. An ellipsoidal surface initialized around the cell of interest acts as a deformable model. The deformable model surface voxels are subjected to various internal and external forces derived from underlying image features as well as externally imposed constraints. The deformable model converges to the optimum cell shape when the vector sum of all the forces acting on the model is zero. The result of segmentation is used to confirm the cell membership of the FISH signals and to reject all the signals that lie outside the cell nuclei. Three‐dimensional region isolation and labeling technique is used to label and count the FISH signals per cell nucleus. A simple study on the effect of different segmentation methods over a quantitative analysis of FISH signals is also presented.

A 3D Surface Tracking Algorithm

This paper presents a 3D surface tracking algorithm which converts volumetric edge data to a surface model. Existing 3D model generation methods either threshold intensity data, which is sensitive to nonuniform illumination, or trace a surface from multiple layers of edge voxels, where a time-consuming heuristic search is used to extract boundaries. The huge search space prevents using it in 3D. Our algorithm directly constructs a surface model from edge voxels. A voxel is identified as being on the surface if its second derivative is negative and changes sign for neighbors in the gradient direction. By this definition, there can only exist one layer of surface voxels, and the tracking algorithm is simply a breadth-first search. Moreover, the definition of surface voxels is not sensitive to gradient directions, thus this approach is robust against noise. The test results on real data are also reported.

Fundamentals of Surface Voxelization

This paper presents the theoretical discrete geometry framework for the voxelization of surfaces. A voxelized object is a 3D discrete representation of a continuous object on a regular grid of voxels. Many important topological properties of a discrete surface cannot be stated solely in terms of connectivity, and thus the concepts of separating, coverage, and tunnel-freeness are introduced. These concepts form the basis for proper voxelization of surfaces.

Evaluation of new algorithms for the interactive measurement of surface area and volume

The maximum unit normal component (MUNC) method used for surface area measurement and the divergence theorem algorithm (DTA) used for volume measurement were evaluated. The accuracy and precision of these methods were investigated at varying signal-to-noise ratios (SNRs), sampling, spatial averaging, and orientation. The accuracy of the MUNC measured surface area, as indicated by the mean error, was 2.0% for seven spherical samples, with SNRs ranging from 5:1 to 39:1. The precision, as indicated by the percent coefficient of variation (% CV) for these samples, was less than 3.0%. Likewise, the accuracy and precision of the DTA measured volume for these samples were both less than 1.0%. MUNC surface area measurement from 23 samples of a computed tomography (CT) image of a wooden sphere (51.44-mm diameter) with x,y voxel size ranging from 1 to 10 mm and z voxel size ranging from 2 to 14 mm yielded an accuracy of 1.3% and a precision of 2.2%. The DTA volume measurements from 18 samples of the wooden sphere with x,y vowel size ranging from 1 to 8 mm and z size ranging from 2 to 14 mm provided an accuracy of 1.2% and a precision of 1.8%. Measurement of surface area for a cylindrical rod scanned by CT in five different orientations, ranging from along each axis to between all three axes, yielded an accuracy of 3.7% and a precision of 2.0%. The volume of the cylindrical rod measured by the DTA method for these orientations produced an accuracy of 4.0% and a precision of 3.7%. The volume measured by DTA compared well with the volume measured by a modified voxel counting method. The MUNC surface area method was superior to counting surface voxels. The accuracy and precision for five interactive surface area and volume measurements, using paired cut planes to select subsets of a computer-generated sphere with radius 25 pixels, were both less than 1.0%.

Surface area estimation of digitized planes

A method for estimating surface area is developed for three-dimensional binary objects. This method is based on assigning surface area weights to the surface volume-elements (voxels) of a binary object. Surface voxels are defined as object voxels which are six-connected to one or more background voxels. Of the many types of surface voxel configurations, three basic types dominate the surface of relatively smooth, solid objects. These basic types are surface voxels with one, two or three adjacent sides exposed to the background. In planar regions, only these three types exist. By assigning weights to the various surface voxel configurations, total surface area can be estimated by summing the area contributed by all surface voxels of an object. Since this method estimates an object's true surface area, the weights selected should be optimized to give unbiased results with minimum mean-square-error. Formulae to calculate bias and mean-square-error are developed for this surface model when applied to randomly oriented and positioned planar regions. Optimal weights computed from these formulae for surface voxels with one, two or three adjacent sides are W1 ≈ 0.8940, W2 ≈ 1.3409 and W3 ≈ 1.5879, respectively, in sample-grid units squared, with a coefficient of variation (CV = σ/μ) of 2.33%. To address the feasibility for area estimation of curved surfaces, e.g. surfaces of objects extracted from biological or medical 3D images, this method is applied to binary spheres. Furthermore, previously published methods for estimating surface area are compared with the method presented.

