TRIPTURES-TRITTURE

SOME REFERENCES ON THE RESEARCH CONDUCTED IN THE '90s ON PERSISTENT TOPOLOGY

Some researchers are interested in the historical development of persistent topology and in the results obtained in the '90s on this topic. As it is well known, persistent topology can be seen as a theory that allows us to measure by suitable distances how much the topological features of a filtered topological space persist when the considered filtration is perturbed by noise.

I compiled this list of references for the interested reader:

• The concept of size function (a.k.a. rank invariant or persistent Betti number function in degree 0) has been introduced in the papers [1,2,3,4,5]. Size functions for loops have been introduced in the papers [1,19].
  


• The representation of size functions by formal series (a.k.a. persistence diagrams) has been introduced in the papers [11,17,21].
  


• The comparison of persistence diagrams in degree 0 by means of a matching distance (a.k.a. bottleneck distance) has been introduced in the papers [11,20].
  


• A stability result for size functions has been illustrated in the papers [2] (Proposition 4.1), [8] (Proposition 1.1), [9] (Proposition 2.6), [10] (Theorem 3.2) and [17] (Theorem 3).
  


• The study of the link between persistent topology in degree 0 and mathematical morphology has been introduced in the papers [9,10]. These papers include some stability results.
  


• Methods for the computation and approximation up to a given error of persistent homology in degree 0 have been proposed in the papers [2,3,4,5,17,18].
  


• The transformation of persistence diagrams in degree 0 into real valued functions by replacing each point with a Gaussian function or another integrable function (an important step for the vectorization of persistence diagrams) has been introduced in the papers [14,15].
  


• The concept of size homotopy group (a.k.a. persistent homotopy group) has been introduced for ℝ^k-valued filtering functions in the paper [16], together with some stability results.
  


• The representation of size functions by complex polynomials has been introduced in the paper [22].
  


• The use of families of size functions associated with families of filtering functions has been introduced in the paper [7].
  


• The use of size functions to compare biological structures has been introduced in the paper [6].
  


• Size functions have been used for shape comparison from their very introduction (cf., e.g., [4,5,6]). However, the systematic use of size functions for pattern recognition and computer vision started in the papers [12,13,14,15].
  

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REFERENCES

1. Patrizio Frosini,
   A distance for similarity classes of submanifolds of a Euclidean space,
   Bulletin of the Australian Mathematical Society, 42, 3 (1990), 407-416.


2. Patrizio Frosini,
   Measuring shapes by size functions,
   Proc. of SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA 1607 (1991), 122-133.


3. Patrizio Frosini,
   Discrete computation of size functions,
   Journal of Combinatorics, Information & System Sciences, 17, 3-4 (1992), 232-250.


4. Claudio Uras, Alessandro Verri,
   Describing and recognizing shape through size functions,
   ICSI Technical Report TR-92-057 (1992).


5. Alessandro Verri, Claudio Uras, Patrizio Frosini, Massimo Ferri,
   On the use of size functions for shape analysis,
   Biological Cybernetics, 70, (1993), 99-107.


6. Massimo Ferri, Sandra Lombardini, Clemente Pallotti,
   Leukocyte classification by size functions,
   Proc. 2nd IEEE Workshop on Applications of Computer Vision, Sarasota, 1994 Dec. 5-7 (1994), 223-229.


7. Patrizio Frosini, Massimo Ferri,
   Range size functions,
   Proceedings of the SPIE's Workshop ``Vision Geometry III'', 2356 (1994), 243-251.


8. Patrizio Frosini,
   Connections between size functions and critical points,
   Mathematical Methods In The Applied Sciences, vol. 19 (1996), 555-569.


9. Claudia Landi, Patrizio Frosini,
   Connections between size functions and morphological transformations,
   Proc. SPIE Vol. 2785, p. 24-32, Vision Systems: New Image Processing Techniques, Philippe Refregier; Ed. (1996).


10. Patrizio Frosini, Claudia Landi,
    Size functions and morphological transformations,
    Acta Applicandae Mathematicae, vol. 49 (1) (1997), 85-104.


