Visualization

1. Visualization Chapter 12

2. Introduction: • In this chapter, we apply almost everything we have done so far to the problem of visualizing large data sets that arise in scientific and medical applications. • Fortunately, we can bring many tools to bear on these problems • We can create geometric objects • We can color them, animate them, or change their orientation. • With all these possibilities, and others, there are multiple ways to visualize a given data set, and we will explore some of the possibilities in this chapter. Chapter 12 -- Visualization

3. 1. Data + Geometry • Data comprise numbers. • They are not geometric objects and thus they cannot be displayed directly by our graphics systems. • Only when we combine data with geometry are we able to create a display and to visualize the underlying information. • A simple definition of (scientific) visualization is: • the merging of data with the display of geometric objects through computer graphics Chapter 12 -- Visualization

4. We consider three types of problems: • scalar visualization: • we are given a three-dimensional volume of scalars, such as X-ray densities. • vector visualization: • we start with a volume of vector data, such as the velocity at each point in a fluid • tensor visualization: • we have a matrix of data at each point, such as the stresses within a mechanical part. Chapter 12 -- Visualization

5. We must confront several difficulties • One is the large size of the data sets. • Many of the most interesting applications involve data that are measured over three spatial dimensions and often also in time. • For example, the medical data such as from an MRI can be at a resolution of 512 x 512 x 200 points. • The visible-woman data set is a volume of densities that consist of 1734 x 512 x 512 data points. • consequently, we cannot assume that any operation on our data, no matter how conceptually simple it appears, can be carried out easily Chapter 12 -- Visualization

6. Second, these data are only data • They are not geometric objects • Thus we must translate these data to create and manipulate geometric objects. • Third, we may have multidimensional data points • For example, • When we measure the flow of a fluid at a point, we are measuring the velocity, a three-dimensional quantity. • When we measure the stress or strain on a solid body at some internal point, we collect a 3x3 matrix of data. • How we convert these data to geometric objects and select the attributes of those objects determines the variety of visualization strategies. Chapter 12 -- Visualization

7. 2. Height Fields and Contours • One of the most basic and important tasks in visualization is visualizing the structure in a mathematical function. • This problem arises in many forms, the biggest difference is • how we are given the function • implicitly, explicitly, or parametrically. • And in how many independent variables define the function. Chapter 12 -- Visualization

8. In this section, we consider two problems that lead to similar visualization methods • One is the visualization of an explicit function of the form: • z = f(x,y) • with two dependent variables • Such functions are sometimes called height fields Chapter 12 -- Visualization

9. The second form is an implicit function in two variables: • g(x,y) = c • where c is a constant. • This equation describes one or more two-dimensional curves in the x,y plane. There may also be no x,y values that satisfy the equation. • This curve is called the contour curve, corresponding to c Chapter 12 -- Visualization

10. 2.1 Meshes • The most common method to display a function f(x,y) is to draw a surface. • If the function f is known only discretely, then we have a set of samples or measurements of experimental data of the form: • zij = f(xi,yj) • We assume that the data are evenly spaced. • If f is known analytically, then we can sample it to obtain a set of discrete data with which to work. Chapter 12 -- Visualization

11. One simple way to generate a surface is through either a triangular or a quadrilateral mesh. • Thus the data define a mesh of either • NM quadrilaterals or • 2NM triangles. • The corresponding OpenGL programs are simple. • The following figure shows a rectangular mesh from height data for a part of Honolulu, Hawaii Chapter 12 -- Visualization

12. There are interesting aspects to and modifications we can make to the OpenGL program • First, if we use all the data, the resulting plot may contain many small polygons • The resulting line density may be annoying and can lead to moiré patterns • Hence we might prefer to sub-sample the data • use every kth point for some k • or by averaging groups of data to obtain a new set of samples with smaller N and M. Chapter 12 -- Visualization

13. Second, the figure we saw was drawn with both black lines and white filled polygons. • The lines are necessary to display the mesh. • The polygons are necessary to hide data behind the mesh. • If we display the data by first drawing the polygons in back and then proceed towards the front, the front polygons automatically hide the polygons further back. • Since the data is well-structured, this is possible. • If hidden surface removal is done in software, this might be faster than turning on z-buffering. Chapter 12 -- Visualization

14. There is one additional trick that we used to display this figure. • If we draw both a polygon and a line loop with the same data, such as: • glcolor3f(1.0,1.0,1.0); • glBegin(GL_QUADS) • ...glVertex3i(...) • glEnd(); • glColor3f(0.0,0.0,0.0); • glBegin(GL_LINE_LOOP); • ...glVertex3i(...) • glEnd(); • then the polygon and the line loop are in the same plane. Chapter 12 -- Visualization

15. Even though the line loop is rendered after the polygon, numerical inaccuracies in the renderer often cause part of the line loop to be blocked by the polygon. • We can enable the polygon offset mode and se the offset parameter as in: • glEnable(GL_POLYGON_OFFSET_FIL) • glPolygonOffset(1.0, 0.5) • These functions move the lines slightly toward the viewer relative to the polygon, so all the desired lines are now visible. Chapter 12 -- Visualization

