ituppsala universitet data representation and fundamental algorithms filip malmberg filip @cb.uu.se

IT Uppsala universitet

Data representation and fundamental algorithms

Filip Malmberg filip@cb.uu.se

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08-01-29 | #2@ UU/IT

Data representation Data is discrete, i.e., we only know the data at

a finite set of locations. Interpolation is used to find data values

between sampled points.

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Data representationData consists of structure (regular or irregular) geometry (Sample locations) attributes (Data at sample locations)

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Uniform grid• Regular structure• Regular geometry

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Image data

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Multi-dimensional images A multi-dimensional image can be considered

as a function f(x,y,z,t,b), where• z: third spatial direction• t: time sequence• b: spectral bands

Different combinations, for example• f(x,y,z,t) is a 4D image representing a time

sequence of volume images

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Volume images There are several different tomographic

devices:• Computed Tomography (CT)• Magnetic Resonance Imaging (MRI)• Positron Emission Tomography (PET)• and others

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Computed tomography CT produces anatomical images (attenuation

maps) Low contrast in soft tissue Bone is visible High resolution

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Magnetic Resonance Imaging MRI produces anatomical images Very good contrast in soft tissue

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Positron Emission TomographyPET shows the concentration of a positron emitting tracer given to patientFunctional images (show physiological parameters) Low resolution

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Rectilinear grid• Regular structure• “Partially” regular geometry

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Structured grid• Regular structure• Irregular geometry

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Unstructured points• Irregular geometry• No structure

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Unstructured grid• The most general form of a dataset• A collection of vertices, edges, faces,

any cell type• Connectivity information must be explicitly

stored

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Example: Polygon mesh

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How are unstructured meshes different than regular grids?

Regular Gridsmesh info accessed implicitly using grid point indicesefficient in both computation and storagetypically use finite difference (FD) discretizationCartesian grids or logically rectangular grids

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How are unstructured meshes different than regular grids?

Unstructured Meshes

•mesh connectivity information must be stored•handles complex geometries and grid adaptivity•typically use finite volume or •finite element (FE) discretization•mesh quality becomes a concern

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Data attributes Scalar Vector

• magnitude and direction Normal

• a vector of magnitude 1• used for lighting

Texture coordinate• mapping data points into a texture space

Tensor

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Operations on datasets• Geometric transformations• Topological transformations• Attribute transformations• Combined transformations

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Scalar algorithms: Color mapping Colour mapping is a common scalar

visualization technique that maps scalar data to colours

The scalar values serve as indices into a colour lookup table

Or mapping through a transfer function

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Colour mapping

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Scalar algorithms: Isocontour/Isosurface Extractions

2D isocontours 3D isosurface

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Surface rendering of volume dataThe data must be converted to an intermediate surface description using geometric primitives

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Surface versus Volume rendering Surface Rendering We assume that data to

be visualized can be modelled by surfaces. Normally, we model the object with geometric primitives such as points, lines, triangles or polygons and use standard Computer Graphics techniques to render the data

Volume Rendering operates on the data itself and takes into account the changing properties inside the object

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Surface reconstruction from volumesThe by far most popular method is the Marching Cubes algorithm (MC) (Lorensen et al., SIGGRAPH 1987)MC uses the grey-scale information in the data to build triangulated surfacesMC works by looking at each element of the volume dataset one at a time to decide whether its vertices are inside or outside the isosurfaceThis decision is made by comparing the values at the vertices of the volume elements with the value which has been defined as the surface valueIf the surface passes through the volume element, the approximate position of the isosurface intersection is calculated, normally using trilinear interpolation

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08-01-29 | #26@ UU/IT

Isocontour vs interpolation

Remember bi-linear interpolation?

p2 p3

p0 p1

P = ?

p4 p5

To know the value of P, we can first compute p4 and p5 and then linearly interpolate P

The problem of extracting an isocontour is the inverse of value interpolation

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08-01-29 | #27@ UU/IT

Isocontour, cont’d

p2 p3

p0 p1

(2) Calculate the positions of P

(3) Connect the points with lines

We can solve the problem based on linear interpolation

(1) Identify edges that contain points P with value C, i.e., if pi < C < pj

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In how many ways can a contour intersect a 2D cell?

