Controlled-Topology Filtering Yotam Gingold & Denis Zorin - - PowerPoint PPT Presentation

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Controlled-Topology Filtering Yotam Gingold & Denis Zorin - - PowerPoint PPT Presentation

Apr 26, 2023 37 likes •799 views

Controlled-Topology Filtering Yotam Gingold & Denis Zorin Motivation Many applications require extraction of isolines & isosurfaces (contours) from scalar functions MRI, CT, terrain data, scientific computing Cow CT Scan Puget

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Controlled-Topology Filtering

Yotam Gingold & Denis Zorin

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Motivation

Many applications require extraction of isolines & isosurfaces (contours) from scalar functions

 MRI, CT, terrain data, scientific computing

Cow CT Scan Puget Sound

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Motivation

Want to filter (smooth, sharpen) all contours at once

 Downsampling  Noise reduction

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Contour-line topology

Number of contours at all isovalues

 In 2D fields, value can be height

Topology changes occur when value equals value of a critical point (min, max, saddle)

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Critical Points

Maxima and minima correspond to hills and valleys Saddles join two hills or valleys

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Features

Correspond to critical contours passing through saddles

 A saddle divides a contour in two  The interior of a contour containing an

unpaired extremum is a feature

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Topological Events

Features appear/disappear in saddle-min/max pairs

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Piecewise linear data

 Scalar values defined on a regular grid  Simplicial mesh guarantees critical points

  • nly at vertices

 Some complex critical points can be stable

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regular min max saddle (simple) saddle (monkey)

Piecewise linear data

Find critical points by comparing value with neighbors

 Break ties with arbitrary, consistent

perturbation

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PL topological events

Only happens when 2 adjacent vertices change relative height

 The edge flips relative to equality

1D

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PL topological events

Nothing Merge/Split Move

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Laplacian Smoothing

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Laplacian Smoothing

Can create features

as in blood vessels Ridge Bridge

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Ridge Bridge (t = 0.0)

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Ridge Bridge (3.5)

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Ridge Bridge (7.0)

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Ridge Bridge (10.5)

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Ridge Bridge (14.0)

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Ridge Bridge (17.5)

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Ridge Bridge (21.0)

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Ridge Bridge (24.5)

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Ridge Bridge (28.0)

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Ridge Bridge (28.0)

2/3

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Ridge Bridge (28.0)

3/3

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Puget Sound (3.5)

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Puget Sound (8.4)

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Puget Sound (13.3)

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Puget Sound (18.2)

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Puget Sound (23.1)

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Puget Sound (28.0)

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Puget Sound (32.9)

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Puget Sound (37.8)

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Puget Sound (42.7)

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Puget Sound (42.7)

2/3

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Puget Sound (42.7)

3/3

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Anisotropic Smoothing

 Can also create features

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Sharpening

 Don’t want to create or destroy features

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Algorithm

  • 1. Obtain proposed values from

the filter function

  • 2. Identify edge flips and sort in

time

  • 3. Detect and prevent disallowed

events

  • 4. Goto step 1
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  • 1. Obtain proposed values

Proposed values at step l+1 Filter function Step size Current values at step l

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  • 2. Identify flips and sort

Function values change linearly from to

 An edge between adjacent vertices flips at

most once

 Must resolve flip time exactly

Otherwise vertex ordering becomes cyclic Every vertex has a unique epsilon perturbation

value 0 t 1

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  • 3. Detect & prevent disallowed events

Process events in order

 If we reach a disallowed event between

vertices , set to values infinitesimally before the event

 Must re-identify events involving v,w  May need to rewind the event queue

value 0 t 1 0 t 1

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1D:

  • 3. Detect & prevent disallowed events

2D:

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Algorithm

  • 1. Obtain proposed values from

the filter function

  • 2. Identify edge flips and sort in

time

  • 3. Detect and prevent disallowed

events

  • 4. Goto step 1
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Results: Ridge Bridge (0.0)

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Results: Ridge Bridge (3.5)

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Results: Ridge Bridge (7.0)

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Results: Ridge Bridge (10.5)

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Results: Ridge Bridge (14.0)

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Results: Ridge Bridge (17.5)

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Results: Ridge Bridge (21.0)

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Results: Ridge Bridge (24.5)

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Results: Ridge Bridge (28.0)

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2/3

Results: Ridge Bridge (28.0)

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3/3

Results: Ridge Bridge (28.0)

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Results: Laplacian Smoothing

uncontrolled controlled

  • riginal

suppressed

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Results: Laplacian Smoothing

uncontrolled controlled

  • riginal

suppressed

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Results: Sharpening

uncontrolled controlled

  • riginal

suppressed

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Results: Sharpening

uncontrolled controlled

  • riginal

suppressed

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Results: Anisotropic Smoothing

uncontrolled controlled

  • riginal

suppressed

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Local artifacts

uncontrolled controlled

Due to slowing time

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No progress guarantee

Terminate when proposed values same as current Never saw topology control prevent progress everywhere

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 Measures the importance of a feature  Common measure is difference in value

between an extremum and its paired saddle

 We can track the time it takes for an extremum

(and its paired saddle) to be annihilated under smoothing

Results: Persistence

1D:

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Results: Persistence

Difference in value Anisotropic diffusion lifetime Cow CT Scan

Features shaded black

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Critical Points Over Time

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Performance

Performance related to number of edge flips and number of undesired topological events Roughly corresponds to number of critical points

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Performance

Laplacian smoothing Puget Sound to one global maximum is 2.2x slower 44% of the time is spent in the first 3 steps

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Future Work

Extend to 3D Predict disallowed events to distribute undesired artifacts

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Conclusion

A simple algorithm that controls topology changes when filtering

Contact: Yotam Gingold <gingold@mrl.nyu.edu> Denis Zorin <dzorin@mrl.nyu.edu> Thanks to: Chris Wu, Chee Yap, Adrian Secord, NYU CS Colleagues, and the reviewers

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Fin

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Filters

Discrete Laplacian smoothing (diffusion) Sharpening Discrete Anisotropic smoothing ([Perona and Malik 1990])

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Feature

Sweep a plane down. Notice how at a maximum a new contour is born! Notice how it merges with another contour at a saddle! Same with minimums and sweeping upwards!

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Feature

Hills and valleys until it gets complicated -- max and mins (pics from original slide)

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Feature

Sweep a plane down data set

 Maximum creates a contour  Minimum creates a contour  Saddle merges two contours  Talk to denis to get terminology consistent

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Feature

The set of contours from an extremum until the saddle which merges the contours with another extremum’s contours, as we sweep a plane upwards/downwards Single contour component from appearance at max/min until merger with another component at saddle as isovalue decreases/increases

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No guaranteed progress

Terminate when proposed values same as current Can topology control prevent progress everywhere? Locally:

uncontrolled controlled

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Critical Points Over Time