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Optical resolution

Optical resolution is the ability of an optical imaging system, such as a microscope or telescope, to distinguish between two closely spaced point sources of light as separate entities, ultimately limited by the wave nature of light and diffraction effects.^[1] This fundamental limit determines the finest detail that can be resolved in an image, preventing perfect reproduction of an object's structure regardless of the system's magnification.^[2] The Rayleigh criterion serves as the conventional standard for defining optical resolution, stating that two point sources are just resolvable when the central maximum of one diffraction pattern falls directly on the first minimum of the other, resulting in a detectable dip in intensity between them.^[1] For circular apertures, this yields an angular resolution θ of approximately 1.22 λ / D, where λ is the wavelength of light and D is the diameter of the aperture, as applies to telescopes observing distant objects.^[1] In microscopy, the lateral resolution d is given by d = 0.61 λ / NA, where NA (numerical aperture) is a measure of the lens's light-gathering capacity, defined as NA = n sin α with n as the refractive index of the medium and α as the half-angle of the maximum cone of light entering the lens.^[2] Higher NA values, achievable through immersion objectives (e.g., oil with n ≈ 1.5), can push resolution limits to around 0.2 micrometers for visible light wavelengths near 500 nm.^[3] Optical resolution plays a critical role across scientific and technological applications, constraining the performance of instruments in fields like biology, astronomy, and materials science.^[2] In biological microscopy, it dictates the visibility of subcellular structures, such as organelles or protein complexes, while in astronomy, it limits the separation of stars or planetary details.^[4] Techniques like super-resolution microscopy (e.g., STED or PALM) circumvent classical limits by exploiting nonlinear optics or localization, achieving resolutions below 50 nm, though these extend beyond traditional diffraction-bound optics.^[5] Factors such as aberration correction and illumination wavelength further influence practical resolution, emphasizing the interplay between optical design and physical constraints.^[6]

Basic Principles

Definition and Significance

Optical resolution refers to the ability of an optical imaging system to distinguish fine spatial details in an object or scene, quantified as the minimum angular or linear separation between two point sources or lines that can be perceived as distinct rather than merged. This capacity depends on fundamental optical properties such as the wavelength of light and the size of the system's aperture, which collectively determine how sharply features can be separated in the resulting image.^[7] The concept of optical resolution was formalized in the 19th century, with pivotal developments by John William Strutt, Lord Rayleigh, in his 1879 paper investigating the limits of optical instruments like spectroscopes and microscopes. Rayleigh's work established resolution as a key performance metric for optical systems, building on earlier ideas from wave optics to address practical limitations in distinguishing closely spaced features.^[8] Optical resolution holds profound significance across diverse applications, as it directly governs image quality and the fidelity of captured data. In scientific instrumentation like microscopes, superior resolution enables the detailed examination of biological specimens, such as cellular structures, which is essential for advancing medical diagnostics and research.^[7] In astronomy, it allows telescopes to separate faint stars or reveal surface details on distant planets, enhancing our understanding of cosmic phenomena.^[9] Similarly, in photography and medical imaging, high resolution ensures sharp visuals and precise lesion detection, impacting everything from artistic expression to clinical accuracy.^[10] Resolution is typically expressed in angular units, such as arcseconds, for wide-field systems like telescopes, where a value of 1 arcsecond might resolve binary stars separated by that angle.^[9] In contrast, microscopic applications use linear units like microns or nanometers, enabling resolutions down to approximately 0.2 microns for visible light to visualize subcellular features.^[7] The ultimate constraint on this capability arises from diffraction, which blurs images beyond a certain scale regardless of optical perfection.

Diffraction Limit

The wave nature of light imposes a fundamental limit on optical resolution through diffraction, which causes the image of a point source to blur into a central bright spot surrounded by concentric rings, known as the Airy disk.^[11] This pattern arises because light waves passing through a finite aperture interfere constructively and destructively, spreading the focused energy rather than concentrating it perfectly at a point.^[12] The radius r of the Airy disk, which defines the size of this blurred spot, is given by the formula

r = 1.22 \frac{\lambda f}{D},

where \lambda is the wavelength of light, f is the focal length of the optical system, and D is the diameter of the aperture.^[12] This expression quantifies the theoretical minimum resolvable detail in a diffraction-limited system. The formula derives from the Fraunhofer diffraction integral applied to a circular aperture, which models the far-field intensity distribution as the square of the first-order Bessel function of the first kind, J_1, with the first zero of this function occurring at approximately 3.83 radians, leading to the factor of 1.22 when normalized.^[13] No optical system can resolve features smaller than this diffraction limit without employing advanced techniques such as super-resolution methods, as the blurring effect is inherent to wave propagation; the limit scales linearly with wavelength \lambda and inversely with aperture diameter D, emphasizing the trade-off between compactness and performance.^[12] For instance, in astronomical telescopes, increasing the aperture diameter D directly enhances resolution by reducing the Airy disk size, enabling the distinction of finer details in distant celestial objects.^[14] Similarly, in microscopy, using shorter wavelengths, such as ultraviolet light, shrinks the diffraction limit and improves resolution for imaging small structures.^[15] The Rayleigh criterion provides a practical measure of resolvability based on this limit, where two points are just resolvable when separated by the Airy disk radius.^[12]

