Pattern recognition using invariants defined from higher order spectra : 2-D image inputs

A new algorithm for extracting features from images for object recognition is described. The algorithm uses higher order spectra to provide desirable invariance properties, to provide noise immunity, and to incorporate nonlinearity into the feature extraction procedure thereby allowing the use of simple classifiers. An image can be reduced to a set of 1D functions via the Radon transform, or alternatively, the Fourier transform of each 1D projection can be obtained from a radial slice of the 2D Fourier transform of the image according to the Fourier slice theorem. A triple product of Fourier coefficients, referred to as the deterministic bispectrum, is computed for each 1D function and is integrated along radial lines in bifrequency space. Phases of the integrated bispectra are shown to be translation- and scale-invariant. Rotation invariance is achieved by a regrouping of these invariants at a constant radius followed by a second stage of invariant extraction. Rotation invariance is thus converted to translation invariance in the second step. Results using synthetic and actual images show that isolated, compact clusters are formed in feature space. These clusters are linearly separable, indicating that the nonlinearity required in the mapping from the input space to the classification space is incorporated well into the feature extraction stage. The use of higher order spectra results in good noise immunity, as verified with synthetic and real images. Classification of images using the higher order spectra-based algorithm compares favorably to classification using the method of moment invariants

Shape discrimination using invariants defined from higher-order spectra

An approach to pattern recognition using invariant parameters based on higher-order spectra is presented. In particular, bispectral invariants are used to classify one-dimensional shapes. The bispectrum, which is translation invariant, is integrated along straight lines passing through the origin in bifrequency space. The phase of the integrated bispectrum is shown to be scale- and amplification-invariant. A minimal set of these invariants is selected as the feature vector for pattern classification. Pattern recognition using higher-order spectral invariants is fast, suited for parallel implementation, and works for signals corrupted by Gaussian noise. The classification technique is shown to distinguish two similar but different bolts given their one-dimensional profiles

A general procedure for the derivation of principal domains of higher-order spectra

A general procedure to determine the principal domain (i.e., nonredundant region of computation) of any higher-order spectrum is presented, using the bispectrum as an example. The procedure is then applied to derive the principal domain of the trispectrum of a real-valued, stationary time series. These results are easily extended to compute the principal domains of other higher-order spectra

Position, rotation, and scale invariant recognition of images using higher-order spectra

A new approach to recognition of images using invariant features based on higher-order spectra is presented. Higher-order spectra are translation invariant because translation produces linear phase shifts which cancel. Scale and amplification invariance are satisfied by the phase of the integral of a higher-order spectrum along a radial line in higher-order frequency space because the contour of integration maps onto itself and both the real and imaginary parts are affected equally by the transformation. Rotation invariance is introduced by deriving invariants from the Radon transform of the image and using the cyclic-shift invariance property of the discrete Fourier transform magnitude. Results on synthetic and actual images show isolated, compact clusters in feature space and high classification accuracies

Vibrational spectroscopic analysis of the mineral crandallite CaAl3(PO4)2(OH)5•(H2O) from the Jenolan Caves, Australia

The mineral crandallite CaAl3(PO4)2(OH)5•(H2O) has been identified in deposits found in the Jenolan Caves, New South Wales, Australia by using a combination of X-ray diffraction and Raman spectroscopic techniques. A comparison is made between the vibrational spectra of crandallite found in the Jenolan Caves and a standard crandallite. Raman and infrared bands are assigned to PO43- and HPO42- stretching and bending modes. The predominant features are the internal vibrations of the PO43 and HPO42- groups. A mechanism for the formation of crandallite is presented and the conditions for the formation are elucidated.

A Raman spectroscopic study of the mono-hydrogen phosphate mineral dorfmanite Na2(PO3OH)•2H2O – and in comparison with brushite

Aspects of the molecular structure of the mineral dorfmanite Na2(PO3OH)•2H2O were determined by Raman spectroscopy. The mineral originated from the Kedykverpakhk Mt., Lovozero, Kola Peninsula, Russia. Raman bands are assigned to the hydrogen phosphate units. The intense Raman band at 949 cm-1 and the less intense band at 866 cm-1 are assigned to the PO3 and POH stretching vibrations. Bands at 991, 1066 and 1141 cm-1 are assigned to the ν3 antisymmetric stretching modes. Raman bands at 393, 413 and 448 cm-1 and 514, 541 and 570 cm-1 are attributed to the ν2 and ν4 bending modes of the HPO4 units, respectively. Raman bands at 3373, 3443 and 3492 cm-1 are assigned to water stretching vibrations. POH stretching vibrations are identified by bands at 2904, 3080 and 3134 cm-1. Raman spectroscopy has proven very useful for the study of the structure of the mineral dorfmanite.

