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PubMed1983

Classification of scattering objects by regular arrays of scatterers using Fourier analysis.

For a class of scattering objects the mean separation between the scatterers is determined. Over a range of frequencies, ultrasonic back-scattered pressure signals are recorded. It is suggested to derive the mean spacing of an unknown scattering object by employing Fourier analysis and comparing the signal with signals from known regular arrays of scatterers. In a computer simulation it is shown that this approach determines the mean spacing very well for random arrays consisting of three-dimensional spherical scatterers.

Fourier Analysis↗
PubMed1997

[Fourier analysis as a mathematical model for evaluating and presenting postoperative corneal topography data after non-mechanical perforating keratoplasty].

BACKGROUND: Videokeratography has given the possibility to obtain information in curvature from a much larger region of the cornea than that covered by keratometry. Fourier analysis as a mathematical model can be used to represent real physical attributes of the cornea and to divide corneal topography in its basic components: the zero-frequency component as the mean ring power, the one-cycle component as a representation of decentration and the two-cycle component as a representation of regular corneal toricity. The purpose of this study was the reconstruction of the corneal refraction after penetrating keratoplasty with a small number of characteristic parameters and the evaluation of the time course of the fourier coefficients as indices for a regular astigmatic cornea in the postkeratoplasty period including suture removal. PATIENTS AND METHODS: Fourty patients (group 1: 20 primary dystrophies, group 2: 20 keratoconus) underwent nonmechanical trephination (excimer laser MEL60, Aesculap-Meditec, Heroldsberg, Germany) in penetrating keratoplasty. All procedures (7.5 mm in dystrophies, 8.0 mm in keratoconus, 8 orientation teeth, double-running 10-0 nylon suture) were performed by one surgeon. At a postoperative gate of 6 weeks, 6 months, before partial suture removal and after complete suture removal, corneal topography (TMS1, Tomey, Tennenlohe, Germany), keratometry, visual acuity and subjective refraction were assessed. Radial approximation with a 5th order polynomial fit of the refractive data on 25 non-centric rings of the TMS, within 256 hemimeridians was performed to get data at equally spaced concentric rings. Fast Fourier transformation of the data sets in the mid periphery (1.4-1.8 mm apical distance) was done to get DC-, one-cycle and two-cycle component. Fourier coefficients were correlated with keratometric readings, subjective refractive values and visual acuity. RESULTS: Spherical equivalent was fairly constant in the postoperative interval before suture removal. After suture removal, a corneal flattening of about 3 diopters occurred. The one-cycle component before suture removal tended to be higher in group 1 compared to group 2 before suture removal. This difference was statistically significant (p = 0.01) after suture removal. Comparing keratometry, calculated meridians by the Tomey software, two-cycle component of the fourier decomposition and subjective refraction, the best correlation (p = 0.02) was observed between two-cycle-component and the refractive cylinder in amplitude and axis after suture removal. Best corrected visual acuity was inversely correlated with the amount of energy of higher harmonics compared to the whole energy before (p = 0.04) and after (p = 0.01) suture removal. CONCLUSIONS: Fourier transformation renders reconstruction of corneal topography data with a marked data reduction and a small error. From fourier coefficients conclusions may be drawn concerning potential best-corrected visual acuity and amplitude/axis of subjective cylinder, even in corneas with severe local irregularities.

Adult↗
PubMed2000

Elliptical fourier analysis of symphyseal shape in great ape mandibles.

The midsagittal profile of the mandibular symphysis has served as both a taxonomic marker and a phylogenetically salient character in debates over hominoid evolution. Nevertheless, the utility of symphyseal shape as an informative attribute for paleobiological reconstructions is suspect. Quantification of shape variation has proven to be particularly problematic; it has long been recognized that conventional linear measurements (and the indices derived from them), while replicable, summarize aspects of shape very poorly because of the vast amount of contour information that is lost in the process. In this study, a type of Fourier analysis is applied to cross-sectional contours of ape mandibles in order to provide a mathematical accounting of shape variation in a "global" sense; that is, by applying the "totality" of contour information in a comparative analysis. Shape variation in the mandibular symphysis is explored through the decomposition of coordinate data into elliptical Fourier coefficients. These coefficients are used to compute average taxonomic distances (ATD) among individuals of chimpanzees, gorillas, and orang-utans. The resulting shape-based distances are summarized via clustering (UPGMA) and ordination (principal coordinates analysis-PCO). Principal coordinate scores are subjected to analysis of variance in univariate and multivariate designs; these data are also applied to discriminant function analyses. Species and sex effects on morphology are statistically significant; however, no significant interaction of these factors is indicated. This would seem to imply that patterns of sexual dimorphism are not distinct among great apes; to the contrary, within-species sex comparisons reveal that significant size and shape dimorphism is present only in Gorilla. Despite significant size dimorphism in Pan and Pongo, significant shape differences between males and females are not apparent in these taxa. These results suggest that it is theoretically possible to sort taxa by a symphyseal shape criterion, but the discriminant function results suggest that there still exists a large potential for error in assigning particular shapes to a given species or sex. Thus, despite real shape differences among these species, the use of symphyseal shape as a character in species identification or in systematic arguments remains limited and problematic.

