Based on protein sequence information, a simple and effective method was used to analyze protein sequence similarity and predict DNA-binding protein. It is absolutely necessary that we generate computational methods of low complexity to accurate infer protein structure, function, and evolution in the rapidly growing number of molecular biology data available. It is important to generate novel computational algorithms for analyzing and comparing protein sequences with the rapidly growing number of molecular biology data available. Based on global and local position representation with the curves of Fermat spiral and normalized moments of inertia of the curve of Fermat spiral, respectively, moreover, composition of 20 amino acids to get the numerical characteristics of protein sequences. It has been applied to analyze the similarity/dissimilarity of nine ND5 proteins, the analysis results are consistent with the biological evolution theory. Furthermore, we employ the Logistic regression with 5-fold cross-validation to establish the prediction of DNA-binding proteins model, which outperformed the DNAbinder, iDNA-prot, DNA-prot and gDNA-prot by 0.0069-0.609 in terms of F-measure, 0.293-0.898 in terms of MCC in unbalanced dataset. These results show that our method, namely FermatS, is effective to compare, recognition and prediction the protein sequences.
This study introduces three-dimensional Null Subtraction Imaging (3D NSI), a nonlinear beamforming framework designed to extend the computationally efficient null-subtraction process to volumetric imaging with matrix arrays. This is accomplished by designing specialized NSI apodizations (a zero-mean window and two DC-offset variants) for 2D receive apertures, enabling simultaneous synthetic beam narrowing and sidelobe suppression in both azimuthal and elevational dimensions. The versatility of the framework was demonstrated on a 1024-element matrix array using three aperture configurations: a fully-addressed circular aperture and two sparse Fermat's spiral apertures, chosen for comparable aperture sizes. By further implementing a spiral no-reuse apodization that enforces non-overlapping element sets across transmit-receive events, we achieved up to a 16-fold increase in acquisition volume rate using only 240 active elements. In computer simulations and tissue-mimicking phantom experiments, 3D NSI achieved an average improvement of 36% in azimuthal and elevational resolutions, along with an approximately 20% higher contrast ratio, compared to the conventional Delay-and-Sum (DAS) beamformer under matched transmit/receive configurations. When implemented with the spiral no-reuse aperture, the 3D NSI framework achieved an acquisition rate exceeding 1000 volumes per second with a computational load less than three times that of DAS, making it a practical solution for real-time 4D imaging. Preliminary in vivo carotid imaging corroborated the efficacy of the framework in biological tissues, indicating that 3D NSI represents a general and efficient beamforming method for matrix arrays with a potential to impact real-time 4D ultrasound imaging in cardiovascular and other dynamic applications.
Reliable manufacturing and high optical performance of plano-convex aspheric lenses necessitate an improved optical design characterized by an optimal refractive index of the lens material and the prevention of the slope surface shape oscillations, which provide an increased angular field of view and low sensitivity to misalignments. To formulate the criteria for the optimal refractive index, we propose an analytical representation of the optimal aspheric surface based on the Fermat's principle in the form of parametric functions of coordinates, allowing the surface to be approximated by the elliptical conic section expanded with high-order aspheric deformation terms starting from the 6th power. There are determined the ranges of optimal refractive indices for materials of plano-convex aspheric lenses: n=1.605…1.665 for an aplanatic design with the Abbe sine condition fulfilled, ensuring low lens sensitivity of the lens to angular misalignment; n=1.735…1.835 for sharp imaging of elongated longitudinal objects in accordance with the Herschel's condition.
Three-dimensional (3D) reconstruction technology has found widespread applications across various domains, including intelligent driving and underwater exploration. But the existing imaging systems and methods still have deficiencies in terms of reconstruction accuracy, detection distance and system volume. Herein, this paper presents a three-dimensional detection and reconstruction method based on a compact Risley-prism 3D imaging system that achieves multi-viewpoint imaging by rotating the Risley prism to adjust the camera's optical axis. A refractive camera model that integrates the pinhole camera model with the vector form of Snell's law is established to precisely describe beam trajectory. A forward projection method suitable for refractive interfaces is developed based on Fermat's principle, and the influence of systematic errors on the reconstruction is analyzed in detail through simulation. Furthermore, a new 3D reconstruction method combining error calibration based on the optimization iteration is introduced to avoid the influence of error and improve reconstruction quality. Experimental results demonstrate that the proposed approach markedly enhances 3D reconstruction accuracy, reducing the Normalized Root Mean Square Error (NRMSE) from 0.9076 to 0.0207.
