Acceleration techniques for cubic interpolation MIP volume rendering

Abstract

Maximum intensity projection (MIP) is a volume visualization technique that is important in modern medical imaging systems. We propose a method to accelerate high-quality MIP volume rendering using cubic interpolation. First, our method skips more regions of volume data that do not affect the output image. To do this, we propose a method of transforming the B-spline interpolation function into a sub-division of Bezier spline interpolation. We generate the B-spline interpolation control points then the Bezier interpolation control points from three dimensional voxel values. The maximum value of each block is approximated using the Bezier interpolation control points due to the convex hull property of the Bezier spline. By accurately approximating the maximum value of each block, we can skip more unnecessary blocks. Second, we propose an efficient method of parallelization when performing volume visualization using a GPU. In order to reduce the number of memory transfers, our method determines the working shape of a warp, a bundle of 32 GPU threads, depending on the viewing direction. As a result, our method achieves a remarkable rendering speed improvement with no loss of image quality compared to previous studies, and performs high-quality MIP volume rendering using cubic interpolation at interactive speed.

Appendices

Appendix 1

1.1 Subdivision of Bezier curve

$$ {\displaystyle \begin{array}{l}{subd}_{6i-1}={Z}_{3i-1}\\ {}{subd}_{6i}=\frac{Z_{3i-1}+{Z}_{3i}}{2}\\ {}\begin{array}{c}{subd}_{6i+1}=\frac{Z_{3i-1}+2{Z}_{3i}+{Z}_{3i+1}}{4}\kern3.75em \\ {}{subd}_{6i+2}=\frac{Z_{3i-1}+3{Z}_{3i}+3{Z}_{3i+1}+{Z}_{3i+2}}{8}\\ {}\begin{array}{l}{subd}_{6i+3}=\frac{Z_{3i}+2{Z}_{3i+1}+{Z}_{3i+2}\kern1.75em }{4}\kern1.5em \\ {}{subd}_{6i+4}=\frac{Z_{3i+1}+{Z}_{3i+2}}{2}\ \\ {}{subd}_{6i+5}={Z}_{3i+2}\end{array}\end{array}\end{array}} $$

Appendix 2

1.1 Proof of Tables 1 and 2

We prove the Table 1. “The most efficient tile shape is w x h if rotation is arctan (h/w).”

Memory transfer time is proportional to the height of rotated tile (HRT). We define the HRT when the tile width is w and the tile height is h as HRT_{w, h} ≔ h cos θ + w sin θ (assuming 0 ≤ θ < π/2). The HRT is minimized when arctan (h/w) = θ, because:

$$ \underset{\theta }{\min }{HRT}_{w,h}=\underset{\theta }{\min }h\cos \theta +w\sin \theta $$

$$ minimum\ at:h\cos \theta =w\sin \theta $$

$$ \tan \theta =\mathrm{h}/\mathrm{w} $$

Now we prove the first row of Table 2. “32x1 tile is efficient when 0 ≤ θ < arctan(1/16)”.

If θ is close to 0, we obviously select 32x1 tile. As θ increases, we select the 32x1 tile if HRT_32,1 < HRT_16,2 and select the 16 × 2 tile if HRT_32,1 > HRT_16,2. The boundary θ value between them is when HRT_32,1 = HRT_16,2. We select the 32x1 tile when 0 ≤ θ < arctan(1/16) because:

$$ {\displaystyle \begin{array}{c}1\cos \theta +32\sin \theta =2\cos \theta +16\sin \theta \\ {}\theta =\arctan \left(1/16\right)\end{array}} $$