The laser speckle interferometry approach provides the possibility of an in situ optical noncontacted measurement for the surface morphology of plasma facing components(PFCs),and the reconstruction image of the PFC su...The laser speckle interferometry approach provides the possibility of an in situ optical noncontacted measurement for the surface morphology of plasma facing components(PFCs),and the reconstruction image of the PFC surface morphology is computed by a numerical model based on a phase unwrapping algorithm.A remote speckle interferometry measurement at a distance of three meters for real divertor tiles retired from EAST was carried out in the laboratory to simulate a real detection condition on EAST.The preliminary surface morphology of the divertor tiles was well reproduced by the reconstructed geometric image.The feasibility and reliability of this approach for the real-time measurement of PFCs have been demonstrated.展开更多
To implement on-line, real-time monitoring for the surface morphology of Plasma-Facing Materials(PFMs) in tokamak, we developed a Laser Speckle Interferometry measurement approach. A laser ablation method was used to ...To implement on-line, real-time monitoring for the surface morphology of Plasma-Facing Materials(PFMs) in tokamak, we developed a Laser Speckle Interferometry measurement approach. A laser ablation method was used to simulate the erosion process during Plasma-Wall Interactions in a tokamak. In the present investigation, we evaluated the results of laser ablation morphology changes on the surface of Mo material reconstructed by four different approaches(Flood-fill, Quality-guided, Discrete Cosine Transform(DCT) and Weighted-DCT). The morphology results measured by the weighted-DCT approach are very close to the measurement results from confocal microscopy with an average error rate within 7%. It is verified that the weighted-DCT algorithm has high accuracy and can efficiently reduce the influence of noise pollution coming from laser ablation, which is used as a proxy for erosion from plasma wall interaction. Additionally, the CPU computer time has been shortened. This is of great significance for the real-time monitoring of PFMs’ morphology in the Experimental Advanced Superconducting Tokamak(EAST) in the future.展开更多
Speckle interferometry is an efficient method to analyze a vibration. In certain conditions, this technique has some outstanding advantage, and need not strict shock--proof condition, compared with the holographic met...Speckle interferometry is an efficient method to analyze a vibration. In certain conditions, this technique has some outstanding advantage, and need not strict shock--proof condition, compared with the holographic method for measuring vibration. Therefore,it is suitable to analyze a vibration with a large amplitude.Real-time interferometry is a rapid and simple method for measuring vibration of a body, gives speckle pattern containing amplitude distribution of body-surface. By means of time-averaged method, the speckle pattern is recorded in Fourier transform plane, or vibration lines are seen directly with eyes, so as to analyze efficiently amplitude, phase, and model of a vibration. This paper deduces the intensity distribution function with real-time method, and gives experimental demonstration of vibration body-the vibration lines with different frequencies.展开更多
The reliability of microsystems is an important issue and for their quality inspection, it is necessary to know the displacements or deformations due to the applied mechanical, thermal, or electrostatic loads. We show...The reliability of microsystems is an important issue and for their quality inspection, it is necessary to know the displacements or deformations due to the applied mechanical, thermal, or electrostatic loads. We show how interferometrical techniques like digital holography and speckle interferometry can be used for the measurement of in plane deformations of microsystems with nanometric accuracy and we give a description of the measurement uncertainties.展开更多
Three dimensional(3D) displacements, which can be translated further into 3D strain, are key parameters tor design, manufacturing and quality control. Using different optical setups, phase-shift methods, and algorit...Three dimensional(3D) displacements, which can be translated further into 3D strain, are key parameters tor design, manufacturing and quality control. Using different optical setups, phase-shift methods, and algorithms, several different 3D electronic speckle pattern interferometry(ESPl) systems for displacement and strain measurements have been achieved and commercialized. This paper provides a review of the recent developments in ESPI systems for 3D displacement and strain measurement. After an overview of the fundamentals of ESP! theory, temporal phase-shift, and spatial phase-shift techniques, 3D deformation measurements by the temporal phase-shift ESPI system, which is suited well for static measurement, and by the spatial phase-shift ESPI system, which is particularly useful for dynamic measurement, are discussed. For each method, the basic theory, a brief derivation and different optical layouts are presented. The state of art application, potential and limitation of the ESPI systems are shown and demonstrated.展开更多
