Contours & Implicit Modelling 4

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1 Brief Recap Contouring & Implicit Modelling Contouring Implicit Functions Visualisation Lecture 8 lecture 6 Marching Cubes lecture 3 visualisation of a Quadric toby.breckon@ed.ac.uk Computer Vision Lab. Institute for Perception, Action & Behaviour School of Informatics Modelling Algorithms Contours & Implicit Modelling 1 change aspects of data rather than data itself (lecture 5) Contours & Implicit Modelling 2 Contouring Contouring A point about contours: A point about contours: How to visualise the tumour? Brain 30 units 50 Contour of value 50 Draw a contour of value 50? Brain Brain 30 units 30 units Bone 80 units 50 Tumour Bone 80 units Poor visualisation of tumour: Need more control over mapping function from value to pixel transparency! Contours & Implicit Modelling 3 Contours & Implicit Modelling 4

2 Segmentation of CT imaging 2D slice as image How to segment MRI data? 3D contour of bone density A Dog s thorax Lungs Trachea Liver Gall Bladder Marching Cubes Greyscale image colourmap How to extract an organ here? Can add multiple contours at different values to see different components (practicals / current assignment) MRI values not directly proportional to density (instead NMR signal intensity) CT value corresponds to density easy to segment Contours & Implicit Modelling 5 Contours & Implicit Modelling 6 Image Processing on MRI Data Example : MRI segmented knee From image processing / computer vision segment image slices noise removal edge detection (changes in colour) region growing start from initial image patch grow outwards including similar regions (smoothing) stop when boundary or distinct change in value reaching thresholding all scalar values in between upper and lower limits Liver outlined in red Courtesy : Brigham & Women's Hospital Image processing not strictly visualisation additional tool in visualisation pipeline see Advanced Vision course Contours & Implicit Modelling 7 Original MRI slice (top, left), Segmentation (top, right), Overlay (bottom). Enhanced Visualisation : data merging original and segmentation Contours & Implicit Modelling 8

3 Segmentation of dog organs Liver outlined in red 3D model of Liver. Triangulated 3D model of dog liver Vertices in one contour need to be matched with those in the next slice to produce triangulated mesh. Problem different number of vertices in each curve Segmentation of organ in each image slice (2D) visualisation of segmented 2D image stack as 3D volume Contours & Implicit Modelling 9 Contours & Implicit Modelling 10 Problem 1 : Grow hole High slice Problem 2 : split contours Middle slice Middle slice Contours & Implicit Modelling 11 Low slice Known as the branching problem. Contours & Implicit Modelling 12

4 Possible Ambiguities OR OR NUAGES : Triangulate Contour Contour connection method needs to handle these cases Solution: NUAGES Triangulate contour vertices using Delaunay Triangulation no inside / outside constraint forms planar convex hull of each contour fill convex hull of contour with tetrahedra discard those outside the contours or with only one edge on each level 3D contour connection = remaining exterior tetrahedra faces Contours & Implicit Modelling 13 Contours & Implicit Modelling 14 NUAGES : Form Tetrahedra NUAGES : Remove Tetrahedra Form tetrahedra by joining to closest vertex on opposite contour tetrahedra t1 and t2 formed with closest opposite vertex enclosed within area of triangle tetrahedra t12 formed by two edges that cross (special case) Contours & Implicit Modelling 15 Remove : Tetrahedra with an edge outside of the contour. Remove : Tetrahedra with only one edge on each contour plane non solid connection Contours & Implicit Modelling 16

5 NUAGES : Results NUAGES : Results Upper Contour Connected : top down view Lower Contour Connected : side view Use exterior triangles of remaining tetrahedra to form outside boundary of the shape N.B. hole appears as form of a point Contours & Implicit Modelling 17 Multiple medical image slices joined by NUAGES sop.inria.fr/prisme/fiches/medical/ Contours & Implicit Modelling 18 Dog Liver : Final Model Summary Contours Completed model despite presence of contour holes/ branches (NUAGES) Contours & Implicit Modelling 19 Some 3D dataset easy to visualise with iso surfaces CT data values correspond to density iso surfaces from Marching Cubes Some datasets have no such correspondence MRI data values segmentation of organs requires image processing image processing performed on slices slices joined to create 3D volumetric model ambiguities need to be resolved (NUAGES) Contours & Implicit Modelling 20

