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Technical Note: Impact on detective quantum efficiency of edge angle determination method by International Electrotechnical Commission methodology for cardiac x-ray image detectors

Purpose: Cardiac x-ray detectors are used to acquire moving images in real-time for angiography and interventional procedures. Detective quantum efficiency (DQE) is not generally measured on these dynamic detectors; the required “for processing” image data and control of x-ray settings have not been accessible. By 2016, USA hospital physicists will have the ability to measure DQE and will likely utilize the International Electrotechnical Commission (IEC) standard for measuring DQE of dynamic x-ray imaging devices. The current IEC standard requires an image of a tilted tungsten edge test object to obtain modulation transfer function (MTF) for DQE calculation. It specifies the range of edge angles to use; however, it does not specify a preferred method to determine this angle for image analysis. The study aimed to answer the question “will my choice in method impact my results?” Four different established edge angle determination methods were compared to investigate the impact on DQE. Methods: Following the IEC standard, edge and flat field images were acquired on a cardiac flat-panel detector to calculate MTF and noise power spectrum, respectively, to determine DQE. Accuracy of the methods in determining the correct angle was ascertained using a simulated edge image with known angulations. Precision of the methods was ascertained using variability of MTF and DQE, calculated via bootstrapping. Results: Three methods provided near equal angles and the same MTF while the fourth, with an angular difference of 6%, had a MTF lower by 3% at 1.5 mm{sup −1} spatial frequency and 8% at 2.5 mm{sup −1}; corresponding DQE differences were 6% at 1.5 mm{sup −1} and 17% at 2.5 mm{sup −1}; differences were greater than standard deviations in the measurements. Conclusions: DQE measurements may vary by a significant amount, depending on the method used to determine the edge angle when following the IEC standard methodology for a cardiac x-ray detector. The most accurate and precise methods are recommended for absolute assessments and reproducible measurements, respectively.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Detective quantum efficiency of photon-counting x-ray detectors

Purpose: Single-photon-counting (SPC) x-ray imaging has the potential to improve image quality and enable novel energy-dependent imaging methods. Similar to conventional detectors, optimizing image SPC quality will require systems that produce the highest possible detective quantum efficiency (DQE). This paper builds on the cascaded-systems analysis (CSA) framework to develop a comprehensive description of the DQE of SPC detectors that implement adaptive binning. Methods: The DQE of SPC systems can be described using the CSA approach by propagating the probability density function (PDF) of the number of image-forming quanta through simple quantum processes. New relationships are developed to describe PDF transfer through serial and parallel cascades to accommodate scatter reabsorption. Results are applied to hypothetical silicon and selenium-based flat-panel SPC detectors including the effects of reabsorption of characteristic/scatter photons from photoelectric and Compton interactions, stochastic conversion of x-ray energy to secondary quanta, depth-dependent charge collection, and electronic noise. Results are compared with a Monte Carlo study. Results: Depth-dependent collection efficiency can result in substantial broadening of photopeaks that in turn may result in reduced DQE at lower x-ray energies (20–45 keV). Double-counting interaction events caused by reabsorption of characteristic/scatter photons may result in falsely inflated image signal-to-noise ratio and potential overestimation of the DQE. Conclusions: The CSA approach is extended to describe signal and noise propagation through photoelectric and Compton interactions in SPC detectors, including the effects of escape and reabsorption of emission/scatter photons. High-performance SPC systems can be achieved but only for certain combinations of secondary conversion gain, depth-dependent collection efficiency, electronic noise, and reabsorption characteristics.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

The detective quantum efficiency of photon-counting x-ray detectors using cascaded-systems analyses

