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As screwed joints, dental restorations may suffer mechanical failures such as screw loosening and implant or prosthetic screw failure due to fatigue. This work is focused on the failure of the implant and develops a numerical methodology to predict its fatigue life under cyclic loading conditions. This methodology is based on the combination of Critical Plane Methods and the Theory of Critical Distances to account for stress multiaxiality and notch effects. The obtained predictions were validated experimentally, which can be used to identify the main geometrical, assembly and operational factors affecting the fatigue behavior of dental implants. As a result, a powerful and efficient design tool for fatigue life prediction of dental implants is presented. This methodology complements a previously presented one focused on the fatigue life prediction of the prosthetic screw, thereby, offering now a complete design tool package regardless the critical component of the dental restoration, predicting accurately the fatigue response of the restoration, with no need for long-term fatigue test campaigns. This is a pioneering work since no other fatigue design methodology for dental implants with such a solid foundation and experimental validation has been published to date., Competing Interests: The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: The research group reports financial support was provided by the University of the Basque Country through the project LTC AENIGME [COLAB19/04] and 10.13039/501100003086Basque Government, Spain [grant number IT1542-22]. Mikel Armentia reports a relationship with BTI ImasD S.L. that includes: employment. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

The aim of this study was to explore and assess office-based ultrasound-guided intraosseous and intra-articular infiltrations of plasma rich in growth factors (PRGF) in patients with moderate and severe knee osteoarthritis (KOA). Seventy-nine patients (30 women and 49 men) with grade 3-4 KOA according to the Kellgren-Lawrence classification participated in the study. All patients were treated with a minimally invasive technique using local anesthesia WALANT (wide-awake local anesthesia no tourniquet) in the ambulatory setting. A PRGF intra-articular infiltration and two intraosseous infiltrations in the tibial plateau and femoral condyle were performed weekly for a total of three sessions. The evaluation of the results was carried out using knee injury and osteoarthritis outcome score (KOOS) at baseline and post-treatment. After a follow-up period of 11 months (median) [interquartile range, 7-14], all the KOOS domains showed statistically significant improvement ( p < 0.001). Moreover, 88% of the patients showed a pain reduction of at least 10 points (minimally clinically important improvement) from pre- to post-treatment. Our retrospective study using the in-office procedure of ultrasound-guided combination of intra-articular and intraosseous infiltrations of PRGF is a safe and efficacious approach for the treatment of grade 3-4 knee osteoarthritis.

Background: Nerve compression syndromes of the upper extremity are a common cause of neuropathic pain and functional impairment. Recently, platelet-rich plasma (PRP) infiltrations have emerged as an effective biological approach to the treatment of this type of injury. The objectives of this retrospective observational study were to assess clinical improvement in patients with median and ulnar nerve entrapment syndrome after undergoing biologically-assisted nerve release surgery with plasma-rich-in-growth-factors (PRGF) technology. Methods: Participants (n = 39) with moderate-to-severe nerve compression syndrome of the upper limb diagnosed by both electromyography and clinical examination, and who were treated with PRGF, were identified from the center’s medical records. The evaluation was based on patient-reported outcomes. Pre- and post-treatment differences in the Visual analog scale (VAS), the Boston carpal tunnel questionnaire (BCTQ), and the Quick-DASH score were assessed. Results: Three study groups were conducted: patients with carpal tunnel syndrome (n = 16), with recurrent carpal tunnel syndrome (n = 8), and with ulnar nerve entrapment (n = 15). The median follow-up was 12 months (interquartile range (IQR), 9−16). In comparison to pre-treatment values, all three study groups obtained statistically significant improvements for the three analyzed scales at the end of the follow-up, with p < 0.001 for all scales in the carpal tunnel syndrome and ulnar nerve entrapment groups and p < 0.01 for all scales in the recurrent carpal tunnel syndrome group. There were no serious adverse effects in the analyzed patients. Conclusion: PRGF-assisted open surgical nerve release treatment (intraneural and perineural liquid PRGF infiltrations and nerve wrapping with PRGF membrane) exerts long-term beneficial effects on pain reduction and functional improvement in the nerve and nerve−muscle unit in patients with upper extremity compression syndromes., Competing Interests: The authors declare that E.A. is the Scientific Director of and S.P. and R.P. are scientists at the BTI Biotechnology Institute, a biomedical company that investigates in the fields of regenerative medicine and PRGF-Endoret technology. The rest of the authors state that they have no conflict of interest.