Analysis of Variance↗
PubMed1999

Size and shape of the human first permanent molar: a Fourier analysis of the occlusal and equatorial outlines.

Form can be viewed as a combination of size and shape. Shape refers to the boundary outline independently from its orientation, relation to reference planes, and dimension (or size). Shape and its changes could be quantified by mathematical methods such as the Fourier series. In this investigation, Fourier analysis has been used to quantify the morphologic characteristics (size and shape) of the outline of the occlusal surface and maximum circumference (equator) in 259 normal, healthy human first permanent maxillary and mandibular molars and to assess the effect of sex. Large within-group variability was found in the Fourier coefficients. Both equatorial and occlusal molar areas were on average larger in male than in female homologous teeth, but the difference was statistically significant only for the equatorial areas. The mean ratios between equatorial and occlusal dental areas were independent from arch (maxillary and mandibular), side, or sex. Both equatorial and occlusal outlines of left and right homologous molars within sex and arch were similar, without size and shape differences. Similarly, no sex differences in shape were found in the comparison of homologous teeth. The method used in the present study could supply information about dental shape in both its entirety and local variations. In particular, the method is extremely sensitive to local variations in dental shape, and it could be usefully employed to compare single teeth to a standard.

Adolescent↗
PubMed1989

Carapace growth of the turtle Chrysemys scripta: a longitudinal study of shape using Fourier analysis.

Because the majority of morphological structures encountered in Biology are irregular in form, conventional metrics composed of distances, angles and ratios, are inefficient shape descriptors. To circumvent this drawback in the application of morphometrics to describe 2 dimensional shapes, an alternative procedure based on Fourier analysis was developed and applied to the turtle carapace. Once size differences were controlled for, the presence of shape changes with age could be demonstrated. This study showed that small systematic differences in the phase angle are associated with increase in carapace asymmetry. The differences were not visually apparent in the original data. These results reinforce the need for precise shape descriptors that are capable of measuring a large percentage of the informational content that is present in all biological forms.

Age Factors↗
PubMed1999

[Fourier analysis of pupil oscillations for measuring central activation in psychosomatic patients].

The aim of the study was to answer the question if there are any differences in the central activation of different groups of psychosomatic patients and patients with eating disorders, which was measured by means of Fourier analysis of pupillary oscillations. A total of 132 patients (110 f, 22 m) with a mean age of 29.69 years (standard deviation: 9.9) participated in the study. In anorectic and bulimic patients high central activation was observed. Different groups of psychosomatic patients showed significant differences in their central nervous activation. In the group of subjects with the ICD-10 diagnosis F 41.3 (mixed anxiety disorders) the highest amplitudes was observed not only in the particular frequency bands but also in the total spectrum (power), which reflects high central activation. Reduced activation was found in subjects with somatoform autonomic function disorder of the upper and lower gastrointestinal tract (F 45.3). The measurement of central activation in psychosomatic disorders could have consequences for therapeutic interventions.

Adult↗
PubMed2006

Fourier analysis of single running suture adjustment in penetrating and deep lamellar keratoplasty.

PURPOSE: To prospectively evaluate clinical and visual results of running suture adjustment after keratoplasty. METHODS: Thirty-seven eyes of 33 patients received running suture adjustment after keratoplasty. Changes in uncorrected visual acuity, distance best spectacle-corrected visual acuity (BSCVA), refraction, and Fourier analysis constituents were assessed before and after suture adjustment. RESULTS: Thirty eyes, which had improvement of distance BSCVA soon after suture adjustment, maintained improvement at 1 year, whereas 7 eyes with decreased distance BSCVA soon after suture adjustment experienced no improvement at 1 year. Patients with poor distance BSCVA at 1 year had significantly higher high order irregularity than patients with better distance BSCVA despite similar cylindrical power outcomes.

Adult↗
PubMed1996

Shape of the human corpus callosum in childhood. Elliptic Fourier analysis on midsagittal magnetic resonance scans.

