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    FAST RETRIEVAL METHO DS FOR IMAGES WITH SIGNIFICAN T VARIATIONS

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    Paul W Fieguth and Riyin Wan

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    Department of Systems Design EngineeringUniversity of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    ABSTRACTFast image retrieval is the key to success for operations onlarge image databases, and a great many techniques havebeen developed for efficient retrieval. However, most ofthese methods are tailored to visual scenes or to imageshaving limited variations. In this paper we investigate thesearching of enormous databases (of up to l o 7 images) forthe matching and identification of precious stones (princi-pally diamonds). Because of the size of the database, wepropose a hierarchy of classifiers, which successively prunecandidate images such that the more complex classifiers arerequired to test only tiny portions of the data. The newclassifier developed here applies a wavelet transform to im-age histograms and is capable of rejecting 99.9% of badmatches.

    1. INTRODUCTIONDiamonds are small and expensive! They look essentiallyalike, except for variations in color, mass, and shape. Di-amond thefts are relatively common and among the mostinfamous of crimes, so law enforcement authorities are veryinterested in a mechanism to determine the identity (that is,whether stolen or not) of a given diamond.America it is estimated that there are more than two mil-lion. Because of the small dimensions and few colors, wecannot rely on these attributes to differentiate all diamonds,however it has been found that the diamond signature(Fig-ure 1) produced under red laser illumination is very nearlyunique. The FBI has already used such images as the basisfor diamond identification, however at present the process isintolerably slow. Furthermore the differences between dia-mond image signatures can be subtle, so the matching accu-racy of standard image-database algorithms is either unac-ceptable in terms of reliability, or in terms of computationaltime. Existing diamond matching algorithms, performing asequential search through the database, would require aboutone month of time.In standard image-database systems, images taken at dif-ferent times, by different people, and using different cam-eras need to be compared. These images may have dif-

    The numberof diamonds is huge and increasing; in North

    0-7803-6297-7/OO/$l0.00 0 2000 IEEE 522

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    Fig. 1. Two red-laser images from the same diamond; thetwo images are very similar, except for a rotation of the di-amond.ferent noise levels or be captured from different perspec-tives; a perfect matching is no longer possible, nor expected.If the variations are small, many techniques can be used.However, if the variations and random noise are significant,most known algorithms cannot be extended. Furthermore,because images cant be sorted into ascending or descend-ing orders, brute-force sequential searches are undertaken

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    Radial Histogram of Figure 1(a) Radial Histogram of Figure 1(b) Radial Histogram of Different GemFig. 2. A comparison of three radial histograms: the first two from the same diamond, the third from a different one. Althoughthe first two histograms are not identical, they share basic similarities different from the third.- bviously problematic for very large sets of images.

    This paper proposes an effective method to match di-amond images, despite significant noise and image varia-tions. We propose a coarse-to-fine matching, to succes-sively strip away images from the database, leading to anextremely fast result.

    2. PROBLEM FORMULATIONGiven a diamond image we need to find all images (if any)in the database which correspond to the same diamond. Thecentre of each image is known, so matching images possessonly an unknown relative rotation, no translation. Each im-age is binary, 512 x 512 pixels, where the spots are foundbased on a predetermined thresholding.

    We have no prior knowledge regarding how images fromdifferent gems should differ, however we can establish mod-els for the variations of spots in images takes from the samegem. Generally, the spot radii vary only very slightly (astandard deviation on the order of 5 pixels), however thenumber of spurious, mismatching points can be as high as40% (38% of the points mismatch between the two imagesin Figure 1).Obviously directly storing and matching 51 2 x 512 pixelimages is inefficient; rather we can use the spot centers torepresent each image. The image matching problem is thustransformed into the matching of two point sets in two-dimensional space. The point-set matching problem has

    been well-studied in the graph theory literaturer4, 51, sothere are many algorithms which could deal with the un -known angular offset between images, however the largefraction of spot mismatches, the enormous number of im-ages in the database, and the required classification speedlead to additional challenges.

    3. EXACT MATCHINGPoint matching is generally computationally expensive. Themajority of time is required to identify matching pairs orto find the rotational alignment. Chang et al. [5 ] use a 2-D cluster approach to find the optimal matching, havinga complexity of O(nm),where n and m are the numbersof points in the two respective images. Mount et al. [4]search the possible transformation space based on branch-and-bound approach. They all conclude their algorithmsare efficient, which is true in general, although they are im-practical for problems of the scale considered here becausethe general algorithms do not take advantage of the uniqueproperties of diamond images.We have developed algorithms specialized to the dia-mond matching problem. Because the unknown parameteris only the rotation about the known origin, polar coordi-nates are a convenient and efficient representation of thespot centres, since a problem involving rotation is trans-formed into a more efficient one of translation. This first al-gorithm performs an exact matching (n o missed detectionsbased on our database):

    e Find the sets of point features PI and P2 of the twoimages to be matched; each point feature is a spotcenter.Choose the geometric center of the image as the ori-gin and represent the feature points using polar coor-dinates:

    e SortPI and P2 as a function of radius r .