6 Implicit Functions for Cutting 1 Modelling Algorithms Implicit Functions Functions of the form F(x,y,z) = c c is an arbitrary constant function implies a surface (at F=0) e.g F(x,y,z) = x2 + y2 +z2 R2 defines a sphere radius R centred at the origin F = 0 lies on the sphere F < 0 lies inside the sphere F > 0 lies outside the sphere Geometric modelling Selecting data (i.e. cutting) F<0 If F < 0 ( clipping everything inside ) remove data Uses : Evaluate F at every point on F>0 grid (in set) Defines 3 regions : Define the implicit function F Contours & Implicit Modelling 21 similarly for outside iso surface of sphere at F=0 Cutting = Modelling Algorithm : changes aspects of data Contours & Implicit Modelling 22 Implicit Functions for Cutting 2 F>0 F<0 Cutting Implicit functions generate scalars 3D contouring algorithm iso surface A topology change occurs in the data structured points to unstructured points and cells Contours & Implicit Modelling 23 e.g. marching cubes to generate F(x,y,z) = 0 and produce cut surface Cutting a 3D dataset with a plane 2D slice colour mapped or contoured visualise data on (nearly) arbitrary surface Cutting polygonal geometry lines generate contour lines of 3D models better visual aid than lighting & shading for 3D shape Contours & Implicit Modelling 24

7 Cutting in VTK Pipeline for 3D contours File containing Structured points. vktimplicitfunction sub classes are vtksphere, vtkplane, vtkquadric, vtkcylinder, vtkcone,... vtkcutter vtkreader Spacing parameters cuts cells with an implicit function 3D Surface vtkpolydatamapper vtkstripper vtktubefilter Cut function vtkclippolydata vtkcutter vtkcontour cuts polygonal data with an implicit function Contours & Implicit Modelling 25 vtkplane vtkpolydatamapper Contours & Implicit Modelling 26 Head contours cutmodel.tcl File containing Polygonal data. vtkreader Spacing parameters Implicit Object Modelling vtkcutter 3D Surface vtkstripper vtkpolydatamapper union, intersection, difference operations e.g. ((F1 F2) F3) F4 e.g. two spheres and cone (right) extract surface with contouring operation (in VTK pipeline) vtkpolydatamapper vtktubefilter Cut function vtkplane Combine multiple implicit functions Contours & Implicit Modelling 27 Constructive Solid Geometry Contours & Implicit Modelling 28

8 Implicit Object Modelling in VTK vtkcone vtkplane vtkplane vtksphere vtksphere Implicit Modelling cone vertplane baseplane icecream bite # combine primitives to build ice-cream cone vtkimplicitboolean thecone thecone SetOperationTypeToIntersection thecone AddFunction cone thecone AddFunction vertplane thecone AddFunction baseplane Modelling using implicit function vtkimplicitboolean thecream thecream SetOperationTypeToDifference thecream AddFunction icecream thecream AddFunction bite defined as Euclidean distance from geometric primitives Contours & Implicit Modelling 29 e.g. lines spelling the word hello (geometry and iso surface of function shown) VTK : vtkimplicitmodeller Contours & Implicit Modelling 30 Implicit Modelling Functions All mathematical functions of form F(x,y,z) = c (hello.tcl) Summary Implicit Functions e.g. quadric Defined as F(x,y,z) = c Data Selection (sub selection) Implicit Object Modelling constructive solid geometry Implicit Modelling reduce data improve speed, limit visualisation surface implied by distance from geometric primitives Visualisation of Mathematical Functions next lecture : volume rendering Contours & Implicit Modelling 31 Contours & Implicit Modelling 32