Purpose: Single-photon counting (SPC) x-ray imaging has the potential to improve image quality and enable new advanced energy-dependent methods. The purpose of this study is to extend cascaded-systems analyses (CSA) to the description of image quality and the detective quantum efficiency (DQE) of SPC systems. Methods: Point-process theory is used to develop a method of propagating the mean signal and Wiener noise-power spectrum through a thresholding stage (required to identify x-ray interaction events). The new transfer relationships are used to describe the zero-frequency DQE of a hypothetical SPC detector including the effects of stochastic conversion of incident photons to secondary quanta, secondary quantum sinks, additive noise, and threshold level. Theoretical results are compared with Monte Carlo calculations assuming the same detector model. Results: Under certain conditions, the CSA approach can be applied to SPC systems with the additional requirement of propagating the probability density function describing the total number of image-forming quanta through each stage of a cascaded model. Theoretical results including DQE show excellent agreement with Monte Carlo calculations under all conditions considered. Conclusions: Application of the CSA method shows that false counts due to additive electronic noise results in both a nonlinear image signal and increased image noise. There is a window of allowable threshold values to achieve a high DQE that depends on conversion gain, secondary quantum sinks, and additive noise.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A Quantum Leap for Dynamic Radiography

Dynamic radiography techniques have been instrumental in advancing the National Nuclear Security Administration's (NNSA) mission since WWII. This paper explores the transformative potential of quantum information science (QIS) to revolutionize dynamic radiography through enhanced image processing, tomographic reconstruction, statistics and uncertainty quantification (UQ), and artificial intelligence integration. By leveraging quantum algorithms to extract previously inaccessible information from existing datasets, this interdisciplinary approach promises unprecedented insights at the intersection of dynamic imaging, artificial intelligence, and quantum information technologies.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

X-ray scatter correction method for dedicated breast computed tomography

Purpose: To improve image quality and accuracy in dedicated breast computed tomography (BCT) by removing the x-ray scatter signal included in the BCT projections. Methods: The previously characterized magnitude and distribution of x-ray scatter in BCT results in both cupping artifacts and reduction of contrast and accuracy in the reconstructions. In this study, an image processing method is proposed that estimates and subtracts the low-frequency x-ray scatter signal included in each BCT projection postacquisition and prereconstruction. The estimation of this signal is performed using simple additional hardware, one additional BCT projection acquisition with negligible radiation dose, and simple image processing software algorithms. The high frequency quantum noise due to the scatter signal is reduced using a noise filter postreconstruction. The dosimetric consequences and validity of the assumptions of this algorithm were determined using Monte Carlo simulations. The feasibility of this method was determined by imaging a breast phantom on a BCT clinical prototype and comparing the corrected reconstructions to the unprocessed reconstructions and to reconstructions obtained from fan-beam acquisitions as a reference standard. One-dimensional profiles of the reconstructions and objective image quality metrics were used to determine the impact of the algorithm. Results: The proposed additional acquisition results in negligible additional radiation dose to the imaged breast ({approx}0.4% of the standard BCT acquisition). The processed phantom reconstruction showed substantially reduced cupping artifacts, increased contrast between adipose and glandular tissue equivalents, higher voxel value accuracy, and no discernible blurring of high frequency features. Conclusions: The proposed scatter correction method for dedicated breast CT is feasible and can result in highly improved image quality. Further optimization and testing, especially with patient images, is necessary to characterize its impact on clinical performance.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

Scintillating Quantum Dots for Imaging X-rays (SQDIX) for Aircraft Inspection

Scintillation is the process currently employed by conventional x-ray detectors to create x-ray images. Scintillating quantum dots or nano-crystals (StQDs) are a novel, nanometer-scale material that upon excitation by x-rays, re-emit the absorbed energy as visible light. StQDs theoretically have higher output efficiency than conventional scintillating materials and are more environmental friendly. This paper will present the characterization of several critical elements in the use of StQDs that have been performed along a path to the use of this technology in wide spread x-ray imaging. Initial work on the SQDIX system has shown great promise to create state-of-the-art sensors using StQDs as a sensor material. In addition, this work also demonstrates a high degree of promise using StQDs in microstructured fiber optics. Using the microstructured fiber as a light guide could greatly increase the capture efficiency a StQDs based imaging sensor.