Speckle patterns are a fundamental tool in a variety of physical and optical applications. Here, we investigate a method of precisely tuning the intensity statistics of random speckle patterns into a desirable pattern...Speckle patterns are a fundamental tool in a variety of physical and optical applications. Here, we investigate a method of precisely tuning the intensity statistics of random speckle patterns into a desirable pattern that possesses the same spatial correlation length and similar statistics distribution. This tuning mechanism relies on the derivation of the transform function and transmission matrix, which achieves different contrasts while maintaining the same average value or energy level. The statistics properties of the generated speckle patterns are further investigated by analyzing the standard deviation under different fitting parameters. Precisely tuning the intensity statistics of random speckle patterns could be useful for both fundamental research and practical applications, such as microscopy, imaging, and optical manipulation.展开更多
A temporal approach to fast shape measurement is presented.In principle,the rotational object method is used in combination with the sequence pulse counting method (SPCM) to determine the height of the object through ...A temporal approach to fast shape measurement is presented.In principle,the rotational object method is used in combination with the sequence pulse counting method (SPCM) to determine the height of the object through calculating the related phase.Two specimens are tested to demonstrate the validity of the approach.One is an object covered by a Chinese character (tea) with a height variety of 0.3 mm,and the other is an object surface with a relatively large fluctuation of 3.5 mm.The experimental results are compared with mechanical measurements.An axis shifting method is also proposed to determine shapes with relatively large fluctuations.Effects of such parameters on the height measurement as incident angle of the dual light beams,tilting angle of the object,and azimuth angle of the measured point are discussed as well.展开更多
When a surface is illuminated by laser,the field of diffusing reflective light can be described by Fresnel-Kirchhoff integration in the region of Fresnel.If the correlation,before and after the change of intensity fie...When a surface is illuminated by laser,the field of diffusing reflective light can be described by Fresnel-Kirchhoff integration in the region of Fresnel.If the correlation,before and after the change of intensity field,is analysed by statistical method,three governing equations for variations of intensity field and speckle movement formulas can be obtained.In these equations,the surface motion,deformation and wavelength change are all considered to cause the speckle movement.展开更多
Combining speckle interferometry with photoelasticity,the absolute retardation fringes in 3-D model are obtained by interference of the polarized rays reflected from the two surfaces of a sandwiched plate inside the m...Combining speckle interferometry with photoelasticity,the absolute retardation fringes in 3-D model are obtained by interference of the polarized rays reflected from the two surfaces of a sandwiched plate inside the model.This method can avoid the troubles introduced by Poisson's ratio μ=0.5 and stress-optical constants ratio A/B=-2 of the photoelastic material in a stress-frozenstate,and yield the re- sults which provide necessary supplementary information for the completely-experimental solution of 3-D stress state.A comparison of experimental results with theoretical ones shoves the feasibility of the method proposed in the present paper.展开更多
The femtosecond temporal speckle field of a random medium is studied theoretically and experimentally. Femtosecond temporal speckle arises from the interference of multiple randomly scattered electric fields. The femt...The femtosecond temporal speckle field of a random medium is studied theoretically and experimentally. Femtosecond temporal speckle arises from the interference of multiple randomly scattered electric fields. The femtosecond temporal speckle field is measured with a cross-correlation frequency-resolved optical gating method. The spatial average of the speckle field yields a smooth transmitted profile. The speckle field is a circular complex Gaussian variable because the scattered light beams from different trajectories have no correlation with each other. The field and the intensity profiles of individual speckle spots fluctuate randomly in time. The ensemble average of the temporal intensity profiles converges, thereby yielding the photon travel time probability distribution function.展开更多
A new electronic speckle pattern interferometry method is proposed to realize in situ deformation measurements.The feature of the method is the combination of a high-speed camera and multiple laser Doppler vibrometers...A new electronic speckle pattern interferometry method is proposed to realize in situ deformation measurements.The feature of the method is the combination of a high-speed camera and multiple laser Doppler vibrometers(LDVs)for synchronous measurements.The high-speed camera is used to record and select effective interferograms,while the LDVs are used to measure the rigid body displacement caused by vibrations.A series of effective interferograms with known shifted phase values are obtained to calculate the deformation phase.The experimental results show that the method performs well in measuring static and dynamic deformations with high accuracy in vibrating environments.展开更多