16. The basic mesh plot can be extended in many ways. • If we display only the polygons then we can add lighting to create a realistic image of a surface in mapping applications. • The required normals can be computed from the vertices of the triangles or quadrilaterals. • Another option is to map a texture to the surface • The texture map might be an image of the terrain from a photograph or other information that might be obtained by digitization of a map. • If we combine these techniques, we can generate a display in which, by changing the position of the light source, we can make the image depend on the time of day. Chapter 12 -- Visualization

17. 2.2 Contour Plots • One of the difficulties with mesh plots is that, although they give a visual impression of the data, it is difficult to make measurements from them. • An alternative method of displaying the height data is to display contours -- curves that correspond to a fixed value of height. • We can use this method to construct topographic maps used by geologists and hikers. • The use of contours is still important, both on its own and as part of other visualization techniques. • We solve the accuracy problem of sampled data by using a technique called marching squares Chapter 12 -- Visualization

18. 2.3 Marching Squares • Suppose: • we have a function f(x,y) that we sample at evenly spaced points on a rectangular array (or lattice) in x and y. • that we would like to find an approximate contour curve • Our strategy is to construct a curve of connected line segments (piecewise linear) • We start with a rectangular cell determined by four grid points Chapter 12 -- Visualization

19. Our algorithm finds the line segments on a cell-by-cell basis using only the values of z at the corners of the cell to determine whether the desired contour passes through the cell. • Consider this case Chapter 12 -- Visualization

20. Then the contour line will go through the box, similar to the figure below. • The principle to consider is: • When there are multiple possible explanations of a phenomenon that are consistent with the data, choose the simplest one. Chapter 12 -- Visualization

21. Returning to our example: • We can now draw the line for this cell. The issue is where do we draw it? • Pick the mid point on each edge and connect them • Interpolate the crossing points. • There are 16 (24)ways that we can color the vertices of a cell using only black and white. • All could arise in our contour problem. • These cases are itemized on the next slide Chapter 12 -- Visualization

22. Chapter 12 -- Visualization

23. If we study these cases, we see that there are two types of symmetry • Rotational (1 & 2) • Inverses (0 & 15) • Once we take symmetry into account, only 4 cases truly are unique: Chapter 12 -- Visualization

24. The first case has no line • The second case is what we just discussed. • The third case has a line from side to side • The fourth case is ambiguous since there are two equally valid interpretations Chapter 12 -- Visualization

25. We have no reason to prefer one over the other • so we could pick one at random. • Always use only one possibility • ... • But as this figure shows, we get different results depending upon the interpretation we choose Chapter 12 -- Visualization

26. Another possibility is to subdivide the cell into four smaller cells, • and then draw our lines. • The code is fairly simple: • int cell(double a, double b, double c, double d) • { • int num=0; • if(a>THRESHOLD) num+=1; • if(b>THRESHOLD) num+=8; • if(c>THRESHOLD) num+=4; • if(d>THRESHOLD) num+=2; • return num; • } Chapter 12 -- Visualization

27. We then can assign the cell to one of the four cases by: • switch(num){ • case 1: case 2: case 4: case 7: case 8: • case 11: case 13: case 14: • draw_one(num, i,j,a,b,c,d); • break; // contour cuts off one corner • case 3: case 6: case 9: case 12: • draw_adjacent(num,i,j,a,b,c,d); • break; // contour crosses cell • case 5: case 10: • draw_opposite(num, i, j, a,b,c,d); • break; // ambiguous cases • case 0: case 15: • break; // no contour in cell • } Chapter 12 -- Visualization

28. void draw_one(int num, int i, int j, double a, double b, • double c, double d) { • // compute the lower left corner of the cell and s_x and s_y here. • switch(num){ • case 1: case 14: • x1=ox; y1 = oy+s_y; • x2= ox+s_x; y2 = oy; • break; • /* the other cases go here • case 2: case 13: ... • case 4: case 11: ... • Case 7: case 8: ... • } • glBegin(GL_LINES); • glVertex2d(x1,y1); • glvertex2d(x2,y2); • glEnd(); • } Chapter 12 -- Visualization

29. The following figure shows two curves corresponding to a single contour value for: • f(x,y) = (x2+y2+a2)2-4a2x2-b4 (with a=0.49, b=0.5) • The function was sampled on a 50x50 grid • The left uses midpoint, the right interpolation. Chapter 12 -- Visualization

30. This next figure shows the Honolulu data set with equally spaced contours: • The Marching squares algorithm just described is nice because all the cells are independent. Chapter 12 -- Visualization

31. 3. Visualizing Surfaces and Scalar Fields • The surface and contour problems just described have three-dimensional extensions. • f(x,y,z) defines a scalar field. • An example is the density inside an object or the absorption of X-rays in the human body. Chapter 12 -- Visualization

