Fourier shape descriptors

Leicester, S.E., Finney, J.L. and Bywater, R.P. (1994b). A Quantitative Representation of Molecular Surface Shape. 11 Protein Classification Using Fourier Shape Descriptors and Classical Scaling. J.Math.Chem., 16,343-365. 

For completeness, we mention that there are several other methodologies for the characterization of the shape of molecular surfaces from local geometric features. The use of Fourier shape descriptors is an interesting alternative, adapted recently for the analysis of macromolecular surfaces. 

Equation (2) defines the value of the size normalized mean radius of the particle. Equation (3) defines the size normalized sum of the squares of the Fourier coefficients. Equation (4) defines the sum and differences of the multiples. It has been shown that these size and shape descriptors can be used to regenerate the original particle profile. This Indicates that the descriptors together contain all of the size and shape Information contained In the original profile. 

A number of methods have been proposed for particle shape analysis, including shape coefficients, shape factors, verbal descriptions, curvature signatures, moment invariants, solid shape descriptors, and mathematical functions (Fourier series... 

A whole series of orthonormal functions can be used to interpret the information. The most familiar and applicable are the Fourier functions. Before being able to compose a particle shape descriptor in the polar system by the use of Fourier functions, one must realize that all that is normally known of a particle is its silhouette or profile. Therefore, methods must be found to interpret information from cuts through the particle or scans of portions of the surface area and connect it with overall shape. It is assumed that the silhouette of any cut or sample of the surface will give all information, such as roughness and other physical parameters, needed to describe the entire particle surface. Thus, unless the silhouette of a particle misses a unique, dominant feature of the particle shape, it will be representative of the particle. By sampling... 

A number of methods have been proposed for particle shape analysis these include verbal description, various shape coefficients and shape factors, curvature signatures, moment invariants, solid shape descriptors, the octal chain code and mathematical functions like Fourier series expansion or fractal dimensions. As in particle size analysis, here one can also detect intense preoccupation with very detailed and accurate description of particle shape, and yet efforts to relate the shape-describing parameters to powder bulk behaviour are relatively scarce.10... 

Descriptors for objects represented by their boundaries may be generated using the representations already described. Simple descriptors, such as perimeter, length, and orientation of the major axis, shape number, and eccentricity, may be readily computed from the boundary data. The ordered set of points on the boundary having two-dimensional coordinates (xt,yt), where 1 = 1,..., NandNis the total number of points on the boundary, can be treated as a one-dimensional complex function xt + iyk- The coefficients of the discrete Fourier transform applied to this function can also be used as a shape descriptor. 

Feature extraction process Evaluate each patch with texture and shape descriptor. As for texture we analyze feature of contrast, correlation, energy and homogeneity of each medical image. We also measure noise and blur for quality characteristics evaluation. For shape descriptor we use Hu moment invariant and Fourier transform analysis. Finally for color feature we used color histogram descriptor. 

Zhang, D. and Lu, G., A comparative study on shape retrieval using Fourier descriptors with different shape signatures , pp. 1— 9... 

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