We theoretically investigated how the arrangements of tilt points affect the suppression of diffraction contrast in tilt-scan-averaged differential phase contrast scanning transmission electron microscopy (tDPC STEM). To examine the effectiveness of tilt-point arrangement, we compared three sampling patterns: conventional 4-fold, 6-fold symmetric concentric patterns and the Vogel spiral pattern based on Fermat's spiral with the golden angle. Multislice simulations were performed for GaN, GaAs, Si, and SrTiO3 crystals. Although all patterns approached similar converged values with sufficiently high tilt numbers, their convergence behaviors differed markedly. The concentric patterns exhibited non-monotonic decay and showed discrete fluctuations in residual diffraction contrast, whereas the Vogel spiral showed a smoother and more monotonic decrease over a wide range of tilt numbers. This behavior was consistently observed across the crystals with different crystal symmetries. These results show that rotationally asymmetric and uniformly distributed tilt sampling is advantageous for the robust diffraction-contrast suppression in DPC STEM, and identify the Vogel spiral as an effective tilt pattern for this purpose.
In many monitoring scenarios, repeated and operator-independent assessments are needed. Wearable ultrasound technology has the potential to continuously provide the vital information traditionally obtained from conventional ultrasound scanners, such as in fetal monitoring for high-risk pregnancies. This work is an engineering study motivated by that setting. A 144-element annular capacitive micromachined ultrasonic transducer (CMUT) is hereby proposed for 3-D ultrasound imaging. The array is characterized by its compact size and cost-effectiveness, with a geometry and low-voltage operation that make it a candidate for future wearable integration. To enhance the imaging performance, we propose the utilization of a Fermat's spiral virtual source (VS) pattern for diverging wave transmission and conduct a performance comparison with other VS patterns and standard techniques, such as focused and plane waves. To facilitate this analysis, a simplified and versatile simulation framework, enhanced by GPU acceleration, has been developed. The validation of the simulation framework aligned closely with expected values (0.002 ≤ MAE ≤ 0.089). VSs following a Fermat's spiral led to a balanced outcome across metrics, outperforming focused wave transmissions for this specific aperture. The proposed transducer presents imaging limitations that could be improved in future developments, but it establishes a foundational framework for the design and fabrication of cost-effective, compact 2-D transducers suitable for 3-D ultrasound imaging, with potential for future integration into wearable devices.
The bonding integrity of corrugated interfaces in clad plates fundamentally governs their macroscopic performance, yet high-precision nondestructive evaluation (NDE) of these geometrically complex bonds remains a critical industrial challenge. Addressing the inspection demands for corrugated clad plates, this study investigates Cu/Al laminates as a representative system. A multi-parameter coupling framework integrating Rayleigh perturbation theory and critical slope criterion is established to enable quantitative prediction of imaging quality for corrugated interfaces. For the first time, the dimensionless parameter cp is proposed as a scale-invariant criterion to characterize imaging failure. Through systematic finite element simulations, the nonlinear coupling effects of corrugation wavelength, amplitude, and interface depth on imaging fidelity are quantified. The mechanism of imaging degradation at corrugation waists is elucidated via the relationship between local interface slope and ultrasonic reflection angle. Using Fermat's principle, the minimum-time acoustic path in a bilayer medium is derived to analyze systematic imaging offsets caused by excitation-receiver separation. Experimental validation is performed on a laser-ultrasonic NDE platform using eight specimens with precisely controlled corrugation parameters fabricated by wire EDM. Results demonstrate excellent agreement with simulations, confirming the predictive accuracy of the theoretical model. The study reveals that when the maximum interface slope exceeds a critical threshold, reflected signals from waist regions fall outside the detection aperture, causing severe imaging quality deterioration. Decreasing wavelength, increasing amplitude, and increasing depth collectively exacerbate signal attenuation and geometric distortion. The proposed framework provides a theoretical foundation for quality assurance of corrugated clad plates and establishes a generalized analytical methodology for NDE of irregular multi-layered interfaces.
The conformal optical systems are tremendously applied in precision-guided weapons. A corrector is traditionally used to compensate for the dynamic aberrations of an ellipsoidal dome. In this paper, what we believe to be a novel method is proposed to design conformal optical system based on Fermat's principle. The corrector needn't be required in the conformal optical system. The optical path distances (OPD) between the incident spherical wavefront and the rotating center of the imaging system are all set to be equal when the rays entering from different look angles, so the ellipsoidal dome acts as a spherical dome. The inner surface of the dome is calculated to ensure an identical OPD. At last, a conformal optical system is designed with a fineness ratio of 1, a diameter of 120 mm, the field of regard of ±60°, an F-number of 1.2 for the mid-wave infrared 3.7-4.8µm. The example demonstrates the feasibility of designing a conformal optical system using Fermat's principle.