11. Claudia Landi, Patrizio Frosini,
    New pseudodistances for the size function space,
    Proc. SPIE Vol. 3168, p. 52-60, Vision Geometry VI, Robert A. Melter, Angela Y. Wu, Longin J. Latecki (eds.), 1997.


12. Claudio Uras, Alessandro Verri,
    Computing size functions from edge maps,
    International Journal of Computer Vision, 23 (1997), 169–183.


13. Costantino Collina, Massimo Ferri, Patrizio Frosini, Eleonora Porcellini,
    SketchUp: Towards qualitative shape data management,
    Proceedings Third Asian Conference on Computer Vision, Lecture Notes in Computer Science 1351, vol. I, R. Chin, T. Pong (editors) Springer-Verlag, Berlin Heidelberg (1998), 338-345.


14. Pietro Donatini, Patrizio Frosini, Alberto Lovato,
    Size functions for signature recognition,
    Proceedings of the SPIE's Workshop ``Vision Geometry VII'', 3454 (1998), 178-183. Please see also this web page.)


15. Massimo Ferri, Patrizio Frosini, Alberto Lovato, Chiara Zambelli,
    Point selection: A new comparison scheme for size functions (With an application to monogram recognition),
    Proceedings Third Asian Conference on Computer Vision, Lecture Notes in Computer Science 1351, vol. I, R. Chin, T. Pong (editors) Springer-Verlag, Berlin Heidelberg (1998), 329-337. (Please see also this web page.)


16. Patrizio Frosini, Michele Mulazzani,
    Size homotopy groups for computation of natural size distances,
    Bulletin of the Belgian Mathematical Society - Simon Stevin, 6 (1999), 455-464.


17. Patrizio Frosini, Claudia Landi,
    Size theory as a topological tool for computer vision,
    Pattern Recognition And Image Analysis, vol. 9 (4) (1999), 596-603.


18. Patrizio Frosini, Massimiliano Pittore,
    New methods for reducing size graphs,
    International Journal of Computer Mathematics, 70 (3) (1999), 505-517.


19. Patrizio Frosini,
    Metric homotopies,
    Atti del Seminario Matematico e Fisico dell'Università di Modena, XLVII, 271-292 (1999).


20. Pietro Donatini, Patrizio Frosini, Claudia Landi,
    Deformation energy for size functions,
    Proceedings Second International Workshop EMMCVPR'99, Lecture Notes in Computer Science 1654, E. R. Hancock, M. Pelillo (editors) Springer-Verlag, Berlin Heidelberg (1999), 44-53.


21. Claudia Landi, Patrizio Frosini,
    Algebraic representation of size functions ,
    Pattern Recognition and Image Understanding, 5th open German-Russian workshop, B. Radig et al. (eds.), Infix, 41-46, 1999.


22. Massimo Ferri, Claudia Landi,
    Representing size functions by complex polynomials ,
    Proc. Math. Met. in Pattern Recognition 9, Moskow, November 16-19, 1999.

For obvious copyright reasons not depending on us, the final versions of the previous papers cannot be made freely available on the Web. However, if you are a colleague interested in them for research purposes, you can download copies for your personal and internal use by clicking on these links (username: visitor ; password: 1234 ):
A distance for similarity classes of submanifolds of a Euclidean space
Measuring shapes by size functions
Discrete computation of size functions
Describing and recognising shape through size functions
On the use of size functions for shape analysis
Leukocyte classification by size functions
Range size functions
Connections between size functions and critical points
Connections between size functions and morphological transformations
Size functions and morphological transformations
New pseudodistances for the size function space
Computing size functions from edge maps
Sketchup Towards qualitative shape data management
Size functions for signature recognition
Point selection: A new comparison scheme for size functions
Size homotopy groups for computation of natural size distances
Size theory as a topological tool for computer vision
New methods for reducing size graphs
Metric homotopies
Deformation energy for size functions
Algebraic representation of size functions
Representing size functions by complex polynomials