Resolution Criteria

The Rayleigh criterion establishes a standard for resolvability in optical systems by specifying that two closely spaced point sources are just distinguishable when the central maximum of the Airy diffraction pattern from one falls on the first minimum of the other. For a circular aperture of diameter D, the minimum angular separation is given by \theta = 1.22 \frac{[\lambda](/page/Wavelength)}{D}, where \lambda is the wavelength of light. At this separation, the combined intensity at the midpoint between the peaks is approximately 73.5% of the individual peak intensity, providing a detectable dip of about 26.5%. This criterion, originally formulated by John William Strutt (Lord Rayleigh) in 1879, is widely used for incoherent point sources in telescopes and astronomy.^[16]^[17] The Sparrow criterion offers a more stringent limit, defining resolvability at the point where the second derivative of the total intensity profile vanishes, yielding a flat plateau between the two peaks rather than a dip. For a circular aperture, the corresponding angular separation is approximately \theta \approx \frac{\lambda}{D}. In microscopic contexts, this translates to a linear resolution of roughly $0.47 \frac{\lambda}{\mathrm{NA}}, where NA is the numerical aperture, allowing detection of finer details than the Rayleigh limit but requiring greater sensitivity to subtle intensity variations. Proposed by Charles M. Sparrow in 1916, this criterion is often applied in astronomical observations and systems with partial coherence.^[18]^[19] For microscopic imaging, the Abbe diffraction limit provides the theoretical minimum resolvable distance for periodic structures under coherent illumination, expressed as d = \frac{\lambda}{2 \mathrm{NA}}. This formula highlights the cutoff of high spatial frequencies due to the objective's light-gathering capability, with NA incorporating the refractive index and aperture angle. Developed by Ernst Abbe in 1873, it serves as a foundational metric for evaluating microscope performance in resolving fine specimen details.^[20]^[7] These criteria differ in application and stringency: the Rayleigh limit suits incoherent imaging of discrete points in telescopes, yielding coarser resolution ($0.61 \frac{\lambda}{\mathrm{NA}} linearly in microscopy), while the Abbe limit targets coherent periodic objects in microscopes, and the Sparrow criterion bridges them with an intermediate value closer to Abbe's (about two-thirds of Rayleigh's separation). The Rayleigh and Abbe approaches assume ideal diffraction patterns, with Rayleigh emphasizing point-source separation and Abbe focusing on grating-like resolvability.^[19]^[21] All three criteria presuppose aberration-free optics, monochromatic incoherent or coherently controlled light, and perfect alignment; real-world systems suffer degradation from aberrations, chromatic dispersion, and environmental noise, often reducing effective resolution below theoretical values.^[7]^[19]

Resolution in Optical Components

Lens Resolution

Lens resolution in optical imaging systems, such as cameras and microscopes, is fundamentally limited by the lens design and inherent imperfections that influence the point spread function (PSF), which describes how a point source is blurred in the image plane. The aperture size determines the amount of light gathered and the diffraction effects, with larger apertures potentially enhancing resolution up to the diffraction limit but often introducing more aberrations if not properly corrected. Focal length affects the magnification and field of view, where longer focal lengths can reduce certain aberrations like spherical aberration for a fixed lens diameter, thereby sharpening the PSF. Aberrations, including spherical and chromatic types, further broaden the PSF: spherical aberration causes peripheral rays to focus at different points than central rays, depending on aperture size, while chromatic aberration disperses wavelengths, leading to color-dependent focus shifts that degrade overall sharpness.^[22]^[23] A key quantitative measure of lens resolution is the modulation transfer function (MTF), which quantifies the contrast transfer at various spatial frequencies relative to low-frequency performance. The MTF is defined as the magnitude of the Fourier transform of the PSF, given by:

\text{MTF}(f) = \left| \int_{-\infty}^{\infty} \text{PSF}(x) \exp(-i 2\pi f x) \, dx \right|

where f is the spatial frequency. Aberrations and defocus reduce the MTF curve's height and shift its cutoff frequency, directly impacting resolvable detail. In practice, lens performance is evaluated using resolution charts featuring periodic line pairs, measured in line pairs per millimeter (lp/mm), where high-end lenses typically achieve resolutions exceeding 100 lp/mm at 50% contrast, enabling fine detail reproduction in professional applications.^[24]^[25] Lens design involves trade-offs to balance resolution with functionality; for instance, zoom lenses often sacrifice peak resolution for versatility in focal length adjustment, resulting in lower MTF values across the zoom range compared to fixed prime lenses. Apochromatic lenses, which correct chromatic aberration for three wavelengths (typically red, green, and blue), minimize color fringing and maintain a narrower PSF, enhancing resolution in color imaging systems. Historically, developments by Zeiss and Leica in the 20th century, including Zeiss's pioneering use of MTF for lens evaluation in 1943, standardized testing protocols and drove advancements in aberration-corrected designs for high-resolution optics.^[26]^[27]^[28]

Detector Resolution

In optical imaging systems, the spatial resolution of detectors, such as image sensors, is fundamentally limited by pixelation, where the pixel pitch—the center-to-center distance between adjacent pixels—defines the sampling grid for capturing optical details. The fill factor, representing the active light-sensitive area within each pixel relative to its total size, further influences how effectively fine spatial variations are recorded, as lower fill factors can lead to reduced sensitivity and interpolation artifacts. The maximum resolvable spatial frequency, known as the Nyquist frequency f_N, is calculated as f_N = \frac{1}{2p}, where p is the pixel pitch in units of length, such as micrometers; this limit arises because frequencies above f_N cannot be distinguished without overlap in the sampled signal.^[29]^[30] The sampling theorem, originally formulated by Whittaker and Shannon, underpins these limits by requiring that the sampling rate exceed twice the highest frequency component in the input signal to prevent aliasing, where high-frequency details masquerade as lower-frequency patterns in the reconstructed image. In practice, for a detector to faithfully capture optical details from the lens, the pixel sampling must satisfy this condition relative to the incoming spatial frequencies; the resolvable frequency is thus bounded by f_{\text{res}} \leq \frac{1}{2p}, ensuring no destructive interference from undersampling. Violation of this theorem introduces moiré patterns or blurred edges, degrading the effective resolution even if the optics are capable of higher performance.^[31]^[32] Detector types significantly affect resolution capabilities, with charge-coupled devices (CCDs) and complementary metal-oxide-semiconductor (CMOS) sensors representing the primary architectures. CCDs excel in high-resolution applications due to their uniform charge transfer and minimal fixed-pattern noise, enabling precise capture of subtle details across large arrays, though they require higher power for charge shifting. In contrast, CMOS sensors integrate amplification and processing at the pixel level, offering lower power consumption and faster readout speeds suitable for compact systems, but they historically suffered from higher pixel-to-pixel variations that could limit uniformity in resolution; modern advancements have narrowed this gap for resolutions up to VGA and beyond. Back-illuminated sensors, common in both CCD and CMOS designs, enhance quantum efficiency by illuminating the photosensitive layer from the rear, bypassing front-side wiring obstructions and increasing light capture by up to 70% in some configurations, which indirectly boosts resolution by improving signal fidelity in low-light conditions.^[33]^[34]^[35] Noise sources in detectors further constrain effective spatial resolution by masking fine details, with shot noise—arising from the statistical fluctuation in photon arrival (following Poisson statistics)—and readout noise from electronic amplification both contributing to uncertainty in pixel values. These effects reduce the signal-to-noise ratio (SNR), where a higher SNR is essential for distinguishing closely spaced features; for instance, shot noise dominates at high light levels, while readout noise becomes prominent in low-signal scenarios, potentially halving the usable resolution if SNR falls below 5:1 for edge detection tasks. Metrics for detector resolution often include pixels per millimeter (e.g., 1000–2000 pixels/mm for high-end sensors) or total megapixels, but these quantify sampling density rather than true resolving power, which remains capped by the optics; as an example, as of 2025, smartphone CMOS sensors typically feature pixel pitches of approximately 0.5 to 1.2 μm, enabling 48–200 megapixel arrays yet limited to optical resolutions around 100–200 line pairs per millimeter in practice.^[36]^[37]^[38]^[39]

