sept20,*2013:**persistenthomology.***homepage.math.uiowa.edu/~idarcy/aptop/sept20persist.pdf ·...
TRANSCRIPT
![Page 1: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/1.jpg)
MATH:7450 (22M:305) Topics in Topology: Scien:fic and Engineering Applica:ons of Algebraic Topology
Sept 20, 2013: Persistent homology.
Fall 2013 course offered through the University of Iowa Division of Con:nuing Educa:on
Isabel K. Darcy, Department of Mathema:cs
Applied Mathema:cal and Computa:onal Sciences, University of Iowa
hSp://www.math.uiowa.edu/~idarcy/AppliedTopology.html
![Page 2: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/2.jpg)
H0 = < a, b, c, d : tc + td, tb + c, ta + tb> H1 = <z1, z2 : t z2, t3z1 + t2z2 >
[ ) [ ) [ ) [ ) [ z1 = ad + cd + t(bc) + t(ab), z2 = ac + t2bc + t2ab
![Page 3: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/3.jpg)
hSp://link.springer.com/ar:cle/10.1007%2Fs00454-‐004-‐1146-‐y
Hk = Zk / ( Bk Zk ) i, p i+p i i U
p-‐persistent kth homology group:
![Page 4: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/4.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à
![Page 5: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/5.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à
Ck = C0 C1 C2 C3 … + + + + +
![Page 6: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/6.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à
Ck = C0 C1 C2 + + +
![Page 7: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/7.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à a + b + bc + abc + acd
Ck = C0 C1 C2 + + +
![Page 8: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/8.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à a + b bc abc + acd
(a + b, bc, abc + acd) + +
Ck = C0 C1 C2 + + +
![Page 9: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/9.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à
Ck = C0 C1 C2 C3 … + + + + +
: Ck à Ck + +
(0, ab, 0) = (a + b, 0, 0) 2 = 0
![Page 10: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/10.jpg)
2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à
Ck = C0 C1 C2 C3 … + + + + +
(0, ab, 0) = (a + b, 0, 0)
(0 ab 0) = a+b 0 0 + + + +
: Ck à Ck + +
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2
1
C2 = Z2[abc, acd] C1 = Z2[ab, bc, ac, ad, cd] C0 = Z2[a, b, c, d]
à
à
à
3
0 0
à
Ck = C0 C1 C2 C3 … + + + + +
: Ck à Ck + +
![Page 12: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/12.jpg)
hSp://link.springer.com/ar:cle/10.1007%2Fs00454-‐004-‐1146-‐y
Hk = Zk / ( Bk Zk ) i, p i+p i i U
p-‐persistent kth homology group:
![Page 13: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/13.jpg)
Cj à Cj à Cj à Cj à Cj à Cj 0 1 2 3 4 5
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C0 à C0 à C0 à C0 à C0 à C0 0 1 2 3 4 5
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
(a, c + ta, tc + t2b, t2c, t4b, t5b, t6b, … )
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
(a, c + ta, tc + t2b, t2c, t4b, t5b, t6b, … )
t � (a, c + ta, tc + t2b, t2c, t4b, t5b, t6b, … ) = (0, ta, tc + t2a, t2c + t3b, t3c, t5b, t6b, t7b, … )
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H0 = < a, b, c, d : tc + td, tb + c, ta + tb> [ ) [ ) [
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
a: Z2[t] b: Z2[t] c: Σ1 Z2[t] d: Σ1 Z2[t]
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
a: Z2[t] b: Z2[t] c: Σ1 Z2[t] d: Σ1 Z2[t]
acd: Σ5 Z2[t]
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
a: Z2[t] = {n0 + n1t + n2t2 + … + nktk : ni in Z2, k in Z+}
= { (n0, n1, n2, …, nk, 0, 0, … ) : ni in Z2, k in Z+}
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
b: Z2[t] = {n0 + n1t + n2t2 + … + nktk : ni in Z2, k in Z+}
= { (n0, n1, n2, …, nk, 0, 0, … ) : ni in Z2, k in Z+}
