Sparse and Overcomplete Representation: Finding Statistical Orders in Natural Images

Transcription

1 Sprse nd Overcomplete Representton: Fndng Sttstcl Orders n Nturl Imges Amr Rez Sffr Azr Insttute for Theoretcl Computer Scence, Grz Unversty of Technology mr@g.tugrz.t

2 Outlne Vsul Cortex. Sprse nd Overcomplete Representton. Nonlner Herrchl Model for Modelng Hgher-order Structures. Dscusson.

3 Vsul Cortex How vsul cortex V1 represents mges? Vsul cortex s n herrchcl nference mchne. Envronment E, Observed dt D: D E, rors: E. Inference: E D D EE/Z.

4 Illuson

5 Illuson

6 Vsul Cortex ercepton s probblstc nference Helmoltz 1867/1962. Redundncy reducton Brlow Theoretcl models for retn Srnvsn et l nd LGN vn Hteren 1993 response propertes.

7 Vsul Cortex Antomcl convergence of bout 100 mllon photoreceptors onto 1 mllon gnglon cells. In ct V1, there re 25 tmes s mny output fbers s there re nput fbers from LGN. In mcque ths rto s on the order of 50:1. Incresed redundncy. Overcomplete representton.

8 Vsul Cortex A menngful representton cn be cheved by fndng code tht hs less ctve unts t ech tme. Sprse representton. Evdences for sprse ctvty n V1. The verge ctvty n prmte cortex s less thn 1 Hz Lenne 2003.

9 Lner Imge Model x + I x Φ ν x Codng Inference E E x I x Φ x 2 I x Φ x + x 2 β S Olshusen 2003

10 Lner Imge Model Olshusen 2002

11 robblstc Formulton + Φ I x x x ν 1 S S e Z 2 1 1,,, 2 N N S S e Z e Z Φ θ θ θ θ θ λ λ I I I I Olshusen 2002

12 Fndng Sprse Codes + Φ I x x x ν, rg mx ˆ θ I + Φ N S 2, log 2 I I λ θ & Olshusen 2003 x x x x x x j N j N j j j Φ Φ C I Φ b S C b λ λ τ & log1 2 σ γ S +

13 Neurl Interpretton Olshusen 2002, 2003

14 Lernng Bss Functons + Φ I x x x ν rg mx ˆ θ θ θ I T N d e Φ I Φ I I I ˆ log ˆ,, λ θ θ θ θ Δ Φ I e Olshusen 2002, 2003 α σ old new L g g Φ g

15 Results Olshusen 2002, 2003

16 Sprsfcton Olshusen 2002

17 Overcomplete Sprse Representton Olshusen 2005

18 Non-Sttonry Sttstcs n Nturl Imges Krkln nd Lewck 2005

19 robblstc Formulton x + I x Φ ν x N0, λ, q z exp λ q log λ Bv log B, v exp j B j v j q Krkln nd Lewck 2003

20 Exmple Krkln nd Lewck 2005

21 Herrchcl Structure Krkln nd Lewck 2005

22 Encodng Vrnce Coeffcents, rg mx, rg mx ˆ v v B B v v v v 0,1, r N v v v [ ] 1 1 sgn + j r j j j N q j j j v r v e u B q B v Bv & Krkln nd Lewck 2003

23 Lernng Vrnce Bss Functons Bˆ rg mx B B I, Φ rg mx B I Φ, B B log I Φ, B log B, vˆ vˆ det Φ ΔB j v j + v j q e q B [ Bv] j Krkln nd Lewck 2003

24 Results: Bss Functons Krkln nd Lewck 2003

25 Results: Bss Functons Krkln nd Lewck 2003

26 Results: Vrnce Bss Functons Krkln nd Lewck 2003

27 Results: Vrnce Bss Functons Krkln nd Lewck 2003

28 Results: Vrnce Bss Functons Krkln nd Lewck 2003

29 Results: Comprson of Bss Functons Krkln nd Lewck 2003

30 Results: Comprson of Bss Functons Krkln nd Lewck 2005

31 References Olshusen, B. A rncples of Imge Representton n Vsul Cortex, In: The Vsul Neuroscence, Eds: Chlup, L. M., Werner, J. S., MIT ress, Hyvärnen, A., Hoyer,. O., Hurr, J., Gutmnn, M Sttstcl models of mges nd erly vson, roceedngs of the Int. Symposum on Adptve Knowledge Representton nd Resonng AKRR2005, Espoo, Fnlnd. Lenne, The cost of cortcl computton, Curr. Bol. 13,

32 References Brlow, H , ossble prncples underlyng the trnsformtons of sensory messges, In: Sensory Communcton, Ed: W. A. Rosenblth, MIT ress, Srnvsn, M. V., Lughln, S. B., Dubs, A. 1982, redctve codng: fresh vew of nhbton n the retn, roc R Soc Lond, B, 216, vn Hteren 1993, Sptotemporl contrst senstvty of erly vson, Vson Reserch, 33,

33 References Olshusen, B. A. 2002, Sprse Codes nd Spkes, In: robblstc Models of the Brn: ercepton nd Neurl Functon, Eds: R.. N. Ro, B. A. Olshusen, M. S. Lewck, MIT ress, pp Olshusen B. A., Feld D. J. 1996, Emergence of Smple-Cell Receptve Feld ropertes by Lernng Sprse Code for Nturl Imges, Nture, 381, pp Olshusen B. A., Feld D. J How close re we to understndng V1? Neurl Computton, n press.

34 References Krkln Y., Lewck M. S., 2005 A herrchcl Byesn model for lernng non-lner sttstcl regulrtes n nonsttonry nturl sgnls, Neurl Computton, 17 2, pp Krkln Y., Lewck M. S., 2003 Lernng hgher-order structures n nturl mges, Network: Computton n Neurl Systems, 14, pp