Temporal Aspects of Resolution

Temporal resolution in optical systems refers to the capability to capture and distinguish rapid temporal changes in a scene, essential for imaging dynamic phenomena such as moving objects or transient events. It is typically quantified by metrics like frames per second (fps) or shutter speed, which determine the rate at which sequential images are acquired. This aspect is particularly critical in video and high-speed imaging, where insufficient temporal resolution can lead to motion artifacts that obscure details. Key factors influencing temporal resolution include sensor readout speed, which governs how quickly image data can be transferred from the detector, and exposure time, the duration each frame is exposed to light. Motion blur arises when objects move during the exposure period, resulting in image displacement calculated as d = v \times t, where d is the displacement, v is the object's velocity, and t is the exposure time. Shorter exposure times minimize this blur but require brighter illumination or more sensitive sensors to maintain signal quality. In applications, temporal resolution varies widely by domain. Cinematography typically employs frame rates of 24 to 120 fps to achieve smooth motion perception for audiences, as seen in standard film production. In contrast, scientific imaging for fluid dynamics often demands kilohertz (kHz) rates—up to several thousand fps—to resolve turbulent flows and shock waves in real time. Trade-offs are inherent in pursuing higher temporal resolution: increasing fps reduces the light gathered per frame, which can elevate noise levels and compromise spatial resolution due to diminished signal-to-noise ratios. This necessitates advancements in sensor technology, such as high quantum efficiency detectors, to balance these constraints. Metrics for evaluating temporal resolution include the temporal modulation transfer function (MTF), which assesses how well the system preserves contrast in periodically varying signals over time, and the just noticeable events per second, indicating the minimum detectable change rate. For biological context, the human visual system exhibits a flicker fusion threshold around 60 Hz, beyond which rapid fluctuations appear continuous.

System-Level and Contextual Resolution

Combined System Resolution

In optical imaging systems, the overall resolution arises from the interplay of multiple components, including lenses, detectors, and other elements in the imaging chain, rather than any single part in isolation. The modulation transfer function (MTF) provides a quantitative framework for assessing this combined performance, as the system's MTF is the product of the individual MTFs of its components, assuming incoherent illumination and linear shift-invariant conditions.^[25] This multiplicative relationship reflects the convolution of point spread functions (PSFs) in the spatial domain, where each component's blurring effect compounds to degrade contrast transfer at higher spatial frequencies. Mathematically, the system MTF can be expressed as

\text{MTF}_\text{sys}(f) = \text{MTF}_\text{lens}(f) \times \text{MTF}_\text{detector}(f) \times \prod \text{MTF}_\text{other}(f),

where f denotes spatial frequency.^[40]^[41] Deconvolution techniques can partially reverse these effects by applying an inverse filter based on the known system MTF, restoring contrast and effectively enhancing resolution, though noise amplification limits its application in practice.^[42]^[43] The lateral resolution of the combined system corresponds to the minimum resolvable feature size in the object plane, determined by the frequency where the system MTF falls to a threshold like 10% or 5%, often yielding a value coarser than that of the best individual component.^[44] In practice, the weakest link dominates due to the multiplicative nature of MTF degradation; for instance, pairing a high-resolution lens capable of 100 lp/mm with a low-resolution sensor limited to 50 lp/mm results in a system resolution no better than approximately 50 lp/mm, rendering the lens's potential underutilized.^[45] In digital cameras, the effective resolution frequently falls short of the sensor's pixel count because optical blur from the lens spreads light across multiple pixels, reducing the usable detail; for example, a 20-megapixel sensor might achieve only 12-15 megapixel equivalent resolution when combined with a typical consumer lens exhibiting MTF drop-off beyond 40 lp/mm.^[44]^[45] Ray-tracing software such as Zemax enables prediction of this system-level performance by modeling the full optical chain, including diffraction, aberrations, and detector sampling, to optimize component matching before fabrication.^[46]

Bandwidth and Electronic Effects

In analog and mixed-signal optical systems, bandwidth B imposes fundamental limits on the transmission of high-frequency signals, which encode fine spatial details essential for resolution. These high frequencies, corresponding to sharp edges and textures in the imaged scene, are attenuated beyond B, effectively acting as a low-pass filter on the signal chain. Consequently, the maximum resolvable spatial frequency f_{\max} scales approximately with B, as higher bandwidth preserves more frequency content for reconstructing detailed images.^[47] This bandwidth constraint manifests in practical effects, such as edge blurring in video systems where amplifiers introduce low-pass filtering, smoothing transitions and reducing perceived sharpness. The system's rise time \tau, a measure of how quickly signals can transition without distortion, is inversely related to bandwidth by \tau \approx 0.35 / B, where slower rise times (lower B) further degrade transient details. Additionally, expanding B broadens the noise bandwidth, amplifying thermal and other noise sources, which forces a trade-off: enhanced resolution comes at the cost of reduced signal-to-noise ratio and overall sensitivity in low-light optical imaging.^[47]^[48]^[49] Historically, the Electronic Industries Alliance (EIA) standards from the 1950s formalized this linkage in analog television, correlating bandwidth directly to horizontal resolution in TV lines (TVL). For instance, NTSC systems with a 4.2 MHz bandwidth achieved approximately 330–340 TVL, based on an empirical factor of about 80 TVL per MHz, enabling standardized assessment of monochrome broadcast quality.^[50] In contemporary high-speed applications, such as oscilloscopes and scientific high-speed cameras, the analog-to-digital converter (ADC) sampling rate governs the effective bandwidth post-detection. To faithfully capture signals without aliasing, the sampling rate must exceed 2.5–5 times the bandwidth, depending on the system's frequency response; for example, a 1 GHz bandwidth typically requires 2.5–5 GS/s sampling to maintain resolution in transient optical events.^[51]^[52]