This article presents an improved method of despeckling the ultrasound medical images.In this paper a modified local statistics mean variance filter method has been proposed.In the proposed method,more consideration i...This article presents an improved method of despeckling the ultrasound medical images.In this paper a modified local statistics mean variance filter method has been proposed.In the proposed method,more consideration is given to local statistics since local statistical features are more important rather than global features.Various parameters like mean square error,peak signal to noise ratio,quality index,and structural similarity index measure are calculated to analyze the quality of the despeckled image.展开更多
Noise reduction is one of the most exciting problems in electronic speckle pattern interferometry. We present a homomorphic partial differential equation filtering method for interferometry fringe patterns. The diffus...Noise reduction is one of the most exciting problems in electronic speckle pattern interferometry. We present a homomorphic partial differential equation filtering method for interferometry fringe patterns. The diffusion speed of the equation is determined based on the fringe density. We test the new method on the computer-simulated fringe pattern and experimentally obtain the fringe pattern, and evaluate its filtering performance. The qualitative and quantitative analysis shows that this technique can filter off the additive and multiplicative noise of the fringe patterns effectively, and avoid blurring high-density fringe. It is more capable of improving the quality of fringe patterns than the classical filtering methods.展开更多
Traditional speckle fringe patterns by electronic speckle pattern interferometry (ESPI) are inherently noisy and of limited visibility, so denoising is the key problem in ESPI. We present the variational denoising m...Traditional speckle fringe patterns by electronic speckle pattern interferometry (ESPI) are inherently noisy and of limited visibility, so denoising is the key problem in ESPI. We present the variational denoising method for ESPI. This method transforms the image denosing to minimizing an appropriate penalized energy function and solving a partial differential equation. We test the proposed method on computer- simulated and experimental speckle correlation fringes, respectively. The results show that this technique is capable of significantly improving the quality of fringe patterns. It works well as a pre-processing for the fringe patterns by ESPI.展开更多
Digital speckle pattern interferometry (DSPI) is a high-precision deformation t technique for planar objects. However, for curved objects, the three-dimensional (3D) shape information is needed in order to obtain ...Digital speckle pattern interferometry (DSPI) is a high-precision deformation t technique for planar objects. However, for curved objects, the three-dimensional (3D) shape information is needed in order to obtain correct deformation measurement in DSPI. Thus, combined shape and deformation measurement techniques of DSPI have been proposed. However, the current techniques are either complex in setup or complicated in operation. Furthermore, the operations of some techniques are too slow for real-time measurement. In this work, we propose a DSPI technique for both 3D shape and out-of-plane deformation measurement. Compared with current techniques, the proposed technique is simple in both setup and operation and is capable of fast deformation measurement. Theoretical analysis and experiments are performed. For a cylinder surface with an arch height of 9 mm, the error of out-of-plane deformation measurement is less than 0.15 μm. The effectiveness of the proposed scheme is verified.展开更多
Electronic speckle pattern interferometry(ESPI) and digital speckle pattern interferometry are wellestablished non-contact measurement methods. They have been widely used to carry out precise deformation mapping. Ho...Electronic speckle pattern interferometry(ESPI) and digital speckle pattern interferometry are wellestablished non-contact measurement methods. They have been widely used to carry out precise deformation mapping. However, the simultaneous two-dimensional(2D) or three-dimensional(3D) deformation measurements using ESPI with phase shifting usually involve complicated and slow equipment. In this Letter, we solve these issues by proposing a modified ESPI system based on double phase modulations with only one laser and one camera. In-plane normal and shear strains are obtained with good quality. This system can also be developed to measure 3D deformation, and it has the potential to carry out faster measurements with a highspeed camera.展开更多
From Helmholtz equation of the harmonic electromagnetic waves, the integral equations of the light field at the medium boundaries are obtained by use of the Green's theorem and are discretized into linear equation...From Helmholtz equation of the harmonic electromagnetic waves, the integral equations of the light field at the medium boundaries are obtained by use of the Green's theorem and are discretized into linear equation set with the values of the light field and its derivative as the unknowns. On solving the linear equation set, we realize the rigorous computations of the light fields at the boundaries. Then the intensities of the light waves scattered by the random self-affine fractal surfaces in the optical near-field are calculated, and the propagation characteristics, the evolutions of the contrast and the intensity probability density function of the near-field speckles are studied in detail. The near-field speckles are much different from the conventional speckles in the diffraction regions and in the imaging systems. There are obvious local fluctuations in the intensity distributions of the near-field speckles and such fluctuations disappear after propagating a distance of one wavelength from the medium surfaces. For the random surfaces with smaller lateral correlation lengths, the speckle contrasts approach the saturation values and the speckle fields approach Gaussian distribution within the near-field, while for the random surfaces with larger lateral correlation lengths, such evolutions become comparatively slow.展开更多
