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5.1 Vis_04 Data Visualization Lecture 5 3D Scalar Visualization Part One: Isosurfacing

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5.2 Vis_04 Three Dimensional Scalar Visualization n This is often called: Volume Visualization because data is defined in a volume. n Major applications are: – medical imaging – computational fluid dynamics n Main approaches are: – Surface extraction (this lecture) – Volume rendering (next lecture)

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5.3 Vis_04 Surface Extraction n Surface Extraction – SLICING - Take a slice through the 3D volume (often orthogonal to one of the axes), reducing it to a 2D problem – ISOSURFACING - Extract a surface of constant value through the volume Note analogous techniques in 2D visualization: 1D cross-sections, and contours (=isolines)

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5.4 Vis_04 Isosurface Examples

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5.5 Vis_04 Notation f 000 f 001 f 100 f 101 f 111 Volume of data voxel vertex Each voxel transformed to unit cube f 011 f 110 f 010

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5.6 Vis_04 Data Enrichment - Nearest Neighbour Interpolation f 000 f 001 f 100 f 101 f 111 f 011 f 110 f 010 Value at any interior point taken as value at nearest vertex Fast Discontinuous

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5.7 Vis_04 Data Enrichment - Trilinear Interpolation f 000 f 001 f 100 f 101 f 111 f 011 f 110 f 010 Trilinear interpolant is: f(x,y,z) = f 000 (1-x)(1-y)(1-z) + f 100 x(1-y)(1-z) + f 010 (1-x)y(1-z) + f 001 (1-x)(1-y)z + f 110 xy(1-z) + f 101 x(1-y)z + f 011 (1-x)yz + f 111 xyz

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5.8 Vis_04 Data Enrichment - Trilinear Interpolation The value at is found by: (i) 4 1D interpolations in x (ii) 2 1D interpolations in y (iii) 1 1D interpolation in z f 101 f 000 f 001 f 100 f 111 f 011 f 110 f 010

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5.9 Vis_04 Isosurfacing

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5.10 Vis_04 Lobster – Increasing the Threshold Level From University of Bonn

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5.11 Vis_04 Isosurface Construction n For simplicity, we shall work with zero level isosurface, and denote positive vertices as There are EIGHT vertices, each can be positive or negative - so there are 2 8 = 256 different cases!

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5.12 Vis_04 These two are easy! There is no portion of the isosurface inside the cube!

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5.13 Vis_04 Isosurface Construction - One Positive Vertex - 1 Intersections with edges found by inverse linear interpolation (as in contouring)

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5.14 Vis_04 Note on Inverse Linear Interpolation n The linear interpolation formula gives value of f at specified point t: f(x*) = f1 + t ( f2 - f1 ) n Inverse linear interpolation gives value of t at which f takes a specified value f* t = (f* - f1)/(f2 –f1) f1 f2 x1 x2 t f1 f2 x1 x2 t f*

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5.15 Vis_04 Isosurface Construction - One Positive Vertex - 2 Joining edge intersections across faces forms a triangle as part of the isosurface

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5.16 Vis_04 Isosurface Construction -Positive Vertices at Opposite Corners

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5.17 Vis_04 Isosurface Construction n One can work through all 256 cases in this way - although it quickly becomes apparent that many cases are similar. n For example: – 2 cases where all are positive, or all negative, give no isosurface – 16 cases where one vertex has opposite sign from all the rest n In fact, there are only 15 topologically distinct configurations

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5.18 Vis_04 Canonical Cases for Isosurfacing The 256 possible configurations can be grouped into these 15 canonical cases on the basis of complementarity (swapping positive and negative) and rotational symmetry The advantage of doing this is for ease of implementation - we just need to code 15 cases not 256

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5.19 Vis_04 Isosurface Construction n In some configurations, just one triangle forms the isosurface n In other configurations... –...there can be several triangles – …or a polygon with 4, 5 or 6 points which can be triangulated n A software implementation will have separate code for each configuration

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5.20 Vis_04 Marching Cubes Algorithm n Step 1: Classify the eight vertices relative to the isosurface value V1V2 V3 V4 V5 V6 V7 V V1V2V3V4V5V6V7V8 8-bit index ; 1+ve;0 -ve Code identifies edges intersected: V1V4; V1V5; V2V3; V2V6; V5V8; V7V8; V4V8

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5.21 Vis_04 Marching Cubes Algorithm n Step 2: Look up table which identifies the canonical configuration n For example: Configuration Configuration Configuration 1 … Configuration Configuration entries in table

