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http://www.ima.umn.edu/videos/?id=856http://ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafive-handout4up.pdf
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http://www.ima.umn.edu/videos/?id=1846http://www.ima.umn.edu/2011-2012/W3.26-30.12/activities/Carlsson-Gunnar/imamachinefinal.pdf
Application to Natural Image StatisticsWith V. de Silva, T. Ishkanov, A. Zomorodian
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An image taken by black and white digital camera can be viewed as a vector, with one coordinate for each pixel
Each pixel has a “gray scale” value, can be thought of as a real number (in reality, takes one of 255 values)
Typical camera uses tens of thousands of pixels, so images lie in a very high dimensional space, call it pixel space, P
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Lee-Mumford-Pedersen [LMP] study only high contrast patches.
Collection: 4.5 x 106 high contrast patches from acollection of images obtained by van Hateren and van der Schaaf
http://www.kyb.mpg.de/de/forschung/fg/bethgegroup/downloads/van-hateren-dataset.html
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Lee-Mumford-Pedersen [LMP] study only high contrast patches.
Collection: 4.5 x 106 high contrast patches from acollection of images obtained by van Hateren and van der Schaaf
Choose how to model your data
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Choose how to model your dataConsult previous methods.
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What to do if you are overwhelmed by the number of possible ways to model your data (or if you have no ideas):
Do what the experts do.
Borrow ideas. Use what others have done.
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Carlsson et al used
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Carlsson et al used
The majority of high-contrast optical patches are concentrated around a 2-dimensional C1 submanifold embedded in the 7-dimensional sphere.
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0.) Start by adding 0-dimensional data points
Persistent Homology: Create the Rips complex
is a point in S7
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For each fixed , e create Rips complex from the data
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are close
is a point in S7
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For each fixed , e create Rips complex from the data
2.) Add all possible simplices of dimensional > 1.
is a point in S7
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For each fixed , e create Rips complex from the dataIn reality used Witness complex (see later slides).
2.) Add all possible simplices of dimensional > 1.
is a point in S7
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Probe the data
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Probe the data
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Can use function on data to probe the data
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Large values of k:measuring density of large neighborhoods of x, Smaller values mean we are using smaller neighborhoods
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Eurographics Symposium on Point-Based Graphics (2004)Topological estimation using witness complexesVin de Silva and Gunnar Carlsson
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Eurographics Symposium on Point-Based Graphics (2004)Topological estimation using witness complexesVin de Silva and Gunnar Carlsson
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From:http://plus.maths.org/content/imaging-maths-inside-klein-bottle
From: http://www.math.osu.edu/~fiedorowicz.1/math655/Klein2.html
Klein Bottle
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M(100, 10) U Qwhere |Q| = 30
On the Local Behavior of Spaces of Natural Images, Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian, International Journal of Computer Vision 2008, pp 1-12.
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http://www.maths.ed.ac.uk/~aar/papers/ghristeat.pdf
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http://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=112392
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Combine your analysis with other tools
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