基金supported by the National Magnetic Confinement Fusion Science Program of China(No.2013GB109005)National Natural Science Foundation of China(Nos.11175035,11475039)Chinesisch-Deutsches Forschungs project(GZ768)
文摘The laser speckle interferometry approach provides the possibility of an in situ optical noncontacted measurement for the surface morphology of plasma facing components(PFCs),and the reconstruction image of the PFC surface morphology is computed by a numerical model based on a phase unwrapping algorithm.A remote speckle interferometry measurement at a distance of three meters for real divertor tiles retired from EAST was carried out in the laboratory to simulate a real detection condition on EAST.The preliminary surface morphology of the divertor tiles was well reproduced by the reconstructed geometric image.The feasibility and reliability of this approach for the real-time measurement of PFCs have been demonstrated.
基金supported by the National Key R&D Program of China (No. 2017YFE0301304)National Natural Science Foundation of China (Nos. 11605023, 11805028, 11705020)+1 种基金China Postdoctoral Science Foundation (Nos. 2017T100172, 2016M591423)the Fundamental Research Funds for the Central Universities (Nos. DUT17RC(4)53, DUT18LK38)
文摘To implement on-line, real-time monitoring for the surface morphology of Plasma-Facing Materials(PFMs) in tokamak, we developed a Laser Speckle Interferometry measurement approach. A laser ablation method was used to simulate the erosion process during Plasma-Wall Interactions in a tokamak. In the present investigation, we evaluated the results of laser ablation morphology changes on the surface of Mo material reconstructed by four different approaches(Flood-fill, Quality-guided, Discrete Cosine Transform(DCT) and Weighted-DCT). The morphology results measured by the weighted-DCT approach are very close to the measurement results from confocal microscopy with an average error rate within 7%. It is verified that the weighted-DCT algorithm has high accuracy and can efficiently reduce the influence of noise pollution coming from laser ablation, which is used as a proxy for erosion from plasma wall interaction. Additionally, the CPU computer time has been shortened. This is of great significance for the real-time monitoring of PFMs’ morphology in the Experimental Advanced Superconducting Tokamak(EAST) in the future.
文摘Speckle interferometry is an efficient method to analyze a vibration. In certain conditions, this technique has some outstanding advantage, and need not strict shock--proof condition, compared with the holographic method for measuring vibration. Therefore,it is suitable to analyze a vibration with a large amplitude.Real-time interferometry is a rapid and simple method for measuring vibration of a body, gives speckle pattern containing amplitude distribution of body-surface. By means of time-averaged method, the speckle pattern is recorded in Fourier transform plane, or vibration lines are seen directly with eyes, so as to analyze efficiently amplitude, phase, and model of a vibration. This paper deduces the intensity distribution function with real-time method, and gives experimental demonstration of vibration body-the vibration lines with different frequencies.
基金supported by the German Research Foundation (DFG) under grants OS111/22 and PA792/4
文摘The reliability of microsystems is an important issue and for their quality inspection, it is necessary to know the displacements or deformations due to the applied mechanical, thermal, or electrostatic loads. We show how interferometrical techniques like digital holography and speckle interferometry can be used for the measurement of in plane deformations of microsystems with nanometric accuracy and we give a description of the measurement uncertainties.
基金supported by National Natural Science Foundation of China(Grant Nos.51275054,51075116)
文摘Three dimensional(3D) displacements, which can be translated further into 3D strain, are key parameters tor design, manufacturing and quality control. Using different optical setups, phase-shift methods, and algorithms, several different 3D electronic speckle pattern interferometry(ESPl) systems for displacement and strain measurements have been achieved and commercialized. This paper provides a review of the recent developments in ESPI systems for 3D displacement and strain measurement. After an overview of the fundamentals of ESP! theory, temporal phase-shift, and spatial phase-shift techniques, 3D deformation measurements by the temporal phase-shift ESPI system, which is suited well for static measurement, and by the spatial phase-shift ESPI system, which is particularly useful for dynamic measurement, are discussed. For each method, the basic theory, a brief derivation and different optical layouts are presented. The state of art application, potential and limitation of the ESPI systems are shown and demonstrated.