Both invertebrates and vertebrates possess a cluster of immediate and local wound-sealing, pathogen-killing, and tissue healing responses known as immunoclotting and immunothrombosis, respectively, to cope with two life-threatening emergencies, namely, bleeding and microbial invasion. Despite their convergence in function, immunoclotting and immunothrombosis are deployed by different blood cells and intravascular multidomain proteins. In vertebrates, these proteins share some domains with intrinsic chemical affinities useful in generating cooperative networks such as pathogen and damage pattern recognition molecules. Moreover, many of the proteins involved in coagulation and fibrinolysis in humans are multifunctional molecules playing roles in other processes from inflammation to healing and beyond. In our modern society, however, the interaction of activated intravascular allosteric proteins with one another and with blood cells entails vulnerabilities posing a biological paradox: intravascular proteins that locally operate as tissue repair enhancers can nevertheless generate pathogenic processes by acting systemically. In this manuscript, we contextualize and frame the coagulation system and hemostasis through an evolutionary time scale, illustrating their role as dual players in the defense against exsanguination and pathogens while significantly influencing wound healing.

Platelet-rich plasma (PRP) is nowadays used in the treatment of different types of cutaneous lesions. However, different compositions can influence clinical outcomes. Among them, the inclusion of leukocytes is controversial. High-throughput proteomics techniques were used to analyze the proteins that are differentially expressed in human dermal fibroblasts (HDFs) after treatment for 24 h with two PRP types, autologous topical serum (Endoret serum-ES) derived from plasma rich in growth factors (PRGF) and leukocyte- and platelet-rich plasma (L-PRP). The identified proteins were then classified by both Gene Ontology and Ingenuity Pathway Analysis. The obtained results show that the compositions of ES and L-PRP differ in such a way that they induce different responses in HDFs. ES-treated HDFs overexpress growth factor-related proteins, leading to protein synthesis, cell proliferation and migration. By contrast, L-PRP treatment induces a response similar to that caused by proinflammatory molecules. These data could explain the contradictory clinical results obtained for the different types of PRP, especially with respect to their leukocyte contents.

Ocular surface health, the cognitive status, psychological health or human neurological disorders, among others, can be assessed by studying eye blinking, which can be differentiated in spontaneous, reflex and voluntary. Its diagnostic potential has provided a great number of works that evaluate their characteristics and variations depending on the subject's condition (sex, tiredness, health, …). The objective of this study was to analyse the differences in blinking kinematics of spontaneous and reflex blinks, distinguishing between direct and consensual reflexes, using a self-developed, non-invasive and image processing-based method. A video-oculography system is proposed using an air jet driven by a syringe to induce reflex and a high-speed camera to record the blinking of both eyes. The light intensity diffused by the eye changes during blinking and peaks when the eyelid closes. Sixty-second sequences were recorded of 25 subjects blinking. Intensity curves were off-line fitted to an exponentially modified Gaussian (EMG) function, whose σ , μ and τ parameters were analysed. A two-way analysis of variance (ANOVA) of these parameters was conducted to test the influence of the subject, the eye and blink type. In the closing phase, direct and consensual corneal reflexes are faster than spontaneous blinking, but there was no significant difference between them, nor between right and left eyes. In the opening phase, the direct corneal reflex was the slowest and significant differences appeared between right and left eyes., Competing Interests: We declare we have no competing interests, (© 2020 The Authors.)

A known technique to obtain subpixel resolution by using object tracking through cross-correlation consists of interpolating the obtained correlation function and then refining peak location. Although the technique provides accurate results, peak location is usually biased toward the closest integer coordinate. This effect is known as the peak-locking error and it strongly limits this calculation technique's experimental accuracy. This error may differ depending on the scene and algorithm used to fit and interpolate the correlation peak, but in general, it may be attributed to a sampling problem and the presence of aliasing. Many studies in the literature analyze this effect in the Fourier domain. Here, we propose an alternative analysis on the spatial domain. According to our interpretation, the peak-locking error may be produced by a non-symmetrical sample distribution, thus provoking a bias in the result. According to this, the peak interpolant function, the size of the local domain and low-pass filters play a relevant role in diminishing the error. Our study explores these effects on different samples taken from the DIC Challenge database, and the results show that, in general, peak fitting with a Gaussian function on a relatively large domain provides the most accurate results.