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5.22 Vis_04 Marching Cubes Algorithm n Step 3: Inverse linear interpolation along the identified edges will locate the intersection points n Step 4: The canonical configuration will determine how the pieces of the isosurface are created (0, 1, 2, 3 or 4 triangles) n Step 5: Pass triangles to renderer for display Algorithm marches from cube to cube between slices, and then from slice to slice to produce a smoothly triangulated surface

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5.23 Vis_04 Homework n Case 12 has three positive vertices on the bottom plane; and one positive vertex on the top plane, directly above the single negative on the bottom plane. n Without looking at the answer…. Try to work out the isosurface!

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5.24 Vis_04 Isosurfacing by Marching Cubes Algorithm n Advantages – isosurfaces good for extracting boundary layers – surface defined as triangles in 3D - well- known rendering techniques available for lighting, shading and viewing... with hardware support n Disadvantages – shows only a slice of data – ambiguities?

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5.25 Vis_04 Ambiguities n Marching cubes suffers from exactly the same problems that we saw in contouring Case 3: Triangles are chosen to slice off the positive vertices - but could they have been drawn another way?

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5.26 Vis_04 Ambiguities on Faces n On the front face, we have exactly the same ambiguity problem we had with contouring n We can determine which pair of intersections to connect by looking at value at saddle point

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5.27 Vis_04 Ambiguities on Faces n Trouble occurs because: – trilinear interpolant is only linear along the edges – on a face, it becomes a bilinear function... and for correct topology we must join the correct pair of intersections Case 3 has two triangle pieces cutting off corners! 6 configurations include ambiguous faces.. but here is another interpretation!

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5.28 Vis_04 Holes in Isosurfaces n Because of the ambiguity, early implementations which did not allow for this could leave holes where cells join Cases 12 and 3 in adjoining cells can cause holes

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5.29 Vis_04 Resolving the Face Ambiguities n Using the saddle point method to determine the correct behaviour on a face – generates sub-cases for each of the 6 ambiguous configurations – which sub-case is chosen depends on the value of the saddle-point on the face – note that some configurations have several ambiguous faces so many subcases arise - eg see config 13! – if we do not extend the 15 cases there is chance of holes appearing in surface

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5.30 Vis_04 Trilinear Interpolant n The trilinear function: f(x,y,z) = f 000 (1-x)(1-y)(1-z) +f 100 x(1-y)(1-z) +f 010 (1-x)y(1-z) + f 001 (1-x)(1-y)z +f 110 xy(1-z) +f 101 x(1-y)z +f 011 (1-x)yz +f 111 xyz is deceptively complex n For example, the isosurface of f(x,y,z) = 0 is a cubic surface

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5.31 Vis_04 Accurate Isosurface of Trilinear Interpolant True isosurface of a trilinear interpolant is a curved surface cf contouring where contours are hyperbola We are approximating by the triangles shown

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5.32 Vis_04 Interior Ambiguities n In some cases there can also be ambiguities in the interior n Consider case where opposite corners are positive n Two possibilities: separated or tunnel

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5.33 Vis_04 Resolving the Interior Ambiguity n Decided by value at body saddle point (f x = f y = f z = 0) – Negative: two separate shells – Positive: tunnel

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5.34 Vis_04 Marching Cubes 33 n Revised marching cubes with correct topology: – 15 basic canonical configurations – Ambiguous faces divide 6 into 18 subcases - totalling 27 cases – Ambiguous interiors (tunnel or no tunnel) divide 6 into 12 cases - totalling 33 cases n Marching Cubes 33

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5.35 Vis_04 Marching Tetrahedra n As in contouring, another solution is to divide into simpler shapes - here they are tetrahedra 24 tetrahedra in all Fit linear function in each tetrahedron: f(x,y,z) = a + bx + cy + dz Isosurface of linear function is triangle Value at centre = average of vertex values

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5.36 Vis_04 Marching Tetrahedra n A disadvantage of the 24 marching tetrahedra is the large number of triangles which are created - slowing down the rendering time n There are versions that just use 5 tetrahedra

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5.37 Vis_04 Credits and References n Original marching cubes algorithm – Lorensen and Cline (1987) n Face ambiguities – Nielson and Hamann (1992) n Interior ambiguities – Chernyaev (1995) n Accurate marching cubes – Lopes and Brodlie (2003) n For references see the Resources web page

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