Discrete ray tracing

Discrete ray tracing, or 3-D raster ray tracing (RRT), which, unlike existing ray tracing methods that use geometric representation for the 3-D scene employs a 3-D discrete raster of voxels for representing the 3-D scene in the same way a 2-D raster of pixels represents a 2-D image, is discussed. Each voxel is a small quantum unit of volume that has numeric values associated with it representing some measurable properties or attributes of the real object or phenomenon at that voxel. It is shown that RRT operates in two phases: preprocessing voxel and discrete ray tracing. In the voxel phase, the geometric model is digitized using 3-D scan-conversion algorithms that convert the continuous representation of the model into a discrete representation within the 3-D raster. In the second phase, RRT employs a discrete variation of the conventional recursive ray tracer in which 3-D discrete rays are traversed through the 3-D raster to find the first surface voxel. Encountering a nontransparent voxel indicates a ray-surface hit. Results obtained by running the RRT software one one 20-MIPS (25-GHz) processor of a Silicon Graphics 4D/240GTX are presented in terms of CPU time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Representing surfaces with voxels

There are a large class of volume visualization applications in which only the surfaces are of interest. We have developed a system that provides an efficient, general framework for forming these surfaces and working with them. The system described here uses the point in 3D space (voxel), rather than a polygon, as the surface primitive. Voxels are an atomic level construct. They cannot be divided into anything smaller. The voxel is shown to be an appropriate surface primitive for a rich set of graphics operations, many of which have analogues in the polygon world. Surfaces are described by what we call a voxel display list (VDL). Substantial time and space savings are achieved by only storing the surface voxels and rendering from this surface description. The operations that can be performed on voxels surfaces include rotation, projection, shading, wireframes and interactive cutting. Many of these operations are quicker and simpler to perform on voxels than on polygons. The VDL is straightforward to create, render from and manipulate. It can run efficiently without using special hardware.

Brain peeling: viewing the inside of a laminar three-dimensional solid

We describe a 3D surface-tracking algorithm that is used to detect the interior laminar surfaces of a solid shell. Each of these surfaces is called a “peel”. Successive peels may be generated, thus representing the solid shell by its tangential layers. This algorithm is based on voxel surface-tracking methods and solves the problems associated with transforming a surface-tracking algorithm into a “brain peeler”. We also discuss the properties of the voxel surfaces produced by this algorithm. Using the connectivity properties of these objects, we are able to convert voxel representations into polyhedral representations without human interaction. We illustrate this work with a high-resolution reconstruction of a monkey visual cortex. Additional application domains of this work are in areas in which there is a natural laminar structure to a 3D solid, such as in geophysics (earth strata).

Shading 3D-Images from CT Using Gray-Level Gradients

For the 3D-reconstruction of organ surfaces from tomograms, a shading method based on the partial volume effect is presented. In contrast to methods based on the depth and/or the angle of the voxel surface, here the gray-level gradient along the surface is used for shading. It is shown, that at least for bone and soft tissue surfaces, the results are superior to conventional shading. This is due to the high dynamic range of the gray levels within a small spatial neighborhood.

Viewing Transformations of Voxel-Based Objects Via Linear Octrees

This article gives three viewing-transformation algorithms for displaying on a screen 3D pictures represented by linear octrees. All the procedures take advantage of the recursive labeling used to identify the successive decomposition of an object into octants. The first algorithm performs transformations directly on the linear octree, while the second and third algorithms determine the 3D border of the given object first and then project onto the screen the surface voxels thus found. All the algorithms perform the viewing transformations in O(RN) time, where R is the resolution of the picture and N is the number of elements in the linear octree. One of the algorithms provides views of the object at different layers of gray level, while another allows internal views.