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
c: Σ1Z2[t] = {n0t + n1t2 + … : ni in Z2}
= { (0, n0, n1, … ) : ni in Z2}
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
<a, b> <c, d, ta, tb> <tc, td, t2a, t2b> … + + + C0 = + i
d: Σ1Z2[t] = {n0t + n1t2 + … : ni in Z2}
= { (0, n0, n1, … ) : ni in Z2}
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H0 = < a, b, c, d : tc + td, tb + c, ta + tb> [ ) [ ) [
![Page 25: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/25.jpg)
H0 = < a, b, c, d : tc + td, tb + c, ta + tb>
[ ) [ ) [
a: Z2[t] = {n0 + n1t + n2t2 + … + nktk : ni in Z2, k in Z+}
= { (n0, n1, n2, …, nk, 0, 0, … ) : ni in Z2, k in Z+}
![Page 26: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/26.jpg)
H0 = < a, b, c, d : tc + td, tb + c, ta + tb>
[ ) [ ) [
Z2[t] = {n0 + n1t + n2t2 + … : ni in Z2}
b: Z2[t]/t = {n0 : n0 in Z2} = Z2
= { (n0, 0, 0, 0,… ) : ni in Z2}
![Page 27: Sept20,*2013:**Persistenthomology.***homepage.math.uiowa.edu/~idarcy/ApTop/Sept20Persist.pdf · hSp:// ... Applicaon*to*Natural*Image*Stas:cs* With*V.*de*Silva,*T.*Ishkanov,A.* Zomorodian*](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a88106c7f8b9ad30c8e36ba/html5/thumbnails/27.jpg)
H0 = < a, b, c, d : tc + td, tb + c, ta + tb>
[ ) [ ) [
Z2[t] = {n0 + n1t + n2t2 + … : ni in Z2}
d: Σ1 Z2[t]/t = Σ1 {n0 : n0 in Z2}
= { (0, n0, 0, 0, 0, …) : n0 in Z2}
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H0 = < a, b, c, d : tc + td, tb + c, ta + tb>
[ ) [ ) [
Z2[t] = {n0 + n1t + n2t2 + … : ni in Z2}
c: Σ1 Z2[t]/1 = { (0, 0, 0, … ) }
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C0 à C0 à C0 à C0 à C0 à C0 à… 0 1 2 3 4 5
a: Z2[t] = {n0 + n1t + n2t2 + … njtj : ni in Z2}
b: Z2[t]/t = {n0 : n0 in Z2} = Z2
c: Σ1 Z2[t]/1 = empty set
d: Σ1 Z2[t]/t = {n0 : n0 in Z2} = {(0, n0, 0, 0, 0, …)}
H0 = < a, b, c, d : tc + td, tb + c, ta + tb>
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H0 = < a, b, c, d : tc + td, tb + c, ta + tb> = Z2[t] Z2[t]/t (Σ1 Z2[t])/t [ ) [ ) [
+ +
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H0 = < a, b, c, d : tc + td, tb + c, ta + tb> H1 = <z1, z2 : t z2, t3z1 + t2z2 >
[ ) [ ) [ ) [ ) [ z1 = ad + cd + t(bc) + t(ab), z2 = ac + t2bc + t2ab
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H0 = < a, b, c, d : tc + td, tb + c, ta + tb> H1 = < z1, z2 : t z2, t3z1 + t2z2 >
[ ) [ ) [ ) (Σ2 Z2[t])/t3 (Σ3 Z2[t])/t [ ) [ z1 = ad + cd + t(bc) + t(ab), z2 = ac + t2bc + t2ab
+
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H1 = < z1, z2 : t z2, t3z1 + t2z2 >
Z2[t] = {n0 + n1t + n2t2 + n2t3 + n2t4 + … : ni in Z2}
z1: (Σ2 Z2[t])/t3
= { (0, 0, n0, n1, n2, 0, 0, 0,… ) : ni in Z2}
[ ) [ )
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H1 = < z1, z2 : t z2, t3z1 + t2z2 >
Z2[t] = {n0 + n1t + n2t2 + n2t3 + n2t4 + … : ni in Z2}
z2: (Σ3 Z2[t])/t
= { (0, 0, 0, n0, 0, 0, 0, … ) : ni in Z2}
[ ) [ )
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Hk = ( Σ i F[t] ) ( Σ j F[t]/(t j) ) + +
i = 1
n α + i = 1
n γ k
In general when calcula:ng homology over the field F
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hSp://www.ima.umn.edu/videos/?id=856 hSp://ima.umn.edu/2008-‐2009/ND6.15-‐26.09/ac:vi:es/Carlsson-‐Gunnar/imafive-‐handout4up.pdf
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hSp://www.ima.umn.edu/videos/?id=1846 hSp://www.ima.umn.edu/2011-‐2012/W3.26-‐30.12/ac:vi:es/Carlsson-‐Gunnar/imamachinefinal.pdf
Applica:on to Natural Image Sta:s:cs With 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.
Collec:on: 4.5 x 106 high contrast patches from a collec:on of images obtained by van Hateren and van der Schaaf
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