The evaluation of eye blinking has been used for the diagnosis of neurological disorders and fatigue. Despite the extensive literature, no objective method has been found to analyze its kinematic and dynamic behavior. A non-contact technique based on the high-speed recording of the light reflected by the eyelid in the blinking process and the off-line processing of the sequence is presented. It allows for objectively determining the start and end of a blink, besides obtaining different physical magnitudes: position, speed, eyelid acceleration as well as the power, work and mechanical impulse developed by the muscles involved in the physiological process. The parameters derived from these magnitudes provide a unique set of features that can be used to biometric authentication. This possibility has been tested with a limited number of subjects with a correct identification rate of up to 99.7%, thus showing the potential application of the method.

The in-situ measurement of thermal stress in beams or continuous welded rails may prevent structural anomalies such as buckling. This study proposed a non-contact monitoring/inspection approach based on the use of a smartphone and a computer vision algorithm to estimate the vibrating characteristics of beams subjected to thermal stress. It is hypothesized that the vibration of a beam can be captured using a smartphone operating at frame rates higher than conventional 30 Hz, and the first few natural frequencies of the beam can be extracted using a computer vision algorithm. In this study, the first mode of vibration was considered and compared to the information obtained with a conventional accelerometer attached to the two structures investigated, namely a thin beam and a thick beam. The results show excellent agreement between the conventional contact method and the non-contact sensing approach proposed here. In the future, these findings may be used to develop a monitoring/inspection smartphone application to assess the axial stress of slender structures, to predict the neutral temperature of continuous welded rails, or to prevent thermal buckling., Competing Interests: The authors declare no conflict of interest.

Purpose: To assess anterior corneal surface stability 12 months following hyperopic LASIK correction with a light propagation algorithm., Setting: Vissum Instituto Oftalmológico de Alicante, Universidad Miguel Hernández, Alicante, Spain., Methods: This retrospective consecutive observational study includes 37 eyes of 37 patients treated with 6th-generation excimer laser platform (Schwind Amaris). Hyperopic LASIK was performed in all of them by the same surgeon (JLA) and completed 12-month follow-up. Corneal topography was analyzed with a light propagation algorithm, to assess the stability of the corneal outcomes along one year of follow-up., Results: Between three and twelve months postoperatively, an objective corneal power (OCP) regression of 0.39 D and 0.41 D was found for 6 mm and 9 mm central corneal zone, respectively. Subjective outcomes at the end of the follow-up period were as follows: 65% of eyes had spherical equivalent within ±0.50 D. 70% of eyes had an uncorrected distance visual acuity 20/20 or better. 86% of eyes had the same or better corrected distance visual acuity. In terms of stability, 0.14 D of regression was found. No statistically significant differences were found for all the study parameters evaluated at different postoperative moments over the 12-month period., Conclusions: Light propagation analysis confirms corneal surface stability following modern hyperopic LASIK with a 6th-generation excimer laser technology over a 12-month period.

Digital Image Correlation (DIC) is a common tool for assessing the movement of objects in a scene. Among others, one of the most popular techniques consists of tracking a dotted texture imitating speckle patterns. In this work, we analyzed the individual dots that form this pattern in order to propose an optimum size, shape, and dynamic range that allows minimizing the tracking error. Tracking was accomplished by using normalized cross-correlation with peak interpolation in order to obtain subpixel accuracy. For the models here used, we show that dot radii of 30-40 px with 150 gray levels are enough to obtain an accurate subpixel tracking resolution. Also, we show that 0.002 px is the performance limit of this technique, being this limit in accordance with the experimentally achievable subpixel limit., Competing Interests: The authors declare no conflict of interest.