Burke, Eric↗

Scintillating Quantum Dots for Imaging X-Rays (SQDIX) for Aircraft Inspection

Scintillation is the process currently employed by conventional X-ray detectors to create X-ray images. Scintillating quantum dots (StQDs) or nano-crystals are novel, nanometer-scale materials that upon excitation by X-rays, re-emit the absorbed energy as visible light. StQDs theoretically have higher output efficiency than conventional scintillating materials and are more environmentally friendly. This paper will present the characterization of several critical elements in the use of StQDs that have been performed along a path to the use of this technology in wide spread X-ray imaging. Initial work on the scintillating quantum dots for imaging X-rays (SQDIX) system has shown great promise to create state-of-the-art sensors using StQDs as a sensor material. In addition, this work also demonstrates a high degree of promise using StQDs in microstructured fiber optics. Using the microstructured fiber as a light guide could greatly increase the capture efficiency of a StQDs based imaging sensor.

Burke, E. R.↗

A quantum algorithm for string matching

Algorithms that search for a pattern within a larger data-set appear ubiquitously in text and image processing. Here, we present an explicit, circuit-level implementation of a quantum pattern-matching algorithm that matches a search string (pattern) of length M inside a longer text of length N. Our algorithm has a time complexity of $\tildeO$($\sqrt{N}$), while the space complexity remains modest at O(N+ M). We report the quantum gate counts relevant for both pre-fault-tolerant and fault-tolerant regimes.

Physics↗

The Role of Quantum Science Concepts in Enhancing Sensing and Imaging Technologies: Applications for Biology: Proceedings of a Workshop