RATIONALE AND OBJECTIVES: The authors previously investigated the midsagittal magnetic resonance images of neurologically intact adults, and analyzed the shape of corpus callosum from a mathematic standpoint. Significant effects of age were demonstrated, without significant sex differences. In the current study, the same mathematic method was applied to analyze the sex and age shape differences of the human corpus callosum in childhood. METHODS: On the midsagittal magnetic resonance images of 84 neurologically intact children (39 boys, 45 girls, aged 4 months to 15 years) the outline of the corpus callosum was identified. Its shape was quantified using elliptic Fourier analysis, which allows for global evaluation of the shape of organs identified by their outlines independently from size, spatial orientation, and relationship to reference planes. Subjects were grouped by sex and age. RESULTS: The shape of the corpus callosum within age and sex classes in childhood was more homogeneous than in adulthood. A significant effect of age was demonstrated by the analysis of variance; however, no significant sex differences were found. CONCLUSIONS: Corpus callosum shape in midsagittal magnetic resonance images was strongly influenced by central nervous system development and aging, but the influence of sex was not sufficient to be detected by current standard magnetic resonance imaging technology, and by the available sample sizes. The current investigation supplies data on the normal callosal shape in the first two decades of life, thus completing previous analysis.

Adolescent↗
PubMed1988

Magnetic field shimming by Fourier analysis.

The homogeneity of the magnetic field of an Oxford 85/310HR magnet was optimized with the use of passive shims (i.e., pieces of steel) and room-temperature electronic shims. Both the characteristics of the passive shims and the current settings for the room-temperature electronic shims were calculated with the goal of minimizing field inhomogeneities as assessed by Fourier analysis of data from circular field plots. The measurement techniques, the methods of data analysis, and the techniques employed in correcting the magnetic field are presented. Difficulties and constructive approaches are suggested in these descriptions. The final results show a field of excellent homogeneity over a 10-cm-diameter spherical volume, with a peak-to-peak variation of 3.2 ppm, which is better than the manufacturer's expected specification of 5 ppm. A water-filled sphere of 10 cm diameter gave a half-height linewidth of 0.14 ppm after a small amount of additional touch-up shimming. The field homogeneity as corrected is much improved over its original state, allowing the acquisition of improved spectra from samples of a size several centimeters in each linear dimension. Thus the field homogeneity is more useful for in vivo spectroscopy, the purpose for which the magnet was intended.

Fourier Analysis↗
PubMed2000

[Cranial and mandibular morphology of the Indrinae. Contribution of Procrustes methods and Fourier analysis].

Although it is universally accepted that the Indridae family is monophyletic, there is no such unanimity about how the extant species of the Indrinae sub-family are related. Cranium and mandible disparity among Indri indri, Propithecus diadema, Propithecus verreauxi and Avahi laniger is analysed by geometrical morphometric methods and interpreted in adaptive terms. We compare results obtained by two techniques: Procrustes superimpositions for skulls and elliptical Fourier analysis for jawbones. In both studies, the three genera are clearly isolated. Avahi laniger is the species with the most paedomorphic cranium while Indri indri has the most 'juvenile' mandible. Cranial morphology is governed principally by orbit volume and body size, while jaw shape is related to diet (enlargement of the gonial region).

Animals↗
PubMed2002

Fourier analysis of trunk displacements: a method to identify the lame limb in trotting horses.

The aim of this paper is to present a method allowing the identification of the lame limb in trotting horses. Using a 3-D kinematic analysis system, 13 sound and 25 lame horses fitted with 4 skin markers placed on the dorsal midline of their trunk were recorded while trotting on a track in the conditions of the routine lameness examination. The vertical displacements of the trunk markers underwent Fourier analysis. Results indicated that these displacements could be represented using only the first and second harmonics. From these two harmonics, indices were then developed. The sensitivity of these indices to the different types of experimental errors was also studied. Results showed that the values of the indices of the lame horses were relatively unaffected by the experimental errors. In lame horses, these indices allowed the quantification of the degree of the lameness, the identification of lame limb with a reliability > 95% and the characterisation of the type of trunk movements. These indices could be easily implemented in a computer program to provide objective information to the clinician or to be used as a first step in the development of an expert system. Moreover, these clinical tools may also be extended to other quadrupedal or bipedal locomotions.

Algorithms↗
PubMed2007

Cerebral lateralisation for facial processing: gender-related cognitive styles determined using Fourier analysis of mean cerebral blood flow velocity in the middle cerebral arteries.