When using ray tracing for optical system design, it is often the case that the designer would like to implement simplified versions of one or more compound lens groups. This could be the case during initial layout when idealized versions of such compound lenses are needed or, perhaps alternatively, to mimic a well-corrected commercially available lens for which the prescription details are unavailable. One option is to use a paraxial thin lens as a proxy for the actual lens group, but doing so will yield a layout that is not consistent with Fermat's principle or the Abbe sine condition. For example, a paraxial lens version of a compound microscope objective typically produces the wrong numerical aperture for a given entrance pupil diameter, and vice versa. A better option is to use a lens model that provides perfect imaging for a specified paraxial magnification and obeys Fermat's principle. A variant of the model can yield a perfect Fourier transform lens. In addition, it is desirable to implement an idealized thick lens in which the principal planes are separated by a user-specified distance. This paper presents such a model, referred to as the Cardinal Lens, with implementation in Zemax OpticStudio via a user-defined surface.
"My favorite name theorem is Fermat's last theorem, probably one of the most complicated mathematical challenges of the last four centuries. I also love that he did not provide any demonstration as it would not have fit the margins … The most important quality of a mentor is adaptability-every student is different, and different is the way to them …" Find out more about Alessandro Bismuto in his Introducing… Profile.
We present two analytical methods for describing light propagation and related characteristics-such as optical power and optical path length-in gradient refractive index (GRIN) lenses of arbitrary geometry. The first, a direct variational technique based on Fermat's principle, yields analytical expressions for light paths in inhomogeneous media. The second, a Taylor series approach adapted to boundary-value problems, provides complementary analytical insight. Both methods enable ray tracing through complete optical systems, including GRIN lenses embedded in media of different refractive indices, with interface conditions imposed via Snell's law. Comparison with numerical solutions of the Euler-Lagrange equations shows excellent agreement, confirming the accuracy of our approach. The results are directly applicable to modeling the human crystalline lens and other biological or engineered GRIN systems.
Varied-line-spacing gratings (VLS-Gs) serve as key optical elements in spectrometers and monochromators. An ongoing challenge for AI-based automatic VLS-Gs design is integrating groove parameter calculation, focal position determination, engineering tolerance analysis, and performance estimation in a self-contained coding environment. In this article, we report a proof-of-concept study on designing a VLS spherical grating using the ray-tracing method. This approach demonstrates several advantages over conventional optical designs. First, the use of the ray-tracing method paves a direct way to calculate the groove parameters without requiring mathematical deduction of the light path function under Fermat's principle. Second, coma-free focal positions can be efficiently determined for different photon energies using Powell's hybrid method. In addition, the ray-tracing method offers novel insights for determining the acceptance angle limit and optimal grating length. This ray-tracing based VLS-G calculation method may prove valuable for future AI-based automatic design of spectrometers and monochromators.
We present a method for modeling and rendering irregular and heterogeneous glass objects, with a specific emphasis on stained glass windows and window works often encountered in architecture from middle age to 18th century. The artisanal production of sheet glass results in glass panels displaying a vast variety of surface and volume irregularities like bubbles, irregular surface or smoothly varying refractive index, all of which contribute to the specific visual aspect of old glass. We propose to account for all the aforementioned effects in a unified framework based on signed distance functions and an analytic solution of the ray tracing equations on tetrahedral volume elements. We demonstrate how to construct an unbiased estimator for the transmitted lighting produced by such panels by using Fermat's principle and results from seismic ray theory. We use texture coordinates to map arbitrary sections of a complex glass panel onto the individual faces of a mesh, allowing the modeling and rendering of complex 3-dimensional objects composed of colored glass facets such as stained glass windows.
The modified Luneburg lens, otherwise known as the Gutman lens, is a gradient-index spherical lens whose focus occurs inside the lens in the stigmatic case (aberration-free). When the focus is set to lie outside the lens, spherical aberration is introduced. Recently, a scaled version of Gutman's equation has been proposed to minimize this aberration and the corresponding lens was referred to as the reduced aberration acoustic Luneburg lens. The minimization process was based on numerical optimization methods. The term "acoustic" is used since it was originally designed with dimensions suitable for propagating acoustic waves. However, this lens is general enough to be analyzed in terms of its maximum radius, allowing it to be scaled to any size and capable of propagating any wavelength. In other words, this lens can be more broadly referred to as the reduced aberration Luneburg lens (RALL). In this paper, we present the complete formalism of the RALL, conducting a detailed and comprehensive analysis of its performance. Establishing the physical characteristics of the proposed scaled Gutman's equation, we perform an aberration analysis using ray tracing based on the Fermat's invariants. Our analysis reveals that the RALL is essentially a Gutman lens with a larger radius. The reduction of spherical aberration is due to the fact that the rays impinge on a trimmed anterior effective area of the lens. In any optical system it is common that the major contribution for spherical aberration comes from rays at the periphery.