Quantum concepts hold the potential to enable significant advances in sensing and imaging technologies that could be vital to the study of biological systems. The workshop Quantum Science Concepts in Enhancing Sensing and Imaging Technologies: Applications for Biology, held online March 8–10, 2021, was organized to examine the research and development needs to advance biological applications of quantum technology. Hosted by the National Academies of Sciences, Engineering, and Medicine, the event brought together experts working on state-of-the-art, quantum-enabled technologies and scientists who are interested in applying these technologies to biological systems. Through talks, panels, and discussions, the workshop facilitated a better understanding of the current and future biological applications of quantum-enabled technologies in fields such as microbiology, molecular biology, cell biology, plant science, mycology, and many others. The workshop was organized around three main themes. The first, quantum in biology, examined quantum concepts that are hypothesized to be important for life processes and that researchers are working to observe through biological imaging and sensing. The second, quantum for biology, addressed ways to use quantum concepts to enhance technologies for biological imaging and sensing. The third, biology for quantum, offered a wider discussion of how the frontiers of biological imaging and sensing could enable future study using quantum concepts, tools, or technologies. Throughout the workshop, participants identified a wide range of emerging approaches and opportunities at the intersection of quantum physics and biological sensing and imaging. During the workshop, there were some differences in how each speaker defined the term quantum. During one of the panels, Prem Kumar offered thoughts on what phenomena are classical versus quantum, explaining that techniques get progressively more quantum as you move from just having superposition to having superposition with measurement and entanglement. Another explanation from Clarice Aiello delineates the definition into several levels. This includes a base level of “quantum-ness,” which reflects that all matter is made of atoms, and when these particles are isolated they behave based on quantum mechanical principles. A second level is related to quantum coherence, where a single quantum object might be found in a coherent superposition state. A final level, which she described as the quantum-entangled level, involves multiple quantum systems which are entangled among themselves. Overall, the workshop touched on concepts such as superposition, entanglement, and squeezing and their potential implications for communication, computing, and simulation, in addition to the workshop’s main focal area, biological sensing and imaging. At the opening of the workshop, Thorsten Ritz of the University of California, Irvine, identified two questions at the heart of quantum biology: Is the machinery of life quantum mechanical, and can quantum mechanics be used to study the machinery of life in new ways? Participants highlighted systems in which researchers have explored these questions, from the vast array of molecular interactions involved in biological processes such as photosynthesis, to the mechanics involved in cellular functions such as differentiation and aggregation, to the role of oscillating magnetic fields in flight orientation among birds. Sensing and imaging technologies are crucial to biological research; these technologies could both enhance the study of quantum effects and be enhanced by quantum concepts. A critical challenge in biological research is to develop imaging and sensing tools that do not damage or interfere with the often fragile and fleeting systems being studied. Attendees discussed a variety of established and emerging technologies that could enhance noninvasive biological imaging, including single- and two-photon spectroscopy, single-molecule spectroscopy, quantum illumination, ghost imaging, and cryo-electron microscopy. One example came from Marlan Scully who gave a keynote address on the first day of the workshop. Scully emphasized the use of different laser technologies, which exhibit coherence and other quantum properties, in moving toward real-world biological applications, such as the detection of SARS-CoV-2. Both tools and theory will play an important role in advancing quantum biology research and applications. Several participants suggested theorists and experimentalists should work in tandem to understand and model biological processes. While physics often reduces systems to their simplest forms for fundamental insights, participants also noted the value of observing and understanding biological systems in all their “messiness,” capturing both the inner workings of biological systems and the complex interactions that occur within and between organisms. In discussions among participants, several attendees stressed the need to match emerging tools with the right scientific questions. Rather than developing quantum technologies as “a hammer looking for a nail,” participants emphasized a focus on exploring the problems these technologies are best suited to address. For example, it is important to consider the size of the phenomenon being studied, the timescales that are important in answering the scientific question, and other relevant considerations. Every tool along the spectrum from classical to quantum involves its own set of trade-offs. For example, Ted Laurence of the Lawrence Livermore National Laboratory said that quantum measurements, such as single photon counting, fluorescence transitions, and lasers, can take longer to produce the same results as classical measurements. These quantum measurements, however, do not require calibration and can enable new research questions to be answered. Prem Kumar, Northwestern University, noted that, despite their promise, quantum approaches should not be used simply for the sake of using quantum, especially in situations where classical approaches better meet the needs of the researcher. Understanding and applying quantum concepts could enable advances in a wide range of application areas including energy, synthetic biology, medicine, and sustainability. For example, Michelle O’Malley, University of California, Santa Barbara, described how improved noninvasive imaging approaches could help capture the complex interactions and functions involved in the breakdown of organic matter by microbial communities and lead to new technologies for capturing valuable products from plant waste. Several other participants discussed needs in tracking the movement of metabolites and molecules in microbial communities for insights into nutrient cycling in environments such as soil. Margaret Ahmad, Sorbonne University, discussed potential opportunities to leverage the magnetic properties of cryptochromes to advance new treatment approaches for diseases such as COVID-19 and cancer. Looking toward the future development of the field, participants discussed challenges to advancing quantum biology that arise from disciplinary disconnects between physicists and biologists. The siloing of academic research disciplines represents a significant barrier to progress. Disconnects in terminology, motivations and priorities, and structural barriers to collaborative work underscore the need for concerted efforts to bridge these divides. Attendees and speakers offered suggestions for resolving these divergences, establishing a shared language, moving the field forward, and fostering meaningful feedback between disciplines. Overall, participants stressed a need for balance, open communication, collaboration, unity, and clear dialogue on trade-offs between quantum and classical approaches. Keiko Torii, The University of Texas at Austin, said that the best collaborations happen when the project provides mutual advantages that can show off everyone’s talents, each team finds the work interesting, and partners develop a camaraderie to pursue new knowledge. To enable near- and long-term opportunities in this space, participants suggested that exploratory, high-risk funding could improve existing instrumentation to explore quantum enhancement collaboratively. They also emphasized the need for collaboration between quantum physicists and sensing/imaging scientists, which could be advanced through a dedicated quantum biology investigator program. People, even more than technology, will be crucial to the future of quantum biology. Participants explored training, education, and workforce needs to further develop this burgeoning field and cultivate the next generation of scientists. While many programs are still in their nascent stages, participants highlighted examples of approaches and programs being developed at various types of institutions to engage students and professional scientists in quantum biology research. Several participants stressed the need for an inclusive approach, spanning disciplines as well as communities to foster a diverse field fueled by the intellectual contributions of a wide range of people, including historically under-resourced schools and students. To increase awareness and excitement about quantum physics and related areas of biology, attendees suggested capitalizing on the “buzz” around quantum. Several participants emphasized the need to start early, introducing students to quantum concepts and their appealing “weirdness” in K–12 education. Engaging students early—before they become entrenched in traditional disciplinary siloes as typically happens in graduate school—could help to galvanize interest in the area and foster a generation of scientists with the interdisciplinary mindset and skills needed to advance this interdisciplinary field. While these efforts could be advanced at many levels and across multiple sectors, several participants suggested a national quantum biology center could be a valuable hub to coordinate and support quantum biology education and workforce development across academia, industry, and government.

