1. http://link.springer.com/article/10.1007%2Fs00454-004-1146-y A filtered complex is an increasing sequence of simplicial complexes: C 0 C 1 C 2 … UUU

2. a, b is in C 0 C 1 C 2 … C 5 {a, b, c} is in C 4 C 5 000 UUU 22 U 0 U http://link.springer.com/article/10.1007%2Fs00454-004-1146-y A filtered complex is an increasing sequence of simplicial complexes: C 0 C 1 C 2 … UUU

3. Filtered complex from data points:

4. Filtered 1d-complex from data points:

5. Filtered Rips complex from data points:

6. A filtered complex is an increasing sequence of simplicial complexes: C 0 C 1 C 2 … UUU

7. Barcode for H 0 H 0 = Z 0 /B 0 = cycles boundaries

8. Barcode for H 1 H 1 = Z 1 /B 1 = cycles boundaries

9. Barcode for H 2 H 2 = Z 2 /B 2 = cycles boundaries

10. Computing Persistent Homology by Afra Zomorodian, Gunnar Carlsson http://link.springer.com/article/10.1007%2Fs00454-004-1146-y H k = Z k / ( B k Z k ) i, p i+pii U