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MEDICAL IMAGING WITH RAY CASTING


MEDICAL IMAGING WITH RAY CASTING
Ing. Daniel Schwarz, PhD student Department of Biomedical Engineering, FEECS BUT, e-mail: schwarzd@dbme.fee.vutbr.cz Supervisor Ing. Ivo Provazník, Ph.D. ABSTRACT This article describes a method for 3D visualization of medical data called ray casting. Some advantages over the standard rendering methods are showed and necessary techniques and tools with several improvements are discussed here. INTRODUCTION There are two different approaches to volume visualization in medical imaging: surface rendering and volume rendering. Both of them have several advantages and disadvantages. Surface rendering algorithms convert volume data to a database of geometric primitives (e.g. polygon mesh), which are then rendered into a 2D shaded projection. A drawback of this approach is a binary segmentation which is implicit for the surface extraction process. Volume rendering techniques do not impose any geometric structure on the volume data. This is a positive fact in the field of medical imaging because the data representing the objects to visualize do not always posseses tangible surfaces that could be extracted. The final 2D image is computed by assigning different visual properties (e.g. color, emission, absorption) to different value ranges within the volume and then by blending this properties into one pixel value on the image plane. Limitations of these techniques are set by hugeness of the volumetric data, which must be arbitrary accessible. Therefore, volume rendering methods are unable to support real-time interaction and manipulation using common hardware. RAY CASTING Ray casting is one of the powerful volume rendering techniques. Figure 1 describes the principle of the ray casting method. Rays are casted from each pixel of the image plane into the volume data. At locations along each ray a sample value and a surface normal approximation are calculated using values of surrounding voxels.

Figure 1:

The ray-casting principle.

Using the sample value and normal, a sample opacity and color are dynamically assigned by a look-up table or in a preprocessing phase. Then a local shading model is applied and the samples along the ray are composed into a pixel value of the final image [1, 4].

We implemented this algorithm with various definable parameters. One of them is the sample distance along the ray. Resampling the scalar function causes that the samples are at non-integer positions. Tri-linear interpolation is used to estimate the sample value. The same technique is also used for gradient estimation. The sampled nature of volumetric data causes that we approximate it by local differencies between sample values in all three dimensions. The resulting vector is called surface normal. The normalized normal vector is one of input vectors of a local shading model. The magnitude of the surface normal can be used during the classification procedure, when an opacity value and a color is assigned to the sample. Local shading model means computing reflected intensity by the sample from the light source (or sources) towards the image plane. The intensity is used for modulating the assigned color. The theory of light transport is much simplified here. The ray from the light source to the sample is not affected by intermediate voxels. On the other hand, this kind of interaction is considered along the ray to the image plane by composition, which will be described further. The Phong illumination model is used here. The reflected intensity by the sample is computed [5]: n I = I a + λ I l ?k d N ? Ll + k s R ? V υ, (1) ε ? ? υ l where Ia is the reflected intensity due to ambient light, Il is the light source intensity, k d is the material diffuse reflection coefficient, k s is the material specular reflection coefficient, n is the material specular reflection exponent. The meaning of the vectors is described in the Figure 2.

(

)

Figure 2:

Vectors used in the Phong illumination model.

As computation of the vector R is rather complicated, we estimate it like a halway vector between the viewing vector V and the normal N. This approximation is accurate in the case that the light source (sources) and the viewpoint are at infinity. Color images are obtained by mixing red, greed and blue intensities. Three lighting equations in the Phong illumination model are used for rendering color images, one for each color [5]: n I r = I aOr + λ I l ?k d Or N ? Ll + k s R ? V υ ε ? ? υ l

Ig Ib

( ) = I O + λ I ? k O N ? L + k (R ? V ) υ , ε ? ? υ = I O + λ I ? k O N ? L + k (R ? V ) υ ε ? ? υ
n a g l d g l s l n a b l d b l s l

(2)

where Or, Og, Ob are components of a color. They are determined during the classification procedure.