基金Project supported by the National Natural Science Foundation of China(Grant No.61702347)the Natural Science Foundation of Hebei Province,China(Grant No.F2017210161)Education Department of Hebei Province,China(Grant No.QN2017132)
文摘Speckle patterns are a fundamental tool in a variety of physical and optical applications. Here, we investigate a method of precisely tuning the intensity statistics of random speckle patterns into a desirable pattern that possesses the same spatial correlation length and similar statistics distribution. This tuning mechanism relies on the derivation of the transform function and transmission matrix, which achieves different contrasts while maintaining the same average value or energy level. The statistics properties of the generated speckle patterns are further investigated by analyzing the standard deviation under different fitting parameters. Precisely tuning the intensity statistics of random speckle patterns could be useful for both fundamental research and practical applications, such as microscopy, imaging, and optical manipulation.
基金supported by the National Natural Science Foundation of China(Grants Nos. 10972113,10732080)the National Basic Research Program of China(Grant Nos. 2007CB936803,2010CB631005)SRFDP(Grant No. 20070003053)
文摘A temporal approach to fast shape measurement is presented.In principle,the rotational object method is used in combination with the sequence pulse counting method (SPCM) to determine the height of the object through calculating the related phase.Two specimens are tested to demonstrate the validity of the approach.One is an object covered by a Chinese character (tea) with a height variety of 0.3 mm,and the other is an object surface with a relatively large fluctuation of 3.5 mm.The experimental results are compared with mechanical measurements.An axis shifting method is also proposed to determine shapes with relatively large fluctuations.Effects of such parameters on the height measurement as incident angle of the dual light beams,tilting angle of the object,and azimuth angle of the measured point are discussed as well.
基金The project supported by National Natural Science Foundation of China
文摘When a surface is illuminated by laser,the field of diffusing reflective light can be described by Fresnel-Kirchhoff integration in the region of Fresnel.If the correlation,before and after the change of intensity field,is analysed by statistical method,three governing equations for variations of intensity field and speckle movement formulas can be obtained.In these equations,the surface motion,deformation and wavelength change are all considered to cause the speckle movement.
基金The project is supported by the National Natural Science Foundation of China.
文摘Combining speckle interferometry with photoelasticity,the absolute retardation fringes in 3-D model are obtained by interference of the polarized rays reflected from the two surfaces of a sandwiched plate inside the model.This method can avoid the troubles introduced by Poisson's ratio μ=0.5 and stress-optical constants ratio A/B=-2 of the photoelastic material in a stress-frozenstate,and yield the re- sults which provide necessary supplementary information for the completely-experimental solution of 3-D stress state.A comparison of experimental results with theoretical ones shoves the feasibility of the method proposed in the present paper.
基金supported by the Scientific Research Foundation of Harbin Institute of Technology at Weihai, China (Grant No. 2008087)
文摘The femtosecond temporal speckle field of a random medium is studied theoretically and experimentally. Femtosecond temporal speckle arises from the interference of multiple randomly scattered electric fields. The femtosecond temporal speckle field is measured with a cross-correlation frequency-resolved optical gating method. The spatial average of the speckle field yields a smooth transmitted profile. The speckle field is a circular complex Gaussian variable because the scattered light beams from different trajectories have no correlation with each other. The field and the intensity profiles of individual speckle spots fluctuate randomly in time. The ensemble average of the temporal intensity profiles converges, thereby yielding the photon travel time probability distribution function.
基金supported by the National Natural Science Foundation of China(Nos.11890683 and 11972235)。
文摘A new electronic speckle pattern interferometry method is proposed to realize in situ deformation measurements.The feature of the method is the combination of a high-speed camera and multiple laser Doppler vibrometers(LDVs)for synchronous measurements.The high-speed camera is used to record and select effective interferograms,while the LDVs are used to measure the rigid body displacement caused by vibrations.A series of effective interferograms with known shifted phase values are obtained to calculate the deformation phase.The experimental results show that the method performs well in measuring static and dynamic deformations with high accuracy in vibrating environments.
