Marching cubes calculates a triangle mesh that represents the surface of an object, based on a threshold t, the so-called isovalue.
The gray values at the edges of the cube are then compared with t. Here, three different cases can occur: i all values are below t, i. In the latter case, the surface is defined by a set of triangles which separates the vertices that are larger than t, i. After the entire dataset is traversed, a complete triangle mesh is created. The triangular mesh filled with a certain color can then be efficiently visualized by optimized algorithms provided by the graphics card.
In addition the illumination of the scene also plays an important role. In order to accelerate the calculation of the surface mesh, it was transferred to the graphics card. Surface rendering with Marching cubes. The volume is traversed by a cube of eight voxels a. Each voxel of the cube is inspected whether its value is inside or outside the surface.
The cube is divided into triangles that form a triangle mesh. The triangle mesh can then be visualized filled with a color and suitable illumination. A transparent interface is very useful to provide a visual context for a combined visualization including further data b. In texture slicing, a volumetric dataset is visualized as a stack of parallel sections arranged next to each other. A 3D texture is created in the graphics card, which contains the volume. Parallel to the image plane, a stack of 2D textures is generated through the volume.
The 2D textures are filled with scalar values by means of trilinear interpolation of the 3D texture containing the volume. Color and opacity can be set by color tables. Texture slicing also takes advantage of the hardware-near implementation, as the graphics cards are designed for the fast use of texture memory and the computation-intensive interpolations are accelerated on the graphics card.
Volume rendering with texture slicing. With the help of maneuverable and scalable boxes, the internal structures of the gray value visualization a as well as the colored visualization b become visible. A 3D fiber orientation model is a 3D vector field that represents the fiber orientation per voxel. A direct way to visualize a vector field is to use glyphs [ 16 ].
Glyphs are small geometric objects that can represent different properties of the vectors by their color and shape, such as position, direction, orientation, and size. A variable and fast method to calculate glyphs is presented in Section 3. The color is an important factor as it can indicate certain properties such as the direction of the vectors Section 3. The visualization of the fiber orientation of an entire brain by glyphs is opaque and thus inaccessible for analysis.
The outer layers occlude the inner structures. Therefore, suitable methods have been developed that provide an insight into the vector field and thus into the fiber architecture of the brain. This includes the combined visualization with an anatomical dataset Section 3. In addition, a 3D atlas can be used for visualization beyond the scope of an anatomical context, as shown in Section 3. For each voxel of the fiber orientation model, the position and orientation of the vector are extracted.
Each vector can then be visualized as a geometric form with variable length and width. The length of the vector l is variable in the range [0, 1]:. The two points can be used to define an undirected line per voxel and to represent the fiber orientation vector. A line represents the position and orientation of a vector. However, in a 3D vector field, the distances of the vectors to each other and the occlusions of the vectors are difficult to distinguish.
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A 3D shape of the glyphs significantly improves the spatial impression of the 3D vector field. Thus, glyph positions and distances between the glyphs can be clearly recognized. A cylinder is the most suitable glyph shape as it models the round shape of the nerve fibers. In computer graphics, circles the base of a cylinder are approximated, in which points on the real circle are calculated and connected with the smallest possible distance.
The parametric equation of a circle can be used for this purpose:. Since this is computationally very demanding, considering the millions of vectors and thus cylinders that have to be calculated, the base areas of the glyphs are defined only by a few points vertices. For example, a cuboid has four vertices, and the cylinder to be represented has six vertices.
The number of vertices can be set individually which provides space for higher-resolution circles. The number of vertices n determines the angular distance between the vertices on the circle:.
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The variable l specifies the length of the glyph. The shape of the glyph is defined with a given radius r and a number of vertices vi.
With the help of this angle and the parametric equation of a circle, the vertices vi can be described:. The length and width of the glyphs can be set interactively by changing the variables r and l. Another important aspect is the color coding of the glyphs, which is another visual indicator for the orientation of the fibers. Thus, fibers with the same orientations are directly recognizable and visually discriminable from other orientations.
The RGB color space is an additive color space based on the three primary colors red, green, and blue. The x, y, and z components of the vector orientation are assigned directly to the three basic colors, i. The HSV color space defines color by the color value hue [0, ] , the color saturation [0, 1] , and the brightness value [0, 1].
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The color value is determined by the x- and y-component of the vector, the z-component influences the brightness of the color, and the saturation is set to the maximum 1. For this purpose, the two angles are calculated from the vector components by means of spherical coordinates:. The color spheres serve as a legend. For example, yellow in the RGB space codes orientations that run diagonally from the bottom left or bottom right, while these orientations in the HSV space are represented by the different colors green and blue.
The colored spheres in the lower right corner of every image are used as legends. To better recognize the 3D structures, lights are used which darken the colors inside the glyphs by creating shadows. To distinguish between inside and outside the glyphs, the surface normals have to be calculated. For each rectangle that approximates the cylinder, one surface normal is calculated using the cross product.
The other two points are used to calculate the next normal. This produces continuous shading as a form of the local lighting model Gouraud shading [ 17 ]. To get an insight into a 3D vector field, clipping boxes are needed [ 18 ]. A clipping box defines a region that is excluded from the visualization in order to reveal the underlying information. The offset and the size of the box can be interactively changed. In order to obtain an anatomical context despite the removal of vector information, it is an advantage to additionally visualize a PLI modality by means of volume rendering.
In the case of 3D textures, two clipping boxes are used to mask out the regions of interest. In order to get an insight into the brain, clipping boxes are used to remove parts of the brain. For an anatomical context, structural data are visualized as surface a and as 3D texture b in combination with the fiber orientation model.
The 3D fiber orientation model contains very dense information, one vector per voxel. Consequently, the visualization is also very dense. Therefore, the visualization may contain too much information; hence, it might not be possible for the viewer to figure out the important information.
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A reduction of the information can help to get a better overview of all orientations in the model. A quick way of reducing information is to remove every x-th vector from the visualization, but this may also lead to the loss of important information. A better option is to group directions [ 7 , 19 ].
The fiber orientation model is divided into cuboids of equal size, so-called super-voxels. For each super-voxel region, a 3D histogram is created, which calculates the frequency of the orientations of the vectors in the super-voxel. For this purpose, a unit sphere is divided into bins, i. The best match of an orientation vector is determined by the maximum scalar product with the central vector of every bin of the sphere.
With the help of the histogram, a direction can now be displayed by super-voxel, using the same algorithms as described in Section 2.
In order to ensure that no information is lost, it is possible to display the strongest directions in super-voxel up to a defined number at the same time. Create a personal account to register for email alerts with links to free full-text articles. Sign in to save your search Sign in to your personal account.