Two contours pass through the cell

In fact, there are only 4 unique topological cases

Contour completely outside cell

One vertex inside, three vertices outside (or vice versa)

Two vertices inside, two vertices outside

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Marching Squares

4 pixels in a 2 x 2 neighbourhood

8 of 16 possible configurations

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Divide-and-conquer Algorithm

• Look at one cell at a time

• Compare the values at 4 vertices with the isovalue C

• Linearly interpolate along the edges

• Connect the interpolated points together

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Boundary representations

Pixel edgesMarching squares

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Marching Cubes

A marching cube is bounded by the 8 voxels in a 2 x 2 x 2 neighbourhood

If object (value=1) and background (value=0) are considered, then there are 28 = 256 possible configurations of the 8 voxels

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Conversion of voxel data to triangulated surfaces

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MC algorithm

Step 1 – Create the cubes using the centres of 8 voxels; four from slice k four from slice k+1

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MC algorithm

Step 2 - Classify each vertex of the cube

1 = inside the object

0 = outside the object

Step 3 - Build an index

The classification gives an 8-bit index number (e.g., 01100111, 00110010) for each cube

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Cube example

Marching cube configuration 001100102 = 50, Case 5, 3 triangles

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MC algorithm

Step 4 –

Get the edge list

For a given index, access a list of cube edges (a lookup table) that contains triangle vertices

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MC algorithm

Step 5 - Interpolate the triangle verticesFor the cube edges, find the intersection points using (linear) interpolation of the density values

The midpoint of the cube edge may be chosen for the triangle vertex

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MC algorithm, final step

Step 6 - Calculate a unit normal for each triangle vertex Estimate a gradient vector G at each cube vertex using central differences of the density data D

Gx(i,j,k) = D(i+1, j, k) - D(i-1, j, k)Gy(i,j,k) = D(i, j+1, k) - D(i, j-1, k)Gz(i,j,k) = D(i, j, k+1) - D(i, j, k-1)

Interpolate the gradient at the intersection, i.e., the triangle vertex

Rendering algorithms use the normals to produce, e.g., Gouraud shading

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Marching Cubes -Algorithm in summary

1. Create a cube2. Classify each vertex3. Build an index4. Get edge list5. Interpolate triangle vertices6. Calculate and interpolate normals

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More cube examples

Case 1 Case 2 Case 8

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Non-boundary cubes

Completely outside object Completely inside object

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Tricky configurations

Case 6

5 triangles

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MC – Pros and cons

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•Straightforward to implement•Fast to generate•Generates many, many triangles •Does not always connect things in a correct way (ambiguities)

•The generated surfaces are not perfectly smooth

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MC – Pros and cons

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Voxel examples: sphere and cube

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MC examples: sphere and cube

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Vector algorithms

Vector data is a 3D representation of direction and magnitudeLike scalar fields, vectors are defined at discrete points

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Vector field visualization challenges

Displaying a vector requires more visual attributes

(u,v,w) has direction and magnitude

Displaying a vector requires more screen space

more than one pixel is required to display an arrow

It becomes more challenging to display a dense vector field

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Glyphs

•A general visualization technique used to represent data using a meaningful shape or pictorial representation

•Each glyph is generally a function of its input data and may change size, orientation, colour, and shape

•Example: oriented arrows, cones, etc.

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Animated particles

Advection of (massless) particles through the vector field

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Streamlines

•Curves that connect particle positions

•Start from one or more seed-points

IT Uppsala universitet

Data representation and fundamental algorithms

Filip Malmberg filip@cb.uu.se