59 BASIC BIOLOGICAL SCIENCES↗

Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam

We present a direct measurement of the Kirkwood-Rihaczek (KR) distribution for spatial properties of a coherent light beam in terms of position and momentum (angle) coordinates. We employ a two-local oscillator (LO) balanced heterodyne detection (BHD) to simultaneously extract the distribution of the transverse position and momentum of a light beam. The two-LO BHD can measure the KR distribution for any complex wave field (including quantum mechanical wave function) without applying tomography methods (inverse Radon transformation). The transformation of the KR distribution to the Wigner, Glauber-Sudarshan P, and Husimi or Q distributions in spatial coordinates are illustrated through experimental data. The KR distribution can provide the local information of a wave field, which is suitable for studying particle properties of a quantum system. Meanwhile, the Wigner function is suitable for studying wave properties such as interference, and hence provides nonlocal information of the wave field. The method developed here can be used for exploring the spatial quantum state for information processing and optical phase-space imaging for biomedical applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Some features of photolithography image formation in partially coherent light

The coherent-noise level in projection images of an opaque-screen sharp edge, formed in the model scheme of photolithography system at different degrees of spatial coherence of screen-illuminating light is studied experimentally. The spatial coherence of laser radiation was reduced by applying a specially developed device, used as a separate functional unit in the system model. The smoothing of the spatial fluctuations of radiation intensity caused by the random spatial inhomogeneity of the initial beam intensity in the obtained images is shown to be highly efficient. (imaging and image processing. holography)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Measuring impact rebound with photography.

To study the rebound of a sphere colliding against a flat wall, a test setup was developed where the sphere is suspended with strings as a pendulum, elevated, and gravity-released to impact the wall. The motion of the sphere was recorded with a highspeed camera and traced with an image-processing program. From the speed of the sphere before and after each collision, the coefficient of restitution was computed, and shown to be a function of impact speed as predicted analytically.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A review of breast tomosynthesis. Part I. The image acquisition process

Mammography is a very well-established imaging modality for the early detection and diagnosis of breast cancer. However, since the introduction of digital imaging to the realm of radiology, more advanced, and especially tomographic imaging methods have been made possible. One of these methods, breast tomosynthesis, has finally been introduced to the clinic for routine everyday use, with potential to in the future replace mammography for screening for breast cancer. In this two part paper, the extensive research performed during the development of breast tomosynthesis is reviewed, with a focus on the research addressing the medical physics aspects of this imaging modality. This first paper will review the research performed on the issues relevant to the image acquisition process, including system design, optimization of geometry and technique, x-ray scatter, and radiation dose. The companion to this paper will review all other aspects of breast tomosynthesis imaging, including the reconstruction process.

60 APPLIED LIFE SCIENCES↗

Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging

We present a theoretical comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging. We first calculate the signal-to-noise ratio of each process in terms of its controllable experimental conditions. We show that a key distinction is that a thermal ghost image always resides on top of a large background; the fluctuations in this background constitutes an intrinsic noise source for thermal ghost imaging. In contrast, there is a negligible intrinsic background to a quantum ghost image. However, for practical reasons involving achievable illumination levels, acquisition times for thermal ghost images are often much shorter than those for quantum ghost images. We provide quantitative predictions for the conditions under which each process provides superior performance. Our conclusion is that each process can provide useful functionality, although under complementary conditions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Measurement and image processing evaluation of surface modifications of dental implants G4 pure titanium created by different techniques