The mixed anion mineral parnauite Cu9[(OH)10|SO4|(AsO4)2]•7H2O – a Raman spectroscopic study

The mixed anion mineral parnauite Cu9[(OH)10|SO4|(AsO4)2].7H2O from two localities namely Cap Garonne Mine, Le Pradet, France and Majuba Hill mine, Pershing County, Nevada, USA has been studied by Raman spectroscopy. The Raman spectrum of the French sample is dominated by an intense band at 975 cm-1 assigned to the ν1 (SO4)2- symmetric stretching mode and Raman bands at 1077 and 1097 cm-1 may be attributed to the ν3 (SO4)2- antisymmetric stretching mode. Two Raman bands 1107 and 1126 cm-1 are assigned to carbonate CO32- symmetric stretching bands and confirms the presence of carbonate in the structure of parnauite. The comparatively sharp band for the Pershing County mineral at 976 cm-1 is assigned to the ν1 (SO4)2- symmetric stretching mode and a broad spectral profile centered upon 1097 cm-1 is attributed to the ν3 (SO4)2- antisymmetric stretching mode. Two intense bands for the Pershing County mineral at 851 and 810 cm-1 are assigned to the ν1 (AsO4)3- symmetric stretching and ν3 (AsO4)3- antisymmetric stretching modes. Two Raman bands for the French mineral observed at 725 and 777 cm-1 are attributed to the ν3 (AsO4)3- antisymmetric stretching mode. For the French mineral, a low intensity Raman band is observed at 869 cm-1 and is assigned to the ν1 (AsO4)3- symmetric stretching vibration. Chemical composition of parnauite remains open and the question may be raised is parnauite a solid solution of two or more minerals such as a copper hydroxy-arsenate and a copper hydroxy sulphate.

Pattern recognition using invariants defined from higher order spectra - one-dimensional inputs

A new approach to pattern recognition using invariant parameters based on higher order spectra is presented. In particular, invariant parameters derived from the bispectrum are used to classify one-dimensional shapes. The bispectrum, which is translation invariant, is integrated along straight lines passing through the origin in bifrequency space. The phase of the integrated bispectrum is shown to be scale and amplification invariant, as well. A minimal set of these invariants is selected as the feature vector for pattern classification, and a minimum distance classifier using a statistical distance measure is used to classify test patterns. The classification technique is shown to distinguish two similar, but different bolts given their one-dimensional profiles. Pattern recognition using higher order spectral invariants is fast, suited for parallel implementation, and has high immunity to additive Gaussian noise. Simulation results show very high classification accuracy, even for low signal-to-noise ratios.

Raman spectroscopy of the multi anion mineral arsentsumebite Pb2Cu(AsO4)(SO4)(OH) and in comparison with tsumebite Pb2Cu(PO4)(SO4)(OH)

The mineral arsentsumebite Pb2Cu(AsO4)(SO4)(OH), a copper arsenate-sulfate hydroxide of the brackebuschite group has been characterised by Raman spectroscopy. The brackebuschite mineral group are a series of monoclinic arsenates, phosphates and vanadates of the general formula A2B(XO4)(OH,H2O), where A may be Ba, Ca, Pb, Sr, while B may be Al, Cu2+,Fe2+, Fe3+, Mn2+, Mn3+, Zn and XO4 may be AsO4, PO4, SO4,VO4. Bands are assigned to the stretching and bending modes of SO42- AsO43- and HOAsO3 units. Raman spectroscopy readily distinguishes between the two minerals arsentsumebite and tsumebite. Raman bands attributed to arsenate are not observed in the Raman spectrum of tsumebite. Phosphate bands found in the Raman spectrum of tsumebite are not found in the Raman spectrum of arsentsumebite. Raman spectroscopy readily distinguishes the two minerals tsumebite and arsentsumebite.

Raman spectroscopy of the multi-anion mineral mallestigite Pb3Sb5+(SO4)(AsO4)(OH)6•3H2O – a mineral of archaeological significance

Some minerals are formed which show poorly defined X-ray diffraction patterns. Vibrational spectroscopy offers one of the few methods for the assessment of the structure of the oxyanions in such minerals. Among this group of minerals is mallestigite with formula Pb3Sb5+(SO4)(AsO4)(OH)6•3H2O. The objective of this research is to determine the molecular structure of the mineral mallestigite using vibrational spectroscopy. Raman and infrared bands are attributed to the AsO43- , SO42- and water stretching vibrations. Mallestigite is a mineral formed in ancient waste dumps such as occurs at Mallestiger, Carinthia, Austria and as such is a mineral of archaeological significance.