Object tracking with subpixel accuracy is of fundamental importance in many fields since it provides optimal performance at relatively low cost. Although there are many theoretical proposals that lead to resolution increments of several orders of magnitude, in practice this resolution is limited by the imaging systems. In this paper we propose and demonstrate through simple numerical models a realistic limit for subpixel accuracy. The final result is that maximum achievable resolution enhancement is connected with the dynamic range of the image, i.e., the detection limit is 1/2 ∧ (nr.bits). The results here presented may aid in proper design of superresolution experiments in microscopy, surveillance, defense, and other fields.

We present a targetless motion tracking method for detecting planar movements with subpixel accuracy. This method is based on the computation and tracking of the intersection of two nonparallel straight-line segments in the image of a moving object in a scene. The method is simple and easy to implement because no complex structures have to be detected. It has been tested and validated using a lab experiment consisting of a vibrating object that was recorded with a high-speed camera working at 1000 fps. We managed to track displacements with an accuracy of hundredths of pixel or even of thousandths of pixel in the case of tracking harmonic vibrations. The method is widely applicable because it can be used for distance measuring amplitude and frequency of vibrations with a vision system.

Background: The pupillary light reflex characterizes the direct and consensual response of the eye to the perceived brightness of a stimulus. It has been used as indicator of both neurological and optic nerve pathologies. As with other eye reflexes, this reflex constitutes an almost instantaneous movement and is linked to activation of the same midbrain area. The latency of the pupillary light reflex is around 200 ms, although the literature also indicates that the fastest eye reflexes last 20 ms. Therefore, a system with sufficiently high spatial and temporal resolutions is required for accurate assessment. In this study, we analyzed the pupillary light reflex to determine whether any small discrepancy exists between the direct and consensual responses, and to ascertain whether any other eye reflex occurs before the pupillary light reflex., Methods: We constructed a binocular video-oculography system two high-speed cameras that simultaneously focused on both eyes. This was then employed to assess the direct and consensual responses of each eye using our own algorithm based on Circular Hough Transform to detect and track the pupil. Time parameters describing the pupillary light reflex were obtained from the radius time-variation. Eight healthy subjects (4 women, 4 men, aged 24-45) participated in this experiment., Results: Our system, which has a resolution of 15 microns and 4 ms, obtained time parameters describing the pupillary light reflex that were similar to those reported in previous studies, with no significant differences between direct and consensual reflexes. Moreover, it revealed an incomplete reflex blink and an upward eye movement at around 100 ms that may correspond to Bell's phenomenon., Conclusions: Direct and consensual pupillary responses do not any significant temporal differences. The system and method described here could prove useful for further assessment of pupillary and blink reflexes. The resolution obtained revealed the existence reported here of an early incomplete blink and an upward eye movement.

Behavior analysis of construction safety systems is of fundamental importance to avoid accidental injuries. Traditionally, measurements of dynamic actions in civil engineering have been done through accelerometers, but high-speed cameras and image processing techniques can play an important role in this area. Here, we propose using morphological image filtering and Hough transform on high-speed video sequence as tools for dynamic measurements on that field. The presented method is applied to obtain the trajectory and acceleration of a cylindrical ballast falling from a building and trapped by a thread net. Results show that safety recommendations given in construction codes can be potentially dangerous for workers.

We present an algorithm to process images of reflected Placido rings captured by a commercial videokeratoscope. Raw data are obtained with no Cartesian-to-polar-coordinate conversion, thus avoiding interpolation and associated numerical artifacts. The method provides a characteristic equation for the device and is able to process around 6 times more corneal data than the commercial software. Our proposal allows complete control over the whole process from the capture of corneal images until the computation of curvature radii.

Analysis of vibrations and displacements is a hot topic in structural engineering. Although there is a wide variety of methods for vibration analysis, direct measurement of displacements in the mid and high frequency range is not well solved and accurate devices tend to be very expensive. Low-cost systems can be achieved by applying adequate image processing algorithms. In this paper, we propose the use of a commercial pocket digital camera, which is able to register more than 420 frames per second (fps) at low resolution, for accurate measuring of small vibrations and displacements. The method is based on tracking elliptical targets with sub-pixel accuracy. Our proposal is demonstrated at a 10 m distance with a spatial resolution of 0.15 mm. A practical application over a simple structure is given, and the main parameters of an attenuated movement of a steel column after an impulsive impact are determined with a spatial accuracy of 4 µm.

