1GATE-540

Reconstruction from Voxels

(GATE-540)

Dr.Çağatay ÜNDEĞER

InstructorMiddle East Technical University, GameTechnologies

&

General ManagerSimBT Inc.

e-mail : cagatay@undeger.com

Game Technologies Program – Middle East Technical University – Spring 2010

Reference: William E. Lorensen and Harvey E. Cline

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Outline

• Reconstruction

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Goals• Develop 3D Analysis Algorithms:

– Reconstruction– Segmentation– Feature Detection– Labeling– Matching – Classification– Retrielval– Recognition– Clustering

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Voxel

• A voxel (volumetric pixel)– A volume element, representing a value on

a regular grid in three dimensional space. • Analogous to a pixel, which represents 2D

image data in a bitmap

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Voxel Data

• Do not typically have their position (their coordinates) explicitly encoded along with their values.

• Stores:– Binary data:

• empty / full– Float data:

• density / color / distance to surface

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Voxel vs Polygon

• Polygons:– Often explicitly represented by the

coordinates of their vertices. – Able to efficiently represent simple 3D

structures with lots of empty or homogeneously-filled space.

• Voxels:– Good at representing regularly-sampled

spaces that are non-homogeneously filled.– Have a limited resolution, as precise data

is only available at the center of each cell.

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Usage of Voxel• Frequently used in the visualization and

analysis of medical and scientific data. • Some volumetric displays use voxels to

visualize models and describe their resolution in 3D dimension (512×512×256 voxels).

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Usage of Voxel• In games and simulations,

– Used for representation of terrain containing overhangs and caves.

– Concave features cannot be represented by heightmaps.

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Visualizing Voxels

• Visualization:– Direct volume rendering– Extraction of polygon iso-surfaces which

follow the contours of given threshold values.

• The marching cubes algorithm is often used for iso-surface extraction.

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Iso-Surface

• A three-dimensional analog of an iso-contour.• A surface that represents points of a constant

value (e.g. pressure, temperature, velocity, density) within a volume of space.

İso-surface

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Marching Cubes

• Marching Cubes is an algorithm which “creates triangle models of constant density surfaces (iso-surfaces) from 3D medical data.”

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Medical Data Acquisition

• Computed Tomography (CT)• Magnetic Resonance (MR) Imagining (MRI)• Single-Photon Emission Computed Tomography

(SPECT)

• Each scanning process results in two dimensional “slices” of data.

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Data Slices

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Marching Cubes Extraction

• Extracts surfaces from adjacent pairs of data slices using cubes.

• Cubes “march” through the pair of slices until the entire surface of both slices has been examined.

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Marching Cubes Overview

• Load slices.• Create a cube from pixels on adjacent slices.• Find vertices on the surfaces.• Determine the intersection edges.• Interpolate the edge intersections.• Calculate vertex normals.• Output triangles and normals.

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How Are Cubes Constructed

• Uses identical squares of four pixels connected between adjacent slices.

• Each cube vertex is examined to see if it lies “on” or “off” the surface.

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How Are The Cubes Used

• Pixels on the slice surfaces determine 3D surfaces.

• 256 surface permutations, but only 15 unique patterns.

• A normal is calculated for each triangle vertex for rendering.

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15 Unique Patterns

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Triangle Creation

• Determine triangles contained by a cube.• Determine which cube edges are intersected.• Interpolate intersection point using pixel

density.• Calculate unit normals for each triangle vertex

using the gradient vector.

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Determining Triangles

• An index to a pre-calculated array of 256 possible polygon configurations (28 = 256) within the cube

• Treat each of the 8 scalar values (cube corners) as a bit in an 8-bit integer.

• 8 scalars (8 bits) determine the actual index to the polygon configuration array.

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Determining Inside/Outside

• Select a iso-value that the surface will pass through.

• If the scalar's value is higher than the iso-value then – The appropriate bit is set to one (inside)

• If it is lower then– The appropriate bit is set to zero (outside)

iso-value = 0.3

0.1 0.8

0.5 0.0

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Determining Intersections

• Determine intersection points to iso-suface by interpolation.

0.1 0.8

0.5

0.3

0.3

0.00.3

0.3iso-value = 0.3

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Determining Intersections

• Gradient of the scalar field at each grid point is also the normal vector of a hypothetical iso-surface passing from that point.

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Determining Intersections

• Interpolate these normals along the edges of each cube to find the normals of the generated vertices.

N

N

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Grid Resolution

• Variations can increase/decrease surface density.

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Examples

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Marching Squares

• 2D version of Marching Cubes• Aims at drawing lines between interpolated

values along the edges of a square considering given weights of the corners and a reference value.

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Conclusion

• Marching Squires / Marching Cubes provides a simple algorithm to translate a series of 2D medical scans into 2D / 3D objects.