We present a novel, to the best of our knowledge, multi-core fiber-based lensless micro-endoscope incorporating several key innovations: a Fermat's golden spiral core layout, a low-index polymer cladding, and a longitudinally varying fiber diameter. This fiber design enables efficient focusing of ultrashort pulses and collection of the fluorescence signal with both diffraction-limited resolution and field of view. These advancements address several long-standing challenges in the development of ultra miniaturized non-linear endoscopes. We characterize the device's performance by evaluating the Strehl ratio, field of view, and memory effect. Finally, we demonstrate its application to biological cellular samples by performing two-photon excited fluorescence imaging in an endoscopic configuration.
Beginning with Fermat's principle, we provide a detailed derivation of the generalized laws of refraction and reflection for a geometry realizing a metasurface. We first solve the Euler-Lagrange equations for a light ray propagating across the metasurface. The ray-path equation is found analytically, and the results are supported by numerical calculations. We get generalized laws of refraction and reflection that have three main features: (i) They are relevant in gradient-index optics and in geometrical optics; (ii) A collection of rays emerges from the metasurface as a result of multiple reflections inside the metasurface; and (iii) The laws, although derived from Fermat's principle, differ from previously published results.
We propose a method to design the exact phase profile of at least one metasurface in a stigmatic singlet that can be made to implement a desired ray mapping. Following the generalized vector law of refraction and Fermat's principle, we can obtain exact solutions for the required lens shape and phase profile of a phase gradient metasurface to respect particular ray conditions (e.g., Abbe sine) as if it were a freeform refractive element. To do so, the method requires solving an implicit ordinary differential equation. We present comparisons with Zemax simulations of illustrative designed lenses to confirm the anticipated optical behaviour.
The leading order monochromatic aberrations are investigated for optical systems, which obey single-plane symmetry and translational invariance. These aberrations are classified from the symmetry principles for the wave aberration function. Fermat's principle is applied for a system made of generic cylindrical surfaces, and a complete set of the aberration coefficients required for calculation of the aberrations for any paraxial ray is obtained. To demonstrate the applied value of the analytical results obtained, the criteria of compensation of the leading aberrations for the cylindrical analog of the Cassegrain telescope were found.
Elliptical Gaussian beams generated by laser diodes (LDs) often exhibit asymmetrical divergence angle distribution, which limits their practical applications. In this study, we propose what we believe is a novel approach to shape and collimate the elliptical output beam from a LD. The design process involves the construction of two freeform reflective surfaces on a reference circle using a three-dimensional point-by-point iterative method, based on the law of conservation of energy, the vector reflection theory, and Fermat's principle. The output beam's maximum divergence angle is effectively compressed to 3.1579 mrad. The design is compact with a folded optical path and antenna size of 368.8c m 3. This paper presents a comprehensive design and optimization process, along with an in-depth analysis of the system's performance, thereby offering novel insights for emerging optical design practitioners.
The heliostat field layout in a central receiver solar thermal power plant has significant optical losses that can ultimately affect the overall output power of the plant. In this paper, an optimized heliostat field layout based on annual efficiency and power of 50 MW for the local coordinates of Quetta, Pakistan, is proposed. The performance of two different heliostat field layouts such as radial staggered and Fermat's spiral distribution are evaluated and different design points in a year are considered for the analysis. The field layouts are then optimized using a rejection sampling based Genetic Algorithm (GA). It considers the output power and mean overall efficiency for vernal equinox, summer solstice, autumnal equinox, and winter solstice as objective functions. The GA optimizes the heliostat field parameters, namely, security distance (DS), tower height (TH), heliostat width to length ratio (WR), and the length of heliostats (LH). The study system was developed in MATLAB for validation. It was observed that for the radial staggered layout, the number of heliostats decreased by 364 and the efficiency was improved by 8.52 % using GA optimization relative to unoptimized results field layout. The annual efficiency for Fermat's spiral configuration was improved by 14.62 % and correspondingly, the number of heliostats decreased by 434.