Fourier descriptors

Korany et al. [28] used Fourier descriptors for the spectrophotometric identification of miconazole and 11 different benzenoid compounds. Fourier descriptor values computed from spectrophotometric measurements were used to compute a purity index. The Fourier descriptors calculated for a set of absorbencies are independent of concentration and is sensitive to the presence of interferents. Such condition was proven by calculating the Fourier descriptor for pure and degraded benzylpenicillin. Absorbance data were measured and recorded for miconazole and for all the 11 compounds. The calculated Fourier descriptor value for these compounds showed significant discrimination between them. Moreover, the reproducibility of the Fourier descriptors was tested by measurement over several successive days and the relative standard deviation obtained was less than 2%. 

Fourier descriptors have been found valuable for the description of rather smooth contours such as the ones of somatic embryos [85-87]. The contour signature defined as R(s) =f(s) where R(s) is the distance between the centroid G and a point of the contour (Fig. 15), of normalized curvilinear abscissa s with respect to an origin O, is holomorph and can be written as ... 

Leicester, S.E., Finney, J.L. and Bywater, R.P. (1988). Description of Molecular Surface Shape Using Fourier Descriptors. J.Mol.Graphics, 6,104-108. 

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

The next challenge is the bin sizes. However, the Fourier approach also comes in handy here. Consider if we construct the Fourier descriptor by applying the following steps ... 

Fig. 7. Illustrations of the dip Fourier descriptors of different subvolumes. We can easily see how different textures are illustrated by different descriptor curves.

Fig. 8. Illustrations of the dip Fourier descriptors before and after rotation of the subvolumes. The dotted curves represent the descriptors before rotation and the solid curves after. We see that only minor changes can be observed, probably due to numeric effects in the rotation.

M. Shridhar, A. Badreldin, High Accuracy Character Recognition Algorithm Using Fourier and Topological Descriptors , Pattern Recognition 17, (1984), 515-524,... 

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... 

For the example in Fig. 2, the Fourier transformed NMR spectra (variables or descriptors being intensity as a function of frequency) were utilized for the creation of the data matrix D. It should be noted that many different descriptors can be used to create D, with the descriptor selection depending on the analysis method and the information to be extracted. For example, in the spectral resolution methods (Section 6), the desired end result is the determination of the true or pure component spectra and relative concentrations present within the samples or mixtures [Eq. (4)]. For this case, the unmodified real spectra Ij co) are commonly used for the chemometric analysis. In contrast, for the non-supervised and supervised methods described in Sections 3 and 4, the classification of a sample into different categories is the desired outcome. For these types of non-supervised and supervised methods the original NMR spectrum can manipulated or transformed to produce new descriptors including... 

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... 

Before we have a quick look at three of the most important transform methods, we should keep the following in mind. The mathematical theory of transformations is usually related to continuous phenomena for instance, Fourier transform is more exactly described as continuous Fourier transform (CFT). Experimental descriptors, such as signals resulting from instrumental analysis, as well as calculated artificial descriptors require an analysis on basis of discrete intervals. Transformations applied to such descriptors are usually indicated by the term discrete, such as the discrete Fourier transform (DFT). Similarly, efficient algorithms for computing those discrete transforms are typically indicated by the term fast, such as fast Fourier transform (FFT). We will focus in the following on the practical application — that is, on discrete transforms and fast transform algorithms. 

Fourier transform is a well-known technique for signal analysis, which breaks down a signal into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a mathematical technique for transforming a descriptor from the spatial domain into a frequency domain. The theory of Fourier transforms is described in several textbooks, such as Boas [53], and is not discussed here in detail. 

Fourier analysis of descriptors is performed by the DFT, which is the sum of the descriptor g r) over all distances r multiplied by a complex exponential. With a descriptor consisting of n components expressed in its discrete form g[x] x is the index of a discrete component), the DFT can be written as... 

The complex exponential can be broken down into real and imaginary sinusoidal components. The results of the transform are the Fourier coefficients g[u] (or g[u,v]) in frequency space. Multiplying the coefficients with a sinusoid of frequency yields the constituent sinusoidal components of the original descriptor. 

Moreover, molecular descriptors different from Free-Wilson descriptors were calculated by transformation of the Free-Wilson matrix through Fourier analysis [Flolik and Halamek, 2002], In this case, Fourier analysis is used to change site- and substituent-oriented binary variables into a few real numbers [Holik and Halamek, 2002]. 

Transformations of a set of molecular descriptors are often performed when there is the need of a —> variable reduction or the need to modify binary vectors, such as site and substituent-oriented variables, into real-valued variable vectors. The milestone of these techniques is the —> Principal Component Analysis (PCA), but also —> Fourier analysis and —> Wavelet analysis are often used, especially for spectra descriptors compression. 

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... 

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. 

S. Leicester, J. Finney, and R. Bywater, /. Math. Chem., 16, 315 (1994). A Quantitative Representation of Molecular Surface Shape. I. Theory and Development of the Method. S. Leicester, J. Finney, and R. Bywater, /. Math. Chem., 16, 343 (1994). A Quantitative Representation of Molecular Surface Shape. II. Protein Classification Using Fourier Shape Descriptors and Classical Scaling. 

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