Facial processing was studied in 16 (eight men and eight women) right-handed healthy participants using a new functional transcranial Doppler technique called functional transcranial Doppler spectroscopy (fTCDS). MFV was recorded simultaneously in both right and left middle cerebral arteries in dark condition and during visual processing of object and facial tasks. fTCDS used Fourier analysis of mean flow velocity (MFV) time series to derive spectral density estimates that correlate with expected mental activity. Men were right lateralised for object and facial perception, while women were left lateralised for facial tasks but showed a right tendency or no lateralisation for object perception. For facial perception, men used a category-specific process-mapping system for right cognitive style, but women used same for the left.

Adult↗
PubMed1998

Foot asymmetry in healthy adults: elliptic fourier analysis of standardized footprints.

The size and shape of paired structures differ in the left and right sides of the body. Shape characteristics should be analyzed separately from size to supply information about the normal variations of human organs. In the present study, the within-subject normal symmetry of footprint shape and size was analyzed from a mathematical standpoint. On the standardized left and right footprints of 46 healthy adults (23 women and 23 men; 19-26 years old), the outline of each foot excluding the toes was identified and its shape was quantified independently from size by the elliptic Fourier analysis. The symmetry in shape was quantified on an intra-subject basis by calculating a morphologic distance D between the mathematical reconstructions of the left and right footprints of each subject. Symmetry in size was assessed by the right-to-left area ratio and found to be very good for all subjects; it ranged from 0.948 to 1.049. The subjects were also grouped by sex, and mean values were calculated. Within-subject symmetry in the footprint shape appeared high, with morphologic distances ranging between 8.94 and 2.66 in men and between 7.15 and 3.09 in women. No consistent associations between footprint symmetry and age, body height and weight, or shoe size were found. On average, women had more symmetric size-standardized footprints than did men (women: mean 4.57, SD 1.14; men: mean 5.46, SD 1.7; p < 0.05). Mean size-independent shapes of male and female left and right footprints were also calculated. Together with the analysis of individual asymmetry, they could be used for the quantitative diagnosis of borderline patients.

Adult↗
PubMed1990

Fourier analysis of the relation between the discharge of quadriceps motor units and periodic mechanical stimulation of cat knee joint receptors.

It has been widely assumed that joint receptors contribute to the reflex regulation of movement and posture, although there have been few clear demonstrations of joint-mediated reflex actions on alpha-motoneurones other than those attributable to flexor reflex afferents. The present study extends our recent work on joint-mediated reflexes by using Fourier analysis of spike train interactions to demonstrate that restricted mechanical activation of a small number (one to five) of knee joint receptors by localized indentation of the joint capsule can modify the firing pattern of quadriceps motoneurones in decerebrated cats. The modulation of this discharge pattern can be reversibly abolished by application of droplets of lignocaine directly to the joint nerve and consequently can be attributed unambiguously to articular afferents. Activation of single joint afferents could on occasion produce changes in motor unit firing patterns, but usually activation of two or more was required before secure effects were observed. Increasing the intensity of indentation (resulting in activation of greater numbers of joint receptors) increases the strength of coupling between joint afferent input and motor unit responses, although the relationship is not linear. The relation between the discharge patterns of pairs of motor units was also examined, and it was found that significant coupling occurred at the stimulus frequency, superimposed on a 'background' coupling from unidentified sources. The phase relationship between pairs of motor units was not affected by the periodic stimulus. However, it was observed that if two motor units were firing independently of one another in the absence of capsule probing, maintained capsular indentation produced a striking synchronization between the discharges of the two motor units. These experiments show strong joint-mediated reflex effects on motor unit firing indicating that joint receptors may have an important role to play in motor control.

Action Potentials↗
PubMed2003

Exploring artificial cranial deformation using elliptic Fourier analysis of Procrustes aligned outlines.

The anatomical effects of artificial cranial deformation on the face and the base have been subject to various metric approaches, including standard linear as well as finite element techniques, and have produced controversial results (Antón [1989] Am. J. Phys. Anthropol. 79:253-267; Kohn et al. [1993] Am. J. Phys. Anthropol. 90:147-158). It can be argued that diverging observations partly result from methodological constraints. The present study compares samples of intentionally deformed and undeformed human crania, using elliptic Fourier analysis (EFA), a morphometric approach which has been shown to be particularly appropriate for characterizing the shape of two-dimensional outlines and associated shape changes. We improve the standard EFA approach by adding a preliminary orientation of the outlines following the rotation parameters of a Procrustes superimposition, using multiple homologous landmarks called control points. The results confirm that circumferentially deformed skulls exhibit modifications of the basioccipital region, together with increased anterior and inferior facial projection. However, the degree to which basioccipital flattening is modified in circumferentially deformed Peruvians was found to be less marked than changes observed in the face. Some of the modifications observed here can be related to morphological trends existing in the population from which our sample was taken. The observation of other modifications may be subject to methodological constraints of standard morphometric approaches.

Adult↗