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Neighborhood sequences for comparing similarity vectors in image retrieval

Neighborhood sequences for comparing similarity vectors in image retrieval. András Hajdu, Tamás Tóth, Krisztán Veréb. 22 August , 200 6. Neighborhood sequences (basic concepts and notation). n  N; q, r  Z n ; m  Z, with 0  m  n; Pr i (q) is the i- th coordinate of q .

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Neighborhood sequences for comparing similarity vectors in image retrieval

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  1. Neighborhood sequences for comparing similarity vectors in imageretrieval András Hajdu, Tamás Tóth, Krisztán Veréb 22August, 2006

  2. Neighborhood sequences (basic concepts and notation) • n  N; q, r  Zn; m  Z, with 0  m  n;Pri(q) is the i-th coordinate of q. • q and r are m-neighbors if |Pri(q) - Pri(r)| ≤ 1, 1 ≤ i ≤ n, and |Pr1(q)-Pr1(r)| + …+ |Prn(q)-Prn(r)| ≤ m. • B is an nD-neighborhoodsequence if B(bi)i  Nwith 1 ≤ bi≤ n, for i  N-re • q = q0 q1 … qj = r is an B-path of length j if qi-1and qiare b(i)-neighbors. • d(q, r; B) is the length of the shortest B-path

  3. Measuring distance in RGB cube by 3D neighborhood sequences • 24bit RGB cube, between black = (0, 0, 0) and white = (255, 255, 255). • Extendable to other color representation and higher dimensions. • Distance values highly depend on the chosen sequence: d(C1,C2;{1})=360, d(C1,C2;{2})=180, d(C1,C2;{3})=120, d(C1,C3;{1})=240, d(C1,C3;{2})=180, d(C1,C3;{3})=180, d(C2,C3;{1})=300, d(C2,C3;{2})=150, d(C2,C3;{3})=150.

  4. Periodic neighborhood sequences • nD neighborhood sequence:B = (Bi)∞i=1 • If for some jN, Bi = Bi+j for all i N, then B is called periodic with period j. • The brief notation of periodic neighborhood sequences:B = B1B2…Bj • If j = 1 then N is called a constant neighborhood sequence: {1} ≡ L1 {3} ≡ L∞, 3D case

  5. „Ultimately periodic” neighborhood sequences • Let B(j) denote the neighborhood sequence obtained by omitting the first j elements of B periodic neighborhood sequence. • The sequence B = (Bi)∞i=1 is called ultimately periodic, if B(k) is periodic for some k N. • If B(k) has period length l–k, then we writeB = B1B2…BkBk+1…Bl

  6. CNS – Classic Neighborhood Sequences • B1, B2, B3 based on the well known 6-, 18-, 26- neighborhoods. • The number in the index shows how many coordinates may change (1-, 2-, 3-neighborhood). • There is no restriction about which coordinate should change, in which direction we can make a move. • B1 = {(0, 0, ±1), (0, ±1, 0), (±1, 0, 0)},B2 = B1 {(0, ±1, ±1), (±1, 0, ±1), (±1, ±1, 0)},B3 = B2 {(±1, ±1, ±1)}. • Naturally algorithms are presented for calculating the distance.

  7. SNS – „Subspace” Neighborhood Sequences • It is explicitly given in which direction we can make a move. • Bx= {(±1, 0, 0)}, By = {(0, ±1, 0)}, Bz = {(0, 0, ±1)},Bxy= {(±1, ±1, 0)}, Byz = {(0, ±1, ±1)}, Bxz = {(±1, 0, ±1)},Bxyz = {(±1, ±1, ±1)}. • Numbers can be used against the names of the axis’s in applications: x  4, y  2, z  1; in case of more axis's these numbers are added. • Distance measurement is trivial.

  8. MNS – „Mixed” Neighborhood Sequences • Classic (CNS) and Explicit (SNS) neighborhood sequences can be combined. • Usable neighborhoods are: B1, B2, B3, Bx, By, Bz, Bxy, Byz, Bxz, Bxyz • We allow both explicit (SNS) and classic neighborhood movements too. • The distance measurement is the hybrid of the methods of CNS and SNS distance measurement.