Biological and Environmental Resolution

Optical resolution in biological systems is fundamentally limited by the spacing and density of photoreceptors in the retina. In the human eye, visual acuity for 20/20 vision corresponds to a resolution of approximately 1 arcminute (60 arcseconds), determined by the minimum separable angle for high-contrast details.^[53] This limit arises primarily from the retinal cone spacing, with foveal cones exhibiting a center-to-center separation of about 0.5 arcminutes, though neural processing and optical aberrations reduce effective acuity to around 1 arcminute.^[54] Across species, biological resolution varies to suit ecological needs. For instance, diurnal raptors like the Harris's hawk achieve higher achromatic spatial resolution of up to 62 cycles per degree, surpassing the human range of 30–60 cycles per degree, enabling superior detail detection for hunting.^[55] In contrast, insect compound eyes prioritize a wide field of view over high resolution; the interommatidial angle (Δφ), which governs acuity, is larger in insects (often 1–5 degrees), trading fine detail for panoramic coverage up to 360 degrees in some species, as seen in honeybees where eye skewness expands the field while reducing sensitivity and resolution.^[56] Environmental factors, particularly atmospheric turbulence, impose additional limits on optical resolution in ground-based observations. In astronomy, turbulence creates a "seeing disk" typically 1–2 arcseconds in diameter under median conditions, blurring point sources due to refractive index variations from temperature fluctuations.^[57] This degradation is quantified by the Fried parameter r_0, the atmospheric coherence length over which wavefront phase errors remain below one radian, with r_0 values of 10–20 cm at visible wavelengths corresponding to such seeing.^[58] To mitigate atmospheric effects, adaptive optics systems employ deformable mirrors that dynamically adjust shape—often thousands of times per second—based on real-time wavefront measurements, restoring near-diffraction-limited performance.^[59] The extent of degradation and correction efficacy is often assessed using the Strehl ratio, defined as the ratio of the observed peak intensity to the ideal diffraction-limited peak, where values below 0.8 indicate significant turbulence-induced loss, dropping to 0.2–0.4 without correction at near-infrared wavelengths.^[60]

Measurement and Evaluation Methods

Test Target Patterns

Test target patterns are traditional physical or printed charts used to empirically evaluate the resolving power of optical systems by imaging periodic or edge features and assessing the visibility of fine details. These patterns provide a straightforward, visual method for determining the spatial frequency at which contrast drops below a discernible threshold, often defined as the Rayleigh criterion where lines are resolvable if separated by at least the Airy disk radius.^[61] Bar patterns, consisting of alternating light and dark lines, represent one of the earliest and most common test targets for optical resolution. The USAF 1951 pattern features binary bars arranged in horizontal and vertical orientations, with six elements per group forming sets of three lines and spaces, covering a resolution range from 0.25 to 228 line pairs per millimeter (lp/mm).^[61] This design allows users to identify the highest resolvable group and element by visual inspection, where the smallest discernible feature corresponds to approximately 2.2 micrometer line widths at the finest scale.^[61] Similarly, the NBS 1952 pattern, developed for photographic lenses and microscope objectives, employs three-bar groups in high- and low-contrast variants, spanning 2.4 to 80 lp/mm in standard configurations, with extendable ranges up to 320 lp/mm through scaling or distance adjustments.^[62] These charts are typically placed at a distance of 26 times the focal length from the lens to minimize edge effects during testing.^[62] The slanted edge method, standardized in ISO 12233 (originally 2000, updated 2024), shifts from bar patterns to a straight edge tilted at an angle (typically 5-10 degrees) relative to the imaging sensor to sample sub-pixel phases. This approach derives the modulation transfer function (MTF) by analyzing the edge spread function, which captures the system's response to a step transition and enables computation of spatial frequency responses without aliasing artifacts from discrete sampling.^[63] Primarily applied to digital cameras, it quantifies resolution in cycles per pixel or lp/mm, providing a more precise metric than visual bar resolution by averaging over multiple phases.^[63] For video and display systems, specialized targets incorporate multiburst signals to assess bandwidth and resolution. The EIA 1956 pattern, designed for black-and-white analog television, includes multiburst zones generating sinusoidal waveforms at frequencies from 200 to 1000 TV lines (transitions per picture height), allowing evaluation of horizontal resolution and transient response in broadcast equipment.^[64] The IEEE 208-1995 standard builds on this with a chart featuring vertical and horizontal bar arrays, plus frequency-specific bursts, to measure end-to-end resolution in camera-display chains, including monitors, by quantifying the frequency where detail fidelity drops significantly.^[65] Random patterns address limitations in periodic targets by using noise-like distributions, such as white noise transparencies with controlled spatial-frequency content, to measure MTF across the entire image field simultaneously. These targets mitigate aliasing bias inherent in regular gratings, where pixel alignment can artificially enhance or degrade perceived resolution, by ensuring shift-invariant results through pixel sizes below the Nyquist frequency.^[66] Printed on films or substrates compatible with visible to infrared wavelengths, they enable comprehensive testing without mechanical scanning.^[66] Despite their utility, test target patterns have notable limitations, including the assumption of ideal high contrast (often 0.9 or greater), which overestimates performance in real-world low-contrast scenarios and ignores phase effects in MTF interpretation.^[67] Additionally, these static charts are increasingly outdated for computational optical systems, where post-processing algorithms like super-resolution can enhance effective resolution beyond physical limits, rendering traditional visual assessments insufficient.^[68]