Foreign substances and organic tissue interaction placed into the jaw in order to eliminate tooth loss involves a highly complex process. Many biological reactions take place as well as the biomechanical forces that influence this formation. Osseointegration denotes to the direct structural and functional association between the living bone and the load-bearing artificial implant's surface. Taking into consideration of the requirements in the manufacturing processes of the implants, surface characterizations with high precise measurement techniques are investigated and thus long-term success of dental implant is emphasized on the importance of these processes in this study. In this research, the detailed surface characterization was performed to identify the dependence of the manufacturing techniques on the surface properties by using the image processing methods and using the scanning electron microscope (SEM) for morphological properties in 3D and Taylor Hobson stylus profilometer for roughness properties in 2D. Three implant surfaces fabricated by different manufacturing techniques were inspected, and a machined surface was included into the study as a reference specimen. The results indicated that different surface treatments were strongly influenced surface morphology. Thus 2D and 3D precise inspection techniques were highlighted on the importance for surface characterization. Different image analyses techniques such as Dark-light technique were used to verify the surface measurement results. The computational phase was performed using image processing toolbox in Matlab with precise evaluation of the roughness for the implant surfaces. The relationship between the number of black and white pixels and surface roughness is presented. FFT image processing and analyses results explicitly imply that the technique is useful in the determination of surface roughness. The results showed that the number of black pixels in the image increases with increase in surface roughness.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Technical Note: Modification of the standard gain correction algorithm to compensate for the number of used reference flat frames in detector performance studies

Purpose: The x-ray performance evaluation of digital x-ray detectors is based on the calculation of the modulation transfer function (MTF), the noise power spectrum (NPS), and the resultant detective quantum efficiency (DQE). The flat images used for the extraction of the NPS should not contain any fixed pattern noise (FPN) to avoid contamination from nonstochastic processes. The ''gold standard'' method used for the reduction of the FPN (i.e., the different gain between pixels) in linear x-ray detectors is based on normalization with an average reference flat-field. However, the noise in the corrected image depends on the number of flat frames used for the average flat image. The aim of this study is to modify the standard gain correction algorithm to make it independent on the used reference flat frames. Methods: Many publications suggest the use of 10-16 reference flat frames, while other studies use higher numbers (e.g., 48 frames) to reduce the propagated noise from the average flat image. This study quantifies experimentally the effect of the number of used reference flat frames on the NPS and DQE values and appropriately modifies the gain correction algorithm to compensate for this effect. Results: It is shown that using the suggested gain correction algorithm a minimum number of reference flat frames (i.e., down to one frame) can be used to eliminate the FPN from the raw flat image. This saves computer memory and time during the x-ray performance evaluation. Conclusions: The authors show that the method presented in the study (a) leads to the maximum DQE value that one would have by using the conventional method and very large number of frames and (b) has been compared to an independent gain correction method based on the subtraction of flat-field images, leading to identical DQE values. They believe this provides robust validation of the proposed method.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Thematic Mapper Image Processing System (TIPS) Processing Status

Radiometric and geometric correction performance is considered. A one quantum level requirement was met in all spectral bands with an occasional band 7 exception. Initial post launch calibration was accomplished. An absolute radiometric calibration is in progress including an investigation of low impact solution to scan striping. Initial and three downstream updates as well as coordination with LANDSAT D' instrument calibration are planned. Methods and problems in measuring geometric correction performance are examined with an emphasis on temporal registration, geodetic registrtion, and the modeling error budget. Planned efforts include eliminating band 2, band 5 chips; continued performance monitoring; calibration of profile, misalignment angles, and filer parameters; studying the need for seasonal geodetic control points; and improvement of the subpixel correlation technique.

Brooks, J.↗

Self-imaging of transparent objects and structures in focusing of spatially phase-modulated laser radiation into a weakly absorbing medium

Self-imaging of transparent objects and structures in focusing of a spatially phase-modulated laser beam into an extended weakly absorbing medium is described. The laser power level that is necessary for effective imaging corresponds to the illuminating beam power when thermal self-defocusing starts evolving in the medium. The effect can be described in terms of the ideology of Zernike's classical phase-contrast method. Edge enhancement in visualised images of transparent objects is experimentally demonstrated. Self-imaging of a microscopic object in the form of transparent letters and long-lived refractive-index fluctuations in liquid glycerol is shown. Due to the adaptivity of the process under consideration, unlike the classical case, self-imaging occurs also in the situations where a beam is displaced (undergoes random walk) as a whole in the Fourier plane, for example, in the presence of thermal flows. (image processing)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