Shape Based Image Retrieval Using Fourier Descriptors Dengsheng
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, guojun. lu@infotech. monash. edu. au 9/26/2020 Monash University 1
Outline Introduction n Shape Signatures n Fourier Descriptors n Retrieval Experiments n Conclusions n 9/26/2020 Monash University 2
Introduction-I --shape feature n What features can we get from a shape? shape perimeter, area, eccentricity, circularity, chaincode… 9/26/2020 Monash University 3
Introduction-II --Classification Shape Contour Structural Syntactic Graph Tree Model-driven Data-driven 9/26/2020 Region Non-Structural Perimeter Compactness Eccentricity Fourier Descriptors Wavelet Descriptors Curvature Scale Space Shape Signature Chain Code Hausdorff Distance Elastic Matching Monash University Area Euler Number Eccentricity Geometric Moments Zernike Moments Pseudo-Zernike Mmts Legendre Moments Grid Method 4
Introduction-III --criteria n Criteria for shape representation Rotation, scale and translation Invariant u Compact & easy to derive u Perceptual similarity u Robust to shape variations u Application Independent u FD satisfies all these criteria n Problem n u 9/26/2020 Different shape signatures are used to derive FD, which is the best? Monash University 5
Shape Signatures n n n n 9/26/2020 Complex Coordinates Central Distance Chordlength Curvature Cumulative Angles Area function Affine FD Monash University 6
Complex Coordinates z(t) = [x(t) – xc] + i[y(t) - yc] 9/26/2020 Monash University 7
Central Distance r(t) = ([x(t) – xc]2+ [y(t) - yc]2)1/2 9/26/2020 Monash University 8
Chordlength n The chord length function r*(t) is derived from shape boundary without using any reference point 9/26/2020 Monash University 9
Cumulative Angular Function n (t) = [ (t) - (0)]mod(2 ) L is the perimeter of the shape boundary 9/26/2020 Monash University 10
Curvature Function n K(t) = (t) - (t-1) w is the jumping step in selecting next pixel 9/26/2020 Monash University 11
Area Function 9/26/2020 Monash University 12
Fourier Descriptors n Fourier transform of the signature s(t) n un, n = 0, 1, …, N-1, are called FD denoted as FDn n Normalised FD Where m=N/2 for central distance, curvature and angular function m=N for complex coordinates 9/26/2020 Monash University 13
Affine Invariants k = 1, 2, … where Xk, Yk are the Fourier coefficients of x(t), y(t) respectively 9/26/2020 Monash University 14
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 9/26/2020 Monash University 15
Convergence Speed-II r(t) k(t) z(t) r*(t) 9/26/2020 Monash University 16
Convergence Speed-III • Ten very complex shapes are selected to simulate the worst convergence cases Signature functions 9/26/2020 Number of normalized spectra greater than 0. 1 r(t) 15 Number of normalized spectra greater than 0. 01 120 r*(t) 40 360 A(t) 20 210 z(t) 10 50 (t) 40 280 (t) k(t) 100 600 Qk 20 100 Monash University 17
FD Indexing n n 9/26/2020 Indexing each shape in the database with its Fourier Descriptors Similarity between a query shape and a target shape in the database is Monash University 18
Retrieval Experiments n n A database consisted of 2700 shapes is created from the contour shape database used in the development of 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. 9/26/2020 Monash University 19
Results 9/26/2020 Monash University 20
Conclusions n n n 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 9/26/2020 Monash University 21
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