Properties of the resulting pixel on the image plane can be computed by integrating the contributions of the sample intensities along then ray s [3]: C (s ) = ν q( s ' )e
s0 s ? α ( s ' ' )ds ' '
s0 s'

ν

ds ' ,

(3)

where q is the emission and α is the absorption. The discretization of this integral leads to the compositing formula:
C = λ qkα k Ο (1 ? α i ) .
k =1 i =0 n k ?1

(4)

The reflected i ntensity I from (1) for a monochrome image or intensities Ir, Ig, Ib from (2) for a color image can be substituted for qk in (4). The opacity α k are mapped from a look-up table which is a transfer function α (f) of the interpolated scalar value. Compositing algorithm is [1]: Cout = Cin + qiα i (1 ? α in ) , (5) α out = α in + α i (1 ? α in ) where Cout , α out are results of the current iteration, qi, α i are the current sample intensity (or color intensities) and the current sample opacity and Cin , α in are accumulations of previous iterations. RAY CASTING WITH MEDICAL IMAGES We used magnetic resonance (MRI) and computer tomography (CT) 3D images (sets of 2D images) from NIH Visible Human data set to visualize a part of a human head with the algorithm described above. More comprehensible images are obtained when the bones and soft tissue are displayed with different optical properties. We do it by using two techniques: pseudocoloring and gradient magnitude modulation. For pseudocoloring different objects we need to know their value ranges in the data. This can be easily done with CT images but in MRI data same values can appear in different tissues. Figure 3 shows histogram of the volume CT data and a simple classification function.

Figure 3:

a) Histogram of CT data of 110 slices of a human head. b) Classification function for discriminating soft tissue and bones. There is a range CT number ∈ <1210,1310> which behaves like a mixture of two materials.

We use this classification function for computing opacity and color look-up tables. After interpolation of the scalar value, an opacity value and a color determined by an RGB triple are

assigned to the sample. Then we use three lightning equations in the Phong illumination model (2). With the simple histogram based segmentation procedure, we can render color images containing either bones or soft tissue, see Figure 4a. For imaging semitransparent images with information about both kinds of tissues, surfaces of visualised objects have to be pointed up and homogeneous regions have to be suppressed. For this purpose we use gradient magnitude modulation. Simple multiplication of the gradient magnitude and the opacity value results in smaller color contribution of the voxels in the homogeneous regions and vice versa. We boost up this process by assigning gradient magnitude value also from a look-up table with partly constant and partly linear transfer function, see Figure 5c. A semitransparent image of a human head is on the Figure 5a.

Figure 4:

a) Image from Ray casting without gradient magnitude modulation. b) Look-up table for opacity values used for rendering image a).

ADAPTIVE TERMINATION Ray casting with pseudocoloring and gradient magnitude modulation is very computation intensive process and the goal of adaptive termination is to shorten the rendering time. It is done by finding the last sample along the ray, which affects the resulting intensity (or color intensities) significantly. We do it using a simple criterion Cout – Cin > Epsilon. CONCLUSION This paper describes simple algorithms of ray casting for volume rendering. We use this method to visualize medical CT images. Therefore, some techniques for enhancing interpretation of the resulting images are used here. For this purpose we use pseudocoloring

and gradient magnitude modulation. Resulting images are semitransparent and they can contain information about more different objects. The whole rendering process is done without need of binary segmentation which is implicit for surface extraction techniques. As the volumetric data must be randomly accessible, this method demands expensive hardware.

Figure 5:

a) Image from Ray casting with gradient magnitude modulation. b) Look-up table for opacity values used for rendering image a). c) Look-up table for gradient magnitude values for rendering image a)

ACKNOWLEDGEMENT This work has been partly supported by Research Programme of Brno University of Technology No. J22/98:262200011 and by the grant projects GACR No. 102/00/P079 and GACR No. 102/99/1228. REFERENCES [1] Watt A., "3D Computer Graphics", Addison-Wesley publishing company 1993. [2] Zara J. a kol., "Pocítacová grafika – principy a algoritmy", Grada, 1992. [3] Ertl T., Westermann R., Grosso R., "Multiresolution and Hierarchical Methods for Visualization of Volume Data", Technical Report 30/1998, Universit?t ErlangenNürnberg, 1998. [4] Noordmans H. J., Voort H., Smeulders A., "Spectral Volume Rendering", IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 3, pp.196-207, 2000. [5] Brook P., "The rendering pipeline", http://www.comp.leeds.ac.uk/cuddles/hyperbks/Rendering, 1997. [6] Pfister H., Hardenbergh J, Knittel J., Lauer H., Seiler L., "The VolumePro real-time ray-casting system", Computer Graphics (SIGGRAPH '99 Proc.), pp.251-260, 1999.


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