Higher-Order spectra of nonlinear polynomial models for Chua's circuit

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.

Vibrational spectroscopy of synthetic stercorite H(NH4)Na(PO4)•4H2O – a comparison with the natural cave mineral

In order to mimic the chemical reactions in cave systems, the analogue of the mineral stercorite H(NH4)Na(PO4)•4H2O has been synthesised. X-ray diffraction of the stercorite analogue matches the stercorite reference pattern. A comparison is made with the vibrational spectra of synthetic stercorite analogue and the natural Cave mineral. The mineral in nature is formed by the reaction of bat guano chemicals on calcite substrates. A single Raman band at 920 cm-1 (Cave) and 922 cm-1 (synthesised) defines the presence of hydrogen phosphate in the mineral. In the synthetic stercorite analogue, additional bands are observed and are attributed to the dihydrogen and phosphate anions. The vibrational spectra of synthetic stercorite only partly match that of the natural stercorite. It is suggested that natural stercorite is more pure than that of synthesised stercorite. Antisymmetric stretching bands are observed in the infrared spectrum at 1052, 1097, 1135 and 1173 cm-1. Raman spectroscopy shows the stercorite mineral is based upon the hydrogen phosphate anion and not the phosphate anion. Raman and infrared bands are found and assigned to PO43-, H2O, OH and NH stretching vibrations. Raman spectroscopy shows the synthetic analogue is similar to the natural mineral. A mechanism for the formation of stercorite is provided.

The molecular structure of the multianion mineral hidalgoite PbAl3(AsO4)(SO4)(OH)6 : implications for arsenic removal from soils

The objective of this research is to determine the molecular structure of the mineral hidalgoite PbAl3(AsO4)(SO4)(OH)6 using vibrational spectroscopy. The mineral is found in old mine sites. Observed bands are assigned to the stretching and bending vibrations of (SO4)2- and (AsO4)3- units, stretching and bending vibrations of hydrogen bonded (OH)- ions and Al3+-(O,OH) units. The approximate range of O-H...O hydrogen bond lengths is inferred from the Raman and infrared spectra. Values of 2.6989 Å, 2.7682 Å, 2.8659 Å were obtained. The formation of hidalgoite may offer a mechanism for the removal of arsenic from the environment.

Vibrational spectroscopic study of the copper silicate mineral kinoite Ca2Cu2Si3O10(OH)4

Kinoite Ca2Cu2Si3O10(OH)4 is a mineral named after a Jesuit missionary. Raman and infrared spectroscopy have been used to characterise the structure of the mineral. The Raman spectrum is characterised by an intense sharp band at 847 cm-1 assigned to the ν1 (A1g) symmetric stretching vibration. Intense sharp bands at 951, 994 and 1000 cm-1 are assigned to the ν3 (Eu, A2u, B1g) SiO4 antisymmetric stretching vibrations. Multiple ν2 SiO4 vibrational modes indicate strong distortion of the SiO4 tetrahedra. Multiple CaO and CuO stretching bands are observed. Raman spectroscopy confirmed by infrared spectroscopy clearly shows that hydroxyl units are involved in the kinoite structure. Based upon the infrared spectra, it is proposed that water is also involved in the kinoite structure. Based upon vibrational spectroscopy, the formula of kinoite is defined as Ca2Cu2Si3O10(OH)4•xH2O.

Infrared and Raman spectroscopic studies of archaeological materials

The use of vibrational spectroscopic techniques to characterise historical artefacts and art works continues to grow and to provide the archaeologist and art historian with significant information with which to understand the nature and activities of previous peoples and civilizations. In addition, conservators can gain knowledge of the composition of artworks or historical objects and so are better equipped to ensure their preservation. Both infrared and Raman have been widely used. Microspectroscopy is the preferred sampling technique as it requires only a very small sample, which often can be recovered. The use of synchrotron radiation in conjunction with IR microspectroscopy is increasing because of the substantial benefits in terms of improved spatial resolution and signal-to-noise ratio. The key trend for the future is the growth in the use of portable instruments, both IR and Raman, which are becoming important because they allow non-destructive measurements to be made in situ, for example at an archaeological site or at a museum.