Surface area estimation of digitized 3D objects using weighted local configurations
Introduction
Digital image analysis aims at measuring properties of the continuous world on the basis of digital images. In many applications, quantitative geometrical measures, such as length and area of objects, are of foremost interest. When working with three-dimensional (3D) digital images, an often-desired measure is the surface area of a digitized object. It should be noticed that measurements of this type can only be estimations, since the interest is seldom in the surface area of the digital object, but rather in the surface area of the original, pre-digitized object.
A good estimator should have a number of properties. It should be accurate and precise, have low algorithmic complexity and computational demands, be robust to noise, and easy to use and implement. Unfortunately, these requirements are often partly contradictory, leading to a trade-off between estimator performance, on one side, and speed and ease of use on the other. The estimator that best fits a given situation may therefore differ from the best choice under other conditions.
In Ref. [1], we presented a surface area estimator that utilizes only local computations and a very small neighborhood (2×2×2) to obtain an estimate that is very fast to calculate and still exhibits good performance in terms of accuracy, precision and robustness. In this paper, the methodology of Ref. [1] is put into a more well defined frame. The performance of the estimator is further evaluated on objects with curved surfaces. By exploiting existing freedom in the parameterization, the bias in the estimate for small balls is reduced by an order of magnitude.
The paper is organized as follows. Section 2 gives a brief overview of related work, both regarding surface area estimation and the conceptually close problem of perimeter estimation. Section 3 lists some basic properties and definitions related to this type of estimates. In Section 4 we give a quick walk-through of the suggested method in the 2D case. Here the optimal result is well known in literature, and the result that we get is in accordance with that. In Section 5 we extend the method to the 3D case, and discuss and suggest solutions to the problems encountered. This includes a numerical optimization for selecting one unique set of area weights. The performance of the estimator is evaluated in Section 6. Section 7 summarizes the results of the simulations, while Section 8 concludes the paper.
Section snippets
Perimeter estimates of digitized 2D objects
The main topic of this paper is surface area estimates of 3D objects. However, many connections to perimeter estimates of 2D objects exist, and starting with the 2D case provides a good introduction for the 3D situation.
When working with 2D digital images, the length of the boundary of a digitized object can be estimated as the cumulative distance from pixel center to pixel center along the border of the object. This is straightforward to accomplish using the Freeman chain code [2], but results
Basic notions
Quantitative analysis of digital images requires estimates that are both accurate, and precise, i.e. the estimates agree well with the true measures on the continuous object (accuracy) and that we get similar values for repeated measurements (precision).
Properties such as length or surface area of an object boundary are invariant to rotation and translation. It is desirable that estimators for the same properties are as invariant to rotation and translation as possible. Unfortunately,
Perimeter estimation
In this section, we present the derivation of the method implemented in the 2D case, for measuring the length of the boundary of a digitized object. This illustrates the general concept of the method and the parameter optimization, which is fairly independent of the dimensionality. For example, we conceive no serious problems in extending the method to four dimensions. For higher dimensions, the neighborhood lookup table will grow too big for practical use, so a different approach may then be
Surface area estimation
Moving up in dimensionality, we can apply the same procedure to find an optimal simple estimator of surface area.
An m-cube (short for Marching Cube) is the cube bounded by the centers of eight voxels of a 2×2×2 neighborhood. Hence, each corner of the m-cube corresponds to a voxel center. Just as in the 2D case, a m-cube can be seen as the dual of the vertex that is shared by its eight surrounding voxels. Correspondingly, each voxel is shared by its eight surrounding m-cubes. In a binary image,
Evaluation
To evaluate the performance of the estimator when applied to curved and non-convex surfaces, we test the method on synthetic objects of known surface area. The used test objects are cubes of side lengths 2–512 voxels, cylinders of height=2×radius, and, to get a non-convex object, thick spherical caps (see Fig. 14) with radiuscavity=(1/2)radiuscap. For comparison, balls of radii 1–256 are also included in the plots. We generate 150,000 cubes, 150,000 cylinders, and 150,000 spherical caps in the
Results
Average relative error and mean AD for the surface area estimation method when applied to digitized objects of increasing resolution are shown in Fig. 15. The error bars indicate minimum and maximum values of the estimate. Surface area estimates for 100,000 digitizations of balls, cubes, cylinders and spherical caps of radius 10 and 100 voxels, respectively, are summarized in Table 3.
The surface of a large ball is a good sampling of planes of all normal directions and the described surface area
Discussion and conclusions
We have presented a method for estimating surface area of binary 3D objects using local computations. The algorithm is appealingly simple and uses only a very small local neighborhood, enabling efficient implementations in hardware and/or in parallel architectures. The estimated surface area is computed as a sum of local area contributions. Optimal area weights for the 2×2×2 configurations of voxels that appear on digital planar surfaces are derived. The method gives an unbiased estimate with
Acknowledgements
We thank Nataša Sladoje, Doc. Ingela Nyström, and Prof. Gunilla Borgefors for their strong scientific support. Thanks also to Xavier Tizon for the brain image.
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