Our aim was to ascertain whether the ultrasonic measurement of longitudinal corneal apex displacements carried out in a proper headrest is a credible method of ocular pulse (OP) detection. To distinguish between longitudinal movements of the eye globe treated as a rigid body and ocular surface expansion caused by the variations of the eye-globe volume, two ultrasound distance sensors were applied to noninvasively measure displacements of cornea and sclera. The same sensors were used to examine the influence of the anterio-posterior movements of a fixed head on the registration of corneal apex pulsation. In both experiments, ECG signals were synchronically recorded. Time, spectral, and coherence analyses obtained for four healthy subjects showed that the ocular surface expansion due to pulsatile ocular blood flow (POBF) is the main component of longitudinal corneal displacement. Ocular surface pulsation is always affected by the head movement. However, there exist some unique properties of signals, which help to distinguish between head and eye movements. A rigid headrest and a bite bar are required to stabilize the head during OP measurement. Ultrasonic technique enables noninvasive and accurate in vivo measurement of corneal pulsation, which could be of interest for indirectly estimating intraocular pressure propagation and POBF component.

Muscle wasting is common in advanced heart failure. Myostatin is an important modulator of muscle catabolism. We measured serum levels of myostatin and its propeptide in patients with chronic heart failure and analyzed their relationships with clinical parameters and prognosis. The study included 70 patients: 30 in New York Heart Association (NYHA) functional class I-II and 40 in class III-IV. Their mean ejection fraction was 32%+/-12%. The mean follow-up time was 17.9+/-1.3 months. Thirteen patients (18.6%) died. No correlation was found between myostatin and myostatin propeptide levels. Nor was the myostatin or myostatin propeptide level correlated with age, sex, left ventricular ejection fraction, symptom duration, or the level of N-terminal probrain natriuretic peptide (NT-proBNP) or tumor necrosis factor-alpha receptor type-2 (TNFalpha R2). Moreover, no relationship was observed between the myostatin or myostatin propeptide level and NYHA functional class or mortality, in contrast to the relationships found with NT-proBNP (P< .001 and P< .001, respectively) and TNFalpha R2 (P=.001 and P=.005, respectively) levels. In conclusion, there was no relationship between the myostatin or myostatin propeptide level and any parameter of disease severity or prognosis in patients with chronic heart failure.

Videokeratometers and Scheimpflug cameras permit accurate estimation of corneal surfaces. From height data it is possible to adjust analytical surfaces that will be later used for aberration calculation. Zernike polynomials are often used as adjusting polynomials, but they have shown to be not precise when describing highly irregular surfaces. We propose a combined zonal and modal method that allows an accurate reconstruction of corneal surfaces from height data, diminishing the influence of smooth areas over irregular zones and vice versa. The surface fitting error is decreased in the considered cases, mainly in the central region, which is more important optically. Therefore, the method can be established as an accurate resampling technique.

The eye is not a centered system. The line of sight connects the fovea with the center of the pupil and is usually tilted in the temporal direction. Thus, off-axis optical aberrations, mainly coma and oblique astigmatism, are introduced at the fovea. Tabernero et al. [J. Opt. Soc. Am. A 24(10), 3274-3283 (2007)] showed that a horizontal tilt of the crystalline lens generates a horizontal coma aberration that is compensated by the oblique light incidence on the eye. Here we suggest that corneal astigmatism may also play a role in compensation of oblique aberrations, and we propose a simple model to analyze such a possibility. A theoretical Kooijman eye model with a slight ( approximately 0.6 D) with-the-rule astigmatism is analyzed. Light rays at different incidence angles to the optical axis are considered, and the corresponding point spread functions (PSFs) at the retina are calculated. A quality criterion is used to determine the incidence angle that provides the narrowest and highest PSF energy peak. We show that the best image is obtained for a tilted incidence angle compatible with mean values of the angle kappa. This suggests that angle kappa, lens tilt, and corneal astigmatism may combine to provide a passive compensation mechanism to minimize aberrations on the fovea.

