Guidance On The Selection Of Central Difference Method Accuracy In Volume Rendering

In many applications, such as medical diagnosis, correctness of volume rendered images is very important. The most commonly used method for gradient calculation in these volume renderings is the Central Difference Method (CDM), due to its ease of implementation and fast computation. In this paper, artifacts from using CDM for gradient calculation in volume rendering are studied. Gradients are, in general, calculated by CDM with second-order accuracy, $mathcal{O}(Delta x^2)$. We first introduce a simple technique to find the equations for any desired order of CDM. We then compare the $mathcal{O}(Delta x^2)$, $mathcal{O}(Delta x^4)$, and $mathcal{O}(Delta x^6)$ accuracy versions, using the $mathcal{O}(Delta x^6)$ version as “ground truth”. Our results show that, unsurprisingly, $mathcal{O}(Delta x^2)$ has a greater number of errors than $mathcal{O}(Delta x^4)$, with some of those errors leading to changes in the appearance of images. In addition, we found that, in our implementation, $mathcal{O}(Delta x^2)$ and $mathcal{O}(Delta x^4)$ had virtually identical computation time. Finally, we discuss conditions where the higher-order versions may in fact produce less accurate images than the standard $mathcal{O}(Delta x^2)$. From these results, we provide guidance to software developers on choosing the appropriate CDM, based upon their use case.

Kazuhiro Nagai, and Paul Rosen. Guidance On The Selection Of Central Difference Method Accuracy In Volume Rendering. Lecture Notes in Computer Science: Advances in Visual Computing, 2015.

@inproceedings{nagai2015guidance,
  title = {Guidance on the Selection of Central Difference Method Accuracy in Volume
    Rendering},
  author = {Nagai, Kazuhiro and Rosen, Paul},
  booktitle = {Lecture Notes in Computer Science: Advances in Visual Computing},
  pages = {328--338},
  year = {2015},
  note = {textit{Presented at the International Symposium on Visual Computing 2015.}},
  abstract = {In many applications, such as medical diagnosis, correctness of volume
    rendered images is very important. The most commonly used method for gradient
    calculation in these volume renderings is the Central Difference Method (CDM), due to
    its ease of implementation and fast computation. In this paper, artifacts from using CDM
    for gradient calculation in volume rendering are studied. Gradients are, in general,
    calculated by CDM with second-order accuracy, $mathcal{O}(Delta x^2)$. We first
    introduce a simple technique to find the equations for any desired order of CDM. We then
    compare the $mathcal{O}(Delta x^2)$, $mathcal{O}(Delta x^4)$, and
    $mathcal{O}(Delta x^6)$ accuracy versions, using the $mathcal{O}(Delta x^6)$ version
    as ``ground truth''. Our results show that, unsurprisingly, $mathcal{O}(Delta x^2)$
    has a greater number of errors than $mathcal{O}(Delta x^4)$, with some of those errors
    leading to changes in the appearance of images. In addition, we found that, in our
    implementation, $mathcal{O}(Delta x^2)$ and $mathcal{O}(Delta x^4)$ had virtually
    identical computation time. Finally, we discuss conditions where the higher-order
    versions may in fact produce less accurate images than the standard $mathcal{O}(Delta
    x^2)$. From these results, we provide guidance to software developers on choosing the
    appropriate CDM, based upon their use case.}
}