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    e Estimate the rotation A0 by brute force. Let C(A0)denote the number of point pairs such that( f l i , 01i) M ( ~ 2 j 7 0 2 j- A @ ) (3 )

    where a matching pair is subject to thresholds rr inradius and r e in angle (3 pixel and 2 degrees respec-tively), and where A0 is sampled in increments:A8 = k * 2 n / 3 6 0 , 15 k 5 36 0 (4)

    e Find the optimum rotation angleAgqt = arg,, maxC(A8) (5)

    e Based on A0,,,, find the percentage M % of match-ing points between the two images:> 60 Same Diamond< 40 Different DiamondM = { 40 - 0 Never Observed (6)

    For images in which the matching percentage between40%and 60%, it is hard reach any conclusion, however none ofthe images in our database ever have a matching percentagein this range; that is, our same-image and different-imageclusters are very well separated, and errors are expected tobe rare. Although this exact algorithm is faster than moregeneral approaches, its computational effort is on the orderof many milliseconds, which is still inadequate for a hugesearch through millions of images, which motivates an ap-proximate classifier, presented next.4. APPROXIMATE MATCHING

    The obvious problem with the previous algorithm is that itattempts exact matching. Instead, one or two appropriately-chosen crude features should allow grossly different dia-monds to be discriminated with only a very minimal effort,leaving only a tiny fraction to be evaluated using more so-phisticated approaches. Because the images from the samediamond possess a random angle, sensible features shouldhave a radial dependence only. In principle, the two-dimen-sional point-matching problem can be reduced to the com-parison of one-dimensional radial histograms (shown in Fig-ure 2). Certainly the details of the histograms will vary,but the coarse-scale radial distribution of points will be verysimilar for images of a given diamond.There are, however, two crucial criteria that any succes-sive matching algorithm has to satisfy:1. We must never have missed detections

    Pr{Class ifj DifferentlSame Diamond] = 0, (7)since an image, once rejected, will never be reconsid-ered by a later classifier.

    2. Secondly, to be useful a crude classifier should rejectlarge fraction of mismatching images:Pr{Classify SamelDifferent Diamond] = E 7 Images definitely differentdl2 < T Defer judgment to finer classifier

    The effectiveness of this algorithm - a rejection rate of over99.9%, coupled with a rejection of about 99.8% based onphysical criteria, implies that only a tiny fraction remains tobe tested using an exact test.

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    #Im ages to AnalyzeM 10000000M 500000

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    Physical Criterion-CaratsRadial Histogram Matching M 833332D Image Pattern MatchingPhysical Criterion-Colour

    5. EXPERIMENTAL RESULTS

    Complexity Rejection RateO( 0-6)s 86.7%O(10-6)s 95%O(10-4)s 99.9%O(10-)s 100%

    The full database matching approach proceeds hierarchi-cally, as illustrated in Figure 3, based on a test with severalhundred images. The fastest, and also most approximate,operations are applied first, stripping the most obvious mis-matches from the database. Obviously the ID histogrammatching problem itself admits numerous algorithms, withdifferent complexity and accuracy. Figure 4 shows the ef-fect of wavelet DB3 with a level 3 approximation: the twoclasses (same and different gems) are well separated, andthe rejection rate is over 99%. Because the wavelet coef-ficients can be computed ahead of time and saved in thedatabase, and the comparison of the wavelet coefficient vec-tors is very fast, therefore the total searching time is reducedto only a few minutes.6. REFERENCES

    [11 M. J. Swain and D. H. Ballard, Color indexing, Inc. J.Computer Usion, 7(1):11-32, 1991.[2] J. Lee and B. W. Dickinson, Multiresolution videoindexing for subband coded video databases, Pmc .SPIE, V. 2185, San Jose, CA, Feb. 1994.[3] M. K. Mandal, T. Aboulnasr, and S. Panchanathan,Image indexing using moments and wavelets, IEEE

    Trans. Consumer Electronics, 42(3):557-565, 1996.[4] D. M. Mount, N. S. Netanyahu and Jacqueline L.Moigne, Efficient algorithms for robust feature match-ing, Pattem Recognition,32( 1):17-38, Jan. 1999[5 ] S . H. Chang, E H. Cheng, W. H. Hsu and G. Z. Wu,Fast algorihtm for point matching: Invariant to trans-lations, rotations and scale changes, Pattem Recogni-

    tion, 30(2):311-320,1997.[6] M. Antonini, M. Barlaud, I? Mathieu and I. Daubechies,Image coding using wavelet transform, IEEE Trans.

    Image Proc., 1(2):205-219, 1992.

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    (a) PDFs of Two ClustersROC FTER DENOlSlNGUSING DE3AT LEVEL 3

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    Fig. 4. Performance of Using DB3 at Level 3: (a) distribu-tions of the difference d lz for identical (left) and different(right) gems; (b) the false-alarm / detection graph, illustrat-ing the high rejection ratio.

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