Interferometric and Wavefront Methods

Interferometric methods enable precise quantification of optical resolution by analyzing phase variations in the wavefront, which directly impact the point spread function and thus the system's ability to resolve fine details. These techniques exploit the interference of coherent light to produce interferograms—contour maps of optical path differences—that reveal aberrations such as defocus, astigmatism, and spherical error, limiting resolution below the diffraction limit. By measuring deviations from an ideal spherical wavefront, interferometry achieves sub-wavelength accuracy, far surpassing geometric test methods.^[69] The foundational instrument for these measurements is the Michelson interferometer, invented by Albert A. Michelson in 1881 and refined in the early 1900s for precision optical testing. This setup splits a light beam into two paths, reflects them off test and reference surfaces, and recombines them to form interference fringes sensitive to path length differences on the order of nanometers. Later adaptations, such as the Twyman-Green interferometer developed around 1916, modified the Michelson configuration specifically for testing plane and spherical optics by using a point source and collimating lens to generate fringes that directly map surface irregularities or transmitted wavefront errors. In Twyman-Green setups, straight fringes indicate aberration-free optics, while distortions in the pattern quantify resolution-degrading defects. Similarly, shearing interferometers, including lateral and radial variants, produce interferograms by introducing a small displacement between two copies of the same wavefront, eliminating the need for a perfect reference and highlighting local phase gradients from aberrations.^[70]^[71]^[72] Wavefront sensing techniques build on interferometry to provide quantitative phase maps. The Shack-Hartmann sensor, an array of microlenses that samples the incoming wavefront into sub-apertures and measures centroid displacements on a detector, determines local slopes and reconstructs the full phase error distribution with high spatial resolution. These phase errors are commonly decomposed using Zernike polynomials, a set of orthogonal functions that represent aberrations in a pupil plane, allowing isolation of low-order (e.g., tilt, defocus) and higher-order terms affecting resolution. Key metrics derived from these analyses include the root mean square (RMS) wavefront error, which averages the phase variance across the aperture in waves, and the Strehl ratio, defined as the ratio of peak intensity in the aberrated point spread function to the diffraction-limited ideal. The Strehl ratio approximates the impact on resolution via the Marechal formula:

S \approx \exp\left( -\left( \frac{2\pi \sigma}{\lambda} \right)^2 \right)

where \sigma is the RMS wavefront error and \lambda is the wavelength; values above 0.8 typically indicate near-diffraction-limited performance.^[73] These methods find essential applications in optical shop testing, where Twyman-Green and shearing interferometers verify the figure and quality of manufactured components like lenses and mirrors to ensure they meet resolution specifications. In astronomy, wavefront sensing with Shack-Hartmann arrays facilitates real-time telescope alignment and adaptive optics correction, compensating for atmospheric aberrations to achieve the full diffraction-limited resolution of large apertures. Such techniques have been pivotal since the mid-20th century, evolving from Michelson's early work to modern dynamic systems that maintain high-fidelity imaging in demanding environments.^[74]^[75]

Modern Digital Techniques

Modern digital techniques for assessing optical resolution leverage computational methods to analyze and enhance the performance of captured images, often surpassing the limitations of traditional optical hardware. These approaches process digital data from sensors to estimate resolution metrics like the modulation transfer function (MTF) without requiring specialized physical test equipment. By applying Fourier analysis to image features, such as slanted edges, algorithms derive the system's frequency response directly from standard photographs.^[76] A prominent method for digital MTF estimation is the slant-edge technique standardized in ISO 12233, which uses a slanted black-white edge in the image to create a supersampled edge spread function (ESF). The process involves projecting pixels along the edge direction into subpixel bins to form the ESF, followed by differentiation to obtain the line spread function (LSF), and then applying Fourier transformation to compute the MTF as a function of spatial frequency. This method allows measurement of resolution in cycles per pixel or line pairs per millimeter, accounting for both lens and detector contributions, and is widely implemented in software tools for camera evaluation. The ISO 12233:2024 edition refines this by incorporating low-contrast edge targets for more robust analysis in real-world conditions, improving accuracy for electronic still-picture cameras.^[63]^[76] Super-resolution techniques extend effective resolution by computationally reconstructing higher-detail images from multiple low-resolution inputs or single frames, with measurement relying on metrics that quantify fidelity to ground truth. Algorithms employing sub-pixel shifting align slightly offset images—often from burst captures or camera motion—to synthesize a higher-resolution output, effectively increasing spatial sampling beyond the sensor's Nyquist limit. Performance is assessed using peak signal-to-noise ratio (PSNR), which measures pixel-level error in decibels (higher values indicate better reconstruction), and structural similarity index (SSIM), which evaluates perceived quality by comparing luminance, contrast, and structure (values closer to 1 denote superior results). For instance, multi-image super-resolution with sub-pixel shifts can yield PSNR improvements of up to 2-3 dB over single-frame baselines in controlled tests.^[77]^[78] In computational imaging systems, resolution enhancement arises from designed light modulation followed by inverse processing, such as in coded aperture cameras where a patterned mask encodes the scene into a blurred shadow on the detector. Deconvolution algorithms then recover the sharp image; the Richardson-Lucy method, an iterative maximum-likelihood estimator for Poisson noise, refines the estimate by alternately updating the object and point spread function, often converging in 10-50 iterations to boost effective resolution by factors of 2-4 in coded systems. Similarly, light-field cameras capture directional light rays via microlens arrays, enabling post-capture refocusing and super-resolution through sub-aperture image fusion, where convolutional neural networks can enhance both spatial and angular resolution by up to 4x in benchmark datasets. These techniques trade hardware simplicity for computational load but achieve resolution gains unattainable with conventional optics alone.^[79]^[80] Post-2010 advances have integrated these methods into consumer devices, particularly through smartphone computational photography. The Google Pixel's handheld multi-frame super-resolution algorithm captures bursts of raw Bayer images, aligning them via optical flow to exploit hand tremor for sub-pixel shifts, then merges them directly into a full-color high-resolution output, achieving PSNR values around 42 dB on standard datasets and enabling 2x lossless zoom. AI-based upscaling, exemplified by the Super-Resolution Convolutional Neural Network (SRCNN) introduced in 2014, trains deep networks to map low-resolution inputs to high-resolution counterparts, improving PSNR by 1-2 dB over traditional interpolation while preserving fine details in single images. These innovations have democratized enhanced resolution, with widespread adoption in mobile cameras by the mid-2020s.^[81]^[82] Despite these gains, modern digital techniques introduce limitations, including computational artifacts like ringing or aliasing from imperfect deconvolution, and they do not represent true optical resolution as they cannot recover information lost to diffraction or sensor noise. Over-reliance on algorithms may inflate perceived sharpness without addressing fundamental physical constraints, necessitating careful validation against optical benchmarks.^[78]