文摘This article presents an improved method of despeckling the ultrasound medical images.In this paper a modified local statistics mean variance filter method has been proposed.In the proposed method,more consideration is given to local statistics since local statistical features are more important rather than global features.Various parameters like mean square error,peak signal to noise ratio,quality index,and structural similarity index measure are calculated to analyze the quality of the despeckled image.
文摘Noise reduction is one of the most exciting problems in electronic speckle pattern interferometry. We present a homomorphic partial differential equation filtering method for interferometry fringe patterns. The diffusion speed of the equation is determined based on the fringe density. We test the new method on the computer-simulated fringe pattern and experimentally obtain the fringe pattern, and evaluate its filtering performance. The qualitative and quantitative analysis shows that this technique can filter off the additive and multiplicative noise of the fringe patterns effectively, and avoid blurring high-density fringe. It is more capable of improving the quality of fringe patterns than the classical filtering methods.
文摘Traditional speckle fringe patterns by electronic speckle pattern interferometry (ESPI) are inherently noisy and of limited visibility, so denoising is the key problem in ESPI. We present the variational denoising method for ESPI. This method transforms the image denosing to minimizing an appropriate penalized energy function and solving a partial differential equation. We test the proposed method on computer- simulated and experimental speckle correlation fringes, respectively. The results show that this technique is capable of significantly improving the quality of fringe patterns. It works well as a pre-processing for the fringe patterns by ESPI.
基金supported by the National Key Research and Development Project of China(No.2016YFF0200700)the National Natural Science Foundation of China(No.61405111)
文摘Digital speckle pattern interferometry (DSPI) is a high-precision deformation t technique for planar objects. However, for curved objects, the three-dimensional (3D) shape information is needed in order to obtain correct deformation measurement in DSPI. Thus, combined shape and deformation measurement techniques of DSPI have been proposed. However, the current techniques are either complex in setup or complicated in operation. Furthermore, the operations of some techniques are too slow for real-time measurement. In this work, we propose a DSPI technique for both 3D shape and out-of-plane deformation measurement. Compared with current techniques, the proposed technique is simple in both setup and operation and is capable of fast deformation measurement. Theoretical analysis and experiments are performed. For a cylinder surface with an arch height of 9 mm, the error of out-of-plane deformation measurement is less than 0.15 μm. The effectiveness of the proposed scheme is verified.
基金financially supported by the ANR Micromorfing Program(ANR-14-CE07-0035)China Scholarship Council(CSC)the Labex Action
文摘Electronic speckle pattern interferometry(ESPI) and digital speckle pattern interferometry are wellestablished non-contact measurement methods. They have been widely used to carry out precise deformation mapping. However, the simultaneous two-dimensional(2D) or three-dimensional(3D) deformation measurements using ESPI with phase shifting usually involve complicated and slow equipment. In this Letter, we solve these issues by proposing a modified ESPI system based on double phase modulations with only one laser and one camera. In-plane normal and shear strains are obtained with good quality. This system can also be developed to measure 3D deformation, and it has the potential to carry out faster measurements with a highspeed camera.
基金sponsored by the National Natural Science Foundation of China(Grant No.69978012)by the National Key Basic Research Special Foundation(NKBRSF)of China(Grant No.G1999075200).
文摘From Helmholtz equation of the harmonic electromagnetic waves, the integral equations of the light field at the medium boundaries are obtained by use of the Green's theorem and are discretized into linear equation set with the values of the light field and its derivative as the unknowns. On solving the linear equation set, we realize the rigorous computations of the light fields at the boundaries. Then the intensities of the light waves scattered by the random self-affine fractal surfaces in the optical near-field are calculated, and the propagation characteristics, the evolutions of the contrast and the intensity probability density function of the near-field speckles are studied in detail. The near-field speckles are much different from the conventional speckles in the diffraction regions and in the imaging systems. There are obvious local fluctuations in the intensity distributions of the near-field speckles and such fluctuations disappear after propagating a distance of one wavelength from the medium surfaces. For the random surfaces with smaller lateral correlation lengths, the speckle contrasts approach the saturation values and the speckle fields approach Gaussian distribution within the near-field, while for the random surfaces with larger lateral correlation lengths, such evolutions become comparatively slow.