^{1}

^{1}

^{2}

^{2}

^{1}

^{2}

This paper proposes a plane-based sampling method to improve the traditional Ray Casting Algorithm (RCA) for the fast reconstruction of a three-dimensional biomedical model from sequential images. In the novel method, the optical properties of all sampling points depend on the intersection points when a ray travels through an equidistant parallel plan cluster of the volume dataset. The results show that the method improves the rendering speed at over three times compared with the conventional algorithm and the image quality is well guaranteed.

Modeling three-dimensional (3D) volume of biomedical tissues from 2D sequential images is an important technique to highly improve the diagnostic accuracy [

Ray Casting Algorithm (RCA) is a direct volume rendering algorithm. The traditional RCA is widely used for it can precisely visualize various medical images with details of boundary and internal information from sequential images, while real-time rendering with traditional RCA is still an obstacle due to its huge computation.

In recent years, numerous techniques have been proposed to accelerate the rendering speed. In general, there are three primary aspects, including hardware-based, parallel, and software-based acceleration algorithms. Liu et al. [

However, both hardware-based and parallel techniques are inseparable from the development of computer hardware. By comparison, software-based algorithms can be quickly transplanted among different machines. What is more, they can show flexibility of the procedure and reflect the thoughts of researchers. Yang et al. [

This paper proposes an improved RCA to speed the rendering process. The main idea is, when the ray travels through one group of equidistant parallel planes of the volume, intersection points are obtained. Then the properties of sampling points between adjacent intersection points can be calculated by the formula of definite proportion and separated points. By this method, a small number of intersection points are considered; meanwhile the method does not sacrifice the sampling density.

The traditional RCA involves two steps: (1) assign optical properties such as color and opacity to all 3D discrete vertexes according to their gray value, and (2) apply a sampling and composing process. For each output image pixel in sequence, do the following.

Cast the ray through the volume from back to front.

Sample the color

Set the color of the current output pixel according to

The rendering time is mainly comprised of four parts in the above-mentioned rendering process [

Traditionally, the optical property of each sampling point depends on eight vertexes of its voxel by trilinear interpolation [

Interpolation for ray casting.

For example, to sample point

In Figure

The property

According to the above equations, 17 additions and 16 multiplications are executed for sampling each point such as

The basic idea of the plan-based sampling method is to acquire all sampling points based on intersection points when ray travels through a group of parallel planes in the volume data field.

The sampling process, specifically, consists of three steps. First, intersections and the corresponding plane are obtained based on some necessary initial conditions. Then the optical property of all the intersection points is obtained by linear interpolation according to vertexes on plane clusters. The optical property of sampling points between intersection points along the ray is computed by definite proportion and separated point formula.

Assuming that the direction vector of ray is

Parallel plane clusters along

From the mathematical derivation, when original position, direction vector, and the extent of volume data are given, all the intersections and associated voxels can be quickly obtained.

In Figure

In the same way, the property

In addition, when one component of the direction vector

In the new RCA sampling process, only intersection points on a plane cluster along one axis need to be considered without converting coordinates. While in the conventional sampling process, the world coordinates of each sampling point are converted into voxel’s local coordinates and computed by trilinear interpolation [

As is shown in Figure

Experiments are carried out on head CT sequences and heart CT sequences. Both sequences are scanned by Siemens spiral CT. The detail information is shown in Table

Comparison of two sampling methods.

Objects and sizes | Head |
Heart |
---|---|---|

Spacing |
0.486 × 0.486 × 0.700 | 0.318 × 0.318 × 2.000 |

Sampling distance (mm) | 0.3 | 0.3 |

Time by the traditional (s) | 58.274 | 7.192 |

Time by the proposed (s) | 17.158 | 2.043 |

Acceleration rate | 3.606 | 3.52 |

The reconstructed results of two datasets are shown in Figures

Head images of ray casting.

Traditional method

Proposed method

Heart images with ray casting.

Traditional method

Proposed method

the details of (a)

the details of (b)

The new sampling method does not consult all 3D vertexes of the volume data. For this reason, it is a question whether the image quality can be guaranteed. It can be seen in Figures

By comparing the amount of computation (

Moreover, it is shown that the acceleration rate of the head images is higher than that of the heart images. The main difference between them is that the spacing of head CT sequences is denser than the heart data. Therefore, the denser the data is, the more efficient the new method is.

This paper presented a novel RCA based on a parallel plan cluster sampling method. The proposed method can efficiently speed up the sampling process at more than three times and still clearly display the boundary and internal information of the volume; thus the image quality is well guaranteed. In addition, the comparison of acceleration rate indicates that the new method is more effective for dataset with denser spacing. The new method can meet the real-time requirements of interactive rendering.

This work was supported by the National Natural Science Foundation of China (61105073, 61173096, and 61103140) and the Science and Technology Department of Zhejiang Province (R1110679 and 2010C33095).