Standards and Interpretation

International standards for measuring optical resolution in imaging systems have evolved to address both traditional and digital technologies. The ISO 12233:2024 standard specifies methods for determining the resolution and spatial frequency response (SFR) of electronic still-picture cameras using test charts and edge analysis techniques.^[63] For display systems, IEEE Std 208-1995 outlines techniques for quantifying camera resolution limits, focusing on video techniques and frequency response in broadcast and surveillance applications. Historically, the NBS 1010a Microcopy Resolution Test Chart from the National Bureau of Standards served as a benchmark for evaluating microcopying systems through bar patterns, but it has been superseded by more modern ISO equivalents.^[83] Interpretation of resolution results often relies on the modulation transfer function (MTF), where the spatial frequency at which MTF drops to 10-20% of its peak value indicates the effective resolution limit, balancing contrast loss with detail discernibility.^[25] This metric provides a quantitative basis for comparing system performance, though its meaning varies by context; in astronomy, resolution emphasizes angular separation under atmospheric seeing conditions, prioritizing wide-field diffraction limits, whereas microscopy focuses on lateral resolution governed by numerical aperture and illumination wavelength for sub-micron specimen details.^[7] Earlier standards like those for video targets from the pre-digital era often overlooked pixel-level artifacts in electronic imaging, leading to gaps in applicability for contemporary systems. Recent efforts by ISO/TC 42, the technical committee for photography, continue to refine these through updates like ISO 12233:2024.^[84] Best practices recommend combining multiple measurement methods—such as edge-based SFR with bar targets—to mitigate biases from single approaches, and reporting results with confidence intervals derived from repeated trials to quantify variability in real-world conditions.^[85] Looking ahead, optical resolution standards are increasingly integrating with machine vision frameworks like ISO 24942 (adopted from EMVA 1288 in 2025), which characterizes sensor performance parameters including dynamic range and noise that indirectly influence effective resolution in automated inspection systems.^[86]^[87]