Kidney failure is an important prognostic factor in patients with heart failure. Renal function is usually evaluated by measuring the serum creatinine level. However, a normal creatinine level can mask established kidney failure. We investigated the prognostic significance of the estimated creatinine clearance rate (Cockcroft formula) in 235 patients with heart failure and a normal serum creatinine level. The two-year mortality rate was significantly higher in patients who had established kidney disease (i.e., a creatinine clearance rate <60 mL/min) than in those who did not (35.1% vs. 10.1%, P<.001). Even when only patients without established kidney failure were analyzed, the creatinine clearance rate had prognostic significance (rate > or = 90 mL/min, mortality 3.2%; rate 89-60 mL/min, mortality 13.9%; P=.02). On Cox regression analysis, which included age, sex, heart failure etiology, left ventricular ejection fraction, diabetes and hypertension, the creatinine clearance rate remained an independent predictor of mortality.

Introduction and Objectives: Obesity is an independent risk factor for congestive heart failure. Paradoxically, improved survival has been observed in obese heart failure patients. The objective of this study was to analyze the relationship between body mass index (BMI) and the 2-year mortality rate in outpatients with heart failure of different etiologies who were attending a heart failure unit., Methods: Baseline BMI and survival status at 2-year follow-up were recorded in 501 patients (73% men, median age 68 years). Heart failure etiology was mainly ischemic heart disease, present in 59%. The patients' median ejection fraction was 30%. They were divided into four groups according to BMI: low weight (<20.5), normal weight (20.5 to <25.5), overweight (25.5 to <30), and obese (>/=30)., Results: The mortality rate at 2 years differed significantly (P< .001) between the groups: 46.7% for low-weight patients, 27.8% for normal-weight patients, 18.7% for overweight patients, and 16% for obese patients. After adjusting for age, sex, heart failure etiology, functional class, ejection fraction, hypertension, diabetes, estimated creatinine clearance rate, plasma hemoglobin level, and treatment received, BMI remained an independent predictor of reduced mortality at 2 years (odds ratio=0.92 [0.88-0.97])., Conclusions: A high BMI has been associated with lower all-cause mortality rates at 2-year follow-up. Our findings in a broad population of patients with heart failure of different etiologies further confirm the existence of a paradoxical relationship between obesity and heart failure outcome.

Platelet-rich preparations constitute a relatively new biotechnology for the stimulation and acceleration of tissue healing and bone regeneration. The versatility and biocompatibility of this approach has stimulated its therapeutic use in numerous medical and scientific fields including dentistry, oral implantology, orthopaedics, ulcer treatment, tissue engineering among others. Here we discuss the important progress that has been accomplished in the field of platelet-rich preparations in the last few years. Some of the most interesting therapeutic applications of this technology are discussed as are some of the limitations, future challenges and directions in the field.

An automatic method for three-dimensional (3-D) shape recognition is proposed. It combines the Fourier transform profilometry technique with a real-time recognition setup such as the joint transform correlator (JTC). A grating is projected onto the object surface resulting in a distorted grating pattern. Since this pattern carries information about the depth and the shape of the object, their comparison provides a method for recognizing 3-D objects in real time. A two-cycle JTC is used for this purpose. Experimental results demonstrate the theory and show the utility of the new proposed method.

An optoelectronic implementation of the nonlinear morphological correlation by use of a threshold-decomposition technique and a joint transform correlator architecture is presented. This nonlinear morphological correlation provides improved image detection compared with standard linear optical pattern-recognition correlation methods. It also offers a more robust detection of low-intensity images in the presence of high-intensity patterns to be rejected.

The expanding use of optical communication by means of optical fibers and the situation of drastically increasing amounts of data to be transmitted urge the exploration of novel systems permitting the transmission of large amounts of spatial information by fiber with smaller spatial resolution. An optical encoding and decoding system is suggested for transmitting one- or two-dimensional images by means of a single-mode fiber. The superresolving system is based on wavelength multiplexing of the input spatial information, which is achieved with diffractive optical elements. Preliminary experimental results demonstrate the capabilities of the suggested method for the one- and two-dimensional cases.

The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelet transform and the fractional Fourier transform. Possible implementations of the new transformation are in image compression, image transmission, transient signal processing, etc. Computer simulations demonstrate the abilities of the novel transform. Optical implementation of this transform is briefly discussed.

A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT's in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from -1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.