  9. Disadvantages of SNS • It is not sure that there is finite distance between two points if we use SNS and we do not change every coordinates directly: e.g.: {Bxz} periodic sequent never reaches the point at (0, 1, 0). • Different sequences may give the same result ifempty steps can be taken (if we could notconverge to the goal the length of the waydo not increase):e.g.: t = (3, 1, 3) d(t) = xz, xz, xz, xz, xz, xz, xz, xz, 3 = d(t) = xz, xz, xz, 3 = 4

  10. Empty step • There are three different possibilities to handle empty steps: • Empty step has none-zero weight (this is our main approach). We may hold our position for some cost at every step. • Empty step has zero weight. It means that we can decide to ignore any such elements of neighborhood sequences that cannot help us to reach a given point. The sequence can be permuted freely then. • We do not let empty steps neither for some cost nor for free, and force to move to some other position at every step.

  11. Generative grammar • The suitable neighborhood sequences used in applications may be too long.e.g.: we want to allow 10 B3 steps after 100 B1 step. • G = <VN, VT, S,H> • VT={N1,N2,N3,Nx,Ny,Nz,Nxy,Nxz,Nyz,Nxyz} • S→ {Ak}S → SSA → SA → aa [1…7]  {a, b, c} k  N  {∞}

  12. Generative grammar - examples • The previous neighborhood sequent with grammar notation: {a100}{c10} • The meaning of the {23}{{45}{71}2} formula given by grammar is: {222 444447 444447} • {a3}{{51}{21}∞} means: {aaa5252525252…} ≡ {aaa52}

  13. Oracle – Image Database • Storing of multimedia data (video, sound, image) is a real task of database managing systems. • Oracle developed a visual information retrieval (VIR) tool, which is standard feature of Oracle since 9i. • The feature vector contains the followings: • Color histogram, • Texture descriptor, • Shape representation, • Their location information • The similarity value derivation is built on the matching values of these properties.

  14. Oracle – feature vectors • Location by itself is not a meaningful search parameter, so we won’t consider location as a feature. • A weight value between 0.0 and 1.0 can be assigned to every feature. This value denotes the relative ”importance” of the feature. • 0.0  not considered at all. • 1.0  highest importance • at the end of the comparison, we have three real values between 0.0 and 100.0 (c = color, t = texture, s = shape), and three normalized weights (w1, w2, w3), where w1+w2+w3 = 1. • The final distance is:d = w1 * c + w2 * t + w3 * s

  15. Experimental results • Database of 1240 distinct images. • Generating Oracle feature vectors (both three feature has same importance) : comparing every image to each other. • The build matrix will be processed. • Looking for images which are similar to the seed image.

  16. Experimental results • Used sequence: {Nxz4}{Ny40} 23 32 37 38 44 45 45 46

  17. Experimental results • Used sequence: {Nx4}{Nz4}{Ny30} 27 36 39 41 44 45 49 49

  18. Experimental results • Used sequence: {Nz4}{Nx4}{Ny30} 36 37 39 41 45 46 47 48

  19. Experimental results • Numerical comparison: id1 id2color shape texture860 850 83.6478 57.8826 2.4824860 851 6.16674 40.7421 2.2384860 852 82.1175 27.6878 1.6104860 853 9.20443 29.4861 1.396860 854 3.97565 28.4126 2.0864860 855 5.33467 26.8312 2.1816860 856 21.5916 33.8737 4.268860 857 2.51894 32.7586 1.598860 858 4.38217 33.5518 3.864860 859 3.01489 18.5306 2.9624

  20. Conclusions • Applicability in arbitrary finite dimensions. • Adaptation in other (non database) applications. • More consideration for digitalizing. • Assigning non-negative weights to each step.

  21. References • A. Fazekas, A. Hajdu, L. Hajdu: Lattice of generalized sequences in nD and ∞D, Publ. Math. Derbecen 60 (2002), 405-427. • A. Hajdu, B. Nagy, Z. Zörgő, “Indexing and segmenting colour images using neighbourhood sequences”, IEEE ICIP 2003, Barcelona, Spain, pp. I/957-960 • A. Hajdu, T. Tóth, K. Veréb and Z. Zörgő: Distance functions in multidimensional image processing applications, KÉPAF 4 (2004), Miskolc-Tapolca. • M.D. Levine, and A.M.Nazif: Dynamic measurement of computer generated imagesegmentations, IEEE trans. PAMI 7, pp. 155-164, 1985.

  22. Thank you for your attention

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