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The Rayleigh criterion defines the limit of resolution in a diffraction-limited system, in other words, when two points of light are distinguishable or resolved ...
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[PDF] The Rayleigh Criterion of Resolution and Light Sources of Different ...
In 1879, John William Strutt (Third Baron Rayleigh) investigated the limit of resolution (resolving or separating power) of some optical instruments, such ...
[9]
Telescope resolution
Resolution is another vital telescope function. Simply put, telescope resolution limit determines how small a detail can be resolved in the image it forms.
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Optical Filters in Medical Imaging: Accurate Diagnoses - UQG
Sep 7, 2023 · Optical filters have the ability to improve the accuracy of diagnostics by enhancing the two most important aspects of medical imaging: contrast and resolution.
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On the Diffraction of an Object-glass with Circular Aperture - ADS
On the Diffraction of an Object-glass with Circular Aperture. Airy, G. B.. Abstract. Publication: Transactions of the Cambridge Philosophical Society.
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https://www.edmundoptics.com/knowledge-center/application-notes/imaging/limitations-on-resolution-and-contrast-the-airy-disk/
[13]
[PDF] Chapter 8: Diffraction [version 1208.1.K] - Caltech PMA
As we shall see, the computation is essentially an exercise in Fraunhofer diffraction theory. The essence of the computation is to idealise the telescope as a ...
[14]
[PDF] Chapter 3 Diffraction
Diffraction gives rise to a “Point Spread. Function”, meaning that a point source on the sky will appear as a blob of a certain size on your image plane. The ...
[15]
Resolution - Nikon's MicroscopyU
The resolving power of a microscope is the most important feature of the optical system and influences the ability to distinguish between fine details of a ...
[16]
Rayleigh and other resolution criteria - Optica Publishing Group
Abstract. The Rayleigh resolution criterion, which states that two objects are just resolved when the maximum in the focal plane of one Airy pattern falls ...<|separator|>
[17]
long focal depth imaging over along range - NICT
Here, we have used the Rayleigh criterion of 73.5 % intensity at the midpoint between the peak intensities of the images of the bars as the criterion of ...<|separator|>
[18]
On Spectroscopic Resolving Power - Astrophysics Data System
SAO/NASA ADS Astronomy Abstract Service. ON SPECTROSCOPIC RESOLVING POWER B~ C. M. SPARROW If a spectroscope is just able to separate two monochromatic ...Missing: original | Show results with:original
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The Diffraction Barrier in Optical Microscopy | Nikon's MicroscopyU
In contrast, the Sparrow resolution limit is defined as the distance between two point sources where the images no longer have a dip in brightness between the ...
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Diffraction limit - Scientific Volume Imaging
This definition of microscope resolution is also often referred to as the Rayleigh Criterion.Missing: original | Show results with:original
[21]
Super-Resolution Tutorial - Education - Advanced Microscopy
In the Sparrow criterion, the sum of the two Airy patterns produces a flat intensity profile; in the Abbe limit, a small dip is still discernible between the ...
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Chapter 1 - Geometrical Optics - SPIE Digital Library
For a given lens size, a longer focal length will result in a smaller spherical aberration. 2. For a given lens size and focal length, a larger-refractive-index ...Missing: PSF | Show results with:PSF
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Aberrations - Book chapter - IOPscience
This chapter introduces the forms and equations of the 3 rd -order aberrations and shows how they depend on aperture and field point.
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None
### Summary of MTF Definition, Equation, and Lens Factors from the Document
[25]
https://www.edmundoptics.com/knowledge-center/application-notes/optics/introduction-to-modulation-transfer-function/
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https://www.edmundoptics.com/f/high-resolution-macro-zoom-lenses/14130/
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Chromatic and Monochromatic Optical Aberrations - Edmund Optics
This type of correction is usually achieved by using an apochromatic lens, which is corrected such that three wavelengths focus at the same point, or a ...
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Lenses Make History - ZEISS Lenspire
Nov 2, 2020 · In 1943, ZEISS was the first lens manufacturer to develop a method for measuring a lenses' image quality using MTF (modulation transfer function) ...Missing: Leica century resolution
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Q Selection for an electro-optical earth imaging system
Sep 12, 2013 · The Nyquist frequency for the image is defined as υNyquist = 1/2p. ... detector, consider two spatial frequency domains. The first domain ...
[30]
The modulation transfer function of a type II superlattice focal plane ...
Jan 20, 2012 · A detector's resolution is limited by its Nyquist spatial frequency (nominally the reciprocal of twice the pixel pitch). At that frequency, ...
[31]
Sampling in imaging systems - Optica Publishing Group
ALIASING. In most contexts, the word aliasing connotes the Whittaker–Shannon sampling theorem or the Nyquist sampling rate. ... image or alias. As a ...
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Aliasing and blurring in 2-D sampled imagery
In this paper we formulate and compute the expected magnitude of image degradation due to aliasing and blurring in continuous reconstructions of discrete data ...
[33]
INTRODUCTION - SPIE Digital Library
CMOS sensors have advantages compared to CCDs: lower power consumption, lower cost, and greater radiation hardness. However, CCDs are the choice for very high- ...
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Comparison of modern CCD and CMOS image sensor technologies ...
Our analysis shows that for low resolution imaging (VGA and below) CCD and CMOS sensor technologies are converging to practically indistinguishable solutions.Missing: differences | Show results with:differences
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Efficiency enhancement in a backside illuminated 1.12 μm pixel ...
Jul 8, 2016 · As r decreases to 0.5 μm (0.25 μm), the peak efficiencies of the pixels with parabolic B-, G-, and R-CFs become 66.6% (70.7%), 60.6% (71.7%), ...
[36]
Characterizing and correcting camera noise in back-illuminated ...
In this paper, we systematically characterize the impact of different types of sCMOS noise on image quality and perform corrections to these types of sCMOS ...
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Comparing CMOS sensor technologies - SPIE
Aug 1, 2013 · The shot noise zone is statistically limited by the particle nature of the flooding photos and electrons that are involved, which is the common ...
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A 64Mpixel CMOS Image Sensor with 0.56μm Unit Pixels Separated ...
Mar 17, 2022 · A 64Mpixel CMOS Image Sensor with 0.56μm Unit Pixels Separated by Front Deep-Trench Isolation. Abstract: CMOS image sensors (CISs) with deep- ...
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Modulation Transfer Function (MTF) in Optical System Design
The MTF of the system is the product of all individual MTF curves, including those of the imaging lens, camera sensor, image capture boards, and video cables.Missing: MTFs | Show results with:MTFs
[40]
Modulation Transfer Function | Nikon's MicroscopyU
... modulation transfer function (MTF) is described by the equation: Formula 3 - Modulation Transfer Function (MTF). MTF = Image Modulation/Object Modulation.
[41]
Incorporation of an experimentally determined MTF for spatial ...
The deconvolution approach can provide improved resolution reconstructions for high-angular-sampling OPT acquisitions and the MTF-mask filtering technique leads ...
[42]
[PDF] MTF CHARACTERIZATION AND DECONVOLUTION OF RAPIDEYE ...
As a result, we have implemented a quantitative MTF deconvolution filter which accurately reconstructs image content and leads to significant visual improvement ...
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Camera Resolution: Combining Detector and Optics Performance
Camera resolution depends upon the optical blur diameter and the detector size. Schade combined these to create an equivalent resolution that is a fun.
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Optimizing system resolution: a practical guide to matching lens and ...
System resolution is optimized by matching lens and sensor MTFs, using the MTF chart to calculate the total system MTF by multiplying lens and sensor MTFs.