32. Visualization of scalar fields is more difficult than the problems we have considered so far for two reasons: • First, three-dimensional problems have more data with which we must work. • Consequently, operations that are routine for 2D data (reading files and performing transformations) present practical difficulties. • Second, when we had a problem with two independent variables, we were able to use the third dimension to visualize scalars. • When we have three, we lack the extra dimension to use for display. • Nevertheless, if we are careful, we can extend our previously developed methods to visualize three-dimensional scalar fields Chapter 12 -- Visualization

33. 3.1 Volumetric Data Sets • Once again, assume that our samples are taken at equally spaced points in x,y, and z • The little boxes are called volume elements or voxels Chapter 12 -- Visualization

34. Even more so than for two-dimensional data, there are multiple ways to display these data sets. • However there are two basic approaches: • Direct Volume Rendering • makes use of every voxel in producing an image • Isosurfaces • use only a subset of the voxels Chapter 12 -- Visualization

35. 3.2 Visualization of Implicit Functions • This is the natural extension of contours to 3D • Consider the implicit function in three dimensions • g(x,y,z) = 0 where g is known analytically. • One way to attack this problem involves a simple form of ray tracing (ray casting) Chapter 12 -- Visualization

36. Finding isosurfaces usually involves working with a discretized version of the problem • replacing g by a set of samples taken over some grid. • Our prime isosurface visualization method is called marching cubes, the three-dimensional version of marching squares. Chapter 12 -- Visualization

37. 4. Isosurfaces and Marching Cubes • Assume that we have a data set, where each voxel is a sample of the scalar field f(x,y,z) • Eight adjacent grid points define a three-dimensional cell as shown here: Chapter 12 -- Visualization

38. We can now look for parts of isosurfaces that pass through these cells, based only on the values at the vertices. • There are 256=28 possible vertex colorings, but once we account for symmetries, there are only 14 unique cases Chapter 12 -- Visualization

39. Using the simplest interpretation of the data, we can generate the points of intersection between the surface and the edges of the cubes by linear interpolation. • Finally we can use the triangle polygons to tessellate these intersections. Chapter 12 -- Visualization

40. Because the algorithm is so easy to parallelize and, like the contour plot, can be table driven, marching cubes is a popular way of displaying three-dimensional data. • There is an ambiguity problem analogous to the one for contour maps Chapter 12 -- Visualization

41. 5. Mesh Simplification • On disadvantage of marching cubes is that the algorithm can generate many more triangles than are really needed to display the isosurface. • One reason can be the resolution of the data set. • We can often create a new mesh with far fewer triangles such that the rendered surfaces are visually indistinguishable. • There are multiple approaches to this mesh simplification problem Chapter 12 -- Visualization

42. One popular approach is triangle decimation. • It seeks to simplify the mesh by removing some edges and vertices. • Example: • The decision as to which triangles are to be removed can be made using criteria such as the local smoothness or the shape of the triangles • The latter is important since long thin triangles do not render well Chapter 12 -- Visualization

43. Other approaches are based upon resampling the surface generated by the original mesh , thus creating a new surface • Probably the most popular technique is Delauney triangulation. • This algorithm creates a triangular mesh in which the circle determined by the vertices of each triangle contains no other vertices. • The complexity of this is O(n log n) Chapter 12 -- Visualization

44. 6. Direct Volume Rendering • The weakness of isosurface rendering is that not all voxels contribute to the final image. • Consequently, we could miss the most important part of the data by selecting the wrong isovalue. • Direct volume rendering constructs images in which all voxels can make a contribution to the image. Chapter 12 -- Visualization

45. Early methods for direct volume rendering treated each voxel as a small cube that was either transparent or completely opaque. • These techniques produced images with serious aliasing artifacts. • With the use of color and opacity, we can avoid or mitigate these problems. Chapter 12 -- Visualization

46. 6.1 Assignment of Color and Opacity • We start by assigning a color and transparency to each voxel. • If the data are from a computer-tomography scan of a person’s head, we might assign colors based upon the X-ray density. • Soft tissues (low density) might be red. • Fatty tissues (medium densities) might be blue • Hard tissues (high densities) might be white • Empty space might be black. Chapter 12 -- Visualization

47. Often, these color assignments are based upon the voxel distribution • Here is an example histogram Chapter 12 -- Visualization

48. 6.2 Splatting • Once colors and opacities are assigned,, we can assign a geometric shape to each object. • Sometimes this is done with a back to front painting. • Consider the voxels in the following figure Chapter 12 -- Visualization

49. Each voxel is assigned a simple shape and this shape is projected onto the image plane. This projected footprint is also known as a splat. • Color plate 28 is an example of this Chapter 12 -- Visualization

50. 6.3 Volume Ray Tracing • An alternative direct-volume rendering technique is front-to-back rendering by raytracing. Chapter 12 -- Visualization