Shape Based Image Retrieval Using Fourier Descriptors Dengsheng Zhang and Guojun Lu Gippsland School of Computing and Information Technology Monash University Churchill, Victoria 3842 Australia dengsheng.zhang,

Outline Introduction Shape Signatures Fourier Descriptors Retrieval Experiments Conclusions

Introduction-I --shape feature shape What features can we get from a shape? perimeter, area, eccentricity, circularity, chaincode…

Introduction-II --Classification Shape ContourRegion Structural Syntactic Graph Tree Model-driven Data-driven Perimeter Compactness Eccentricity Fourier Descriptors Wavelet Descriptors Curvature Scale Space Shape Signature Chain Code Hausdorff Distance Elastic Matching Non-Structural Area Euler Number Eccentricity Geometric Moments Zernike Moments Pseudo-Zernike Mmts Legendre Moments Grid Method

Introduction-III --criteria Criteria for shape representation  Rotation, scale and translation Invariant  Compact & easy to derive  Perceptual similarity  Robust to shape variations  Application Independent FD satisfies all these criteria Problem  Different shape signatures are used to derive FD, which is the best?

Shape Signatures Complex Coordinates Central Distance Chordlength Curvature Cumulative Angles Area function Affine FD

Complex Coordinates z(t) = [x(t) – x c ] + i[y(t) - y c ]

Central Distance r(t) = ([x(t) – x c ] 2 + [y(t) - y c ] 2 ) 1/2

Chordlength The chord length function r*(t) is derived from shape boundary without using any reference point

Cumulative Angular Function  (t) = [  (t) -  (0)]mod(2  ) L is the perimeter of the shape boundary

Curvature Function K(t) =  (t) -  (t-1) w is the jumping step in selecting next pixel

Area Function

Fourier Descriptors Fourier transform of the signature s(t) u n, n = 0, 1, …, N-1, are called FD denoted as FD n Normalised FD Where m=N/2 for central distance, curvature and angular function m=N for complex coordinates

Affine Invariants k =  1,  2, … where X k, Y k are the Fourier coefficients of x(t), y(t) respectively

Convergence Speed-I Finite number of coefficients are used to approximate the signal. The partial Fourier sum of degree n of u(t) is given by For piecewise smooth function u(t), there exists a one-to-one correspondence between u(t) and the limit of their Fourier series expansion For shape retrieval application, the number of coefficients to represent a shape should not be large, therefore, the convergence speed of the Fourier series derived from the signature function is crucial

Convergence Speed-II r(t)r(t)  (t) (t) r*(t)z(t)z(t) k(t)k(t)  (t) (t)

Convergence Speed-III Ten very complex shapes are selected to simulate the worst convergence cases Signature functions Number of normalized spectra greater than 0.1 Number of normalized spectra greater than 0.01 r(t)15120 r*(t)40360 A(t)20210 z(t)1050  (t)  (t)  k(t) QkQk 20100

FD Indexing Indexing each shape in the database with its Fourier Descriptors Similarity between a query shape and a target shape in the database is

Retrieval Experiments A database consisted of 2700 shapes is created from the contour shape database used in the development of MPEG-7. MPEG-7 contour shape database is consisted of set A, B and C. Set A has 421 shapes, set B has 1400 shapes which are generated from set A through scaling, affine transform and arbitrary deformation and defection. Set C has 1300 shapes, it is a database of marine fishes. Performance measurement: precision and recall Precision P is the ratio of the number of relevant retrieved shapes r to the total number of retrieved shapes n. Recall R is the ration of the number of relevant retrieved shapes r to the total number m of relevant shapes in the whole database.

Results

Conclusions A comparison has been made between FDs derived from different shape signatures, FDs with affine FDs In terms of overall performance, FDs derived from central distance outperforms all the other FDs Curvature and angular function are not suitable for shape signature to derive FDs due to slow convergence Affine FD is designed for polygon shape, it does not perform well on generic shape Indexing data structure will be studied in the future research Comparison with other shape descriptors