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Ansys Zemax OpticStudio | Optical Design and Analysis Software
Get everything you need to simulate, optimize, and tolerance your optical designs. Design complex optical systems for a wide range of applications.
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Bandwidth Versus Video Resolution - Analog Devices
Jul 22, 2002 · This article discusses the key relationship between video resolution and the required bandwidth to accurately process and display that video signal.
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Where does the formula BW = 0.35 / t10%-90% come from? - Tektronix
Risetime is the time measured from Vout,90% - Vout,10%. The response for an RC network is Vout = Vin (1 – e-t/t) where t= RC. From the above equation we find t ...Missing: B | Show results with:B
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Analysis of the resolution-bandwidth-noise trade-off in wavelength-based photonic analog-to-digital converters
### Summary of Resolution-Bandwidth-Noise Trade-Off in Photonic ADCs
[49]
Horizontal resolution: Pixels or lines | TV Tech - TVTechnology.com
Apr 1, 2005 · The horizontal resolution factor for a 4.28MHz bandwidth is: 339/4.28MHz = 79.2 lines/MHz rounded to 80 lines/MHz. In countries using the NTSC ...
[50]
Evaluating Oscilloscope Bandwidth, Sample Rate, and ... - Tektronix
Learn to evaluate an oscilloscope's frequency, bandwidth, and sample rate. Our guide explains how to select the right scope for your measurement ...Missing: cameras | Show results with:cameras
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Understanding Digital Oscilloscope Sample Rate and Analog ...
Mar 24, 2020 · With a Gaussian frequency response oscilloscope, we usually need the real-time sampling rate to be 4-5 times the oscilloscope bandwidth.Measuring Analog Signals · Measuring Digital Signals · Sampling RateMissing: cameras | Show results with:cameras
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Bill Keel's Lecture Notes - Astronomical Techniques - The Human Eye
The eye is considered to have a resolution element of order 1 arcminute, and resolution near 3. Some people can just make out the crescent of Venus (1 arcmin ...
[53]
Visual Acuity - Ento Key
Jun 29, 2024 · Cones in the fovea have a center-to-center separation of about 0.5 minutes of arc (0.008 degrees), which fits nicely with the observed acuity ...
[54]
High resolution of colour vision, but low contrast sensitivity in a ...
Aug 29, 2018 · Our study reveals that Harris's hawks have high spatial resolving power for both achromatic and chromatic vision, suggesting the importance of colour vision ...
[55]
Measuring compound eye optics with microscope and microCT ...
Mar 7, 2023 · In contrast to our own camera-type eyes, compound eyes reveal their resolution, sensitivity, and field of view externally, provided they have ...Results · Microscope Images · Measuring Ommatidia Using...
[56]
astronomical seeing, part 1: the nature of turbulence - Handprint.com
Thermal turbulence causes image perturbations on the order of 10–5 to 10–4 radians (2 to 20 arcseconds); the radius of the forward scatter caused by diffusion ...
[57]
Effect of atmospheric turbulence on the telescope image (seeing error)
In terms of r0, the FWHM of long-exposure atmospheric PSF (seeing) is given by FWHML=0.98λ/r0, in radians (FWHML=202,140λ/r0 in arc seconds), with λ being the ...Missing: disk | Show results with:disk
[58]
None
### Summary of Adaptive Optics Using Deformable Mirrors for Atmospheric Correction
[59]
AO tutorial 1: turbulence
The image, compared to object, is smoothed, its resolution is degraded. However, for a given telescope diameter $D$ this degradation is the least possible.
[60]
Resolution Test Targets - Thorlabs
Thorlabs offers resolution test targets with 1951 USAF, NBS 1952, and NBS 1963A patterns. Targets are also available with sector star (also known as Siemens ...
[61]
[PDF] Method for determining the resolving power of photographic lenses
NBS Resolution Test Chart of 1952. 7.1. Description. In the design of this ... A Micro Tessar lens or a microscope objective is best suited for this type ...
[62]
ISO 12233:2017
### Summary of ISO 12233:2017
[63]
EIA-1956 Resolution Video Test Chart QA-70 | Imatest
In stockThe EIA-1956 chart is obsolete, based on EIA 1956, includes linear/pie shapes, 200-1000 lines, and is used for tele-cine camera measurements.Missing: multiburst | Show results with:multiburst
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IEEE Video Resolution Chart (QA-71) - Applied Image
30-day returnsThe IEEE Video Resolution Chart conforms to standard STD 208-1995, 'Measurement of Resolution of Camera Systems'. This standard is used to test frequency ...Missing: monitor | Show results with:monitor
[65]
[PDF] Random transparency targets for modulation transfer function ...
Our random-target method does not require mechanical scanning and tests the entire image plane at one time. The random-target method creates random test ...
[66]
https://pages.charlotte.edu/glenn-boreman/wp-content/uploads/sites/146/2014/06/Daniels_OE_4_1995.pdf
[67]
None
Nothing is retrieved...<|separator|>
[68]
[PDF] Introduction to Interferometric Optical Testing
• Vibration insensitive, quantitative interferometer. • Surface figure measurement (nm resolution). • Snap shot of surface height. • Acquisition of “phase ...
[69]
Michelson interferometer | Definition, Description, & Facts - Britannica
Sep 27, 2025 · It was invented in 1881 by the American physicist A.A. Michelson. The instrument consists of a half-silvered mirror that divides a light beam ...
[70]
What is a Twyman-Green Interferometer? - 4D Technology
A Twyman-Green interferometer is a laser interferometer configuration used for measuring the surface shape and transmitted wavefront quality of optical ...
[71]
Shearing Interferometers - Thorlabs
Thorlabs' Shearing Interferometers are designed to qualitatively test the collimation of a coherent beam of light.
[72]
https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=2970
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Optical Shop Testing | Wiley Online Books
Nov 2, 2006 · A single book descriptions of all tests carried out in the optical shop that are applicable to optical components and systems.
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Effect of telescope alignment on a stellar interferometer
For a ground-based stellar interferometer, we investigate the effect of wave-front distortions that are due to telescope alignment errors and other factors.
[75]
https://opg.optica.org/abstract.cfm?uri=ao-38-28-6055
[76]
Multi-image hybrid super-resolution reconstruction via interpolation ...
Apr 24, 2023 · MISR is used to fuse multiple LR images with a sub-pixel shifts to produce a HR image with high-precision features.
[77]
A comprehensive review of deep learning-based single image super ...
This survey is an effort to provide a detailed survey of recent progress in single-image super-resolution in the perspective of deep learning
[78]
Efficient deconvolution methods for astronomical imaging
Introduction. The Richardson-Lucy (RL) algorithm (Richardson 1972; Lucy 1974) is a renowned iterative method for image deconvolution in astronomy and other ...
[79]
[PDF] Light Field Super-Resolution: A Benchmark - CVF Open Access
For portable light field cameras, the micro-lens-array placed between the main lens and the sensor plane virtually splits the main lens into sub-apertures, ...
[80]
[PDF] Handheld Multi-Frame Super-Resolution - arXiv
Fig. 1. We present a multi-frame super-resolution algorithm that supplants the need for demosaicing in a camera pipeline by merging a burst of raw images.
[81]
Image Super-Resolution Using Deep Convolutional Networks - arXiv
We propose a deep learning method for single image super-resolution (SR). Our method directly learns an end-to-end mapping between the low/high-resolution ...
[82]
ISO 12233:2023 - Resolution and spatial frequency responses
This document specifies methods for measuring the resolution and the spatial frequency response (SFR) of electronic still-picture cameras.
[83]
[PDF] nbs standard reference material 1010 - microcopy resolution test chart
The resolution test chart is is- sued to assist in standardizing the evaluation of the performance of microcopying sys- tems. The test patterns are made up of ...Missing: superseded | Show results with:superseded
[84]
ISO/TC 42 - Photography
Standardization primarily, but not exclusively in the field of still picture imaging - chemical and electronic - including, but not limited to:Missing: 2020s | Show results with:2020s
[85]
[PDF] A Good Practices Guide for Digital Image Correlation - iDICs
Note 1: Optical resolution is typically measured from images of a resolution tar- get. 72. Page 89. Resolution, Spatial [pixel]: The minimum distance between ...
[86]
[PDF] EMVA Standard 1288
Nov 29, 2010 · The European Machine Vision Association owns the ”EMVA, standard 1288 compliant” logo. ... Resolution of the sensor's active area: width x height ...