Information is often described as a reduction of uncertainty associated with a restriction of possible choices. Despite appearing in Hartley’s foundational work on information theory, there is a surprising lack of a formal treatment of this interpretation in terms of exclusions. This paper addresses the gap by providing an explicit characterisation of information in terms of probability mass exclusions. It then demonstrates that different exclusions can yield the same amount of information and discusses the insight this provides about how information is shared amongst random variables—lack of progress in this area is a key barrier preventing us from understanding how information is distributed in complex systems. The paper closes by deriving a decomposition of the mutual information which can distinguish between differing exclusions; this provides surprising insight into the nature of directed information. At each selection there are eliminated all of the other symbols which might have been chosen. As the selections proceed more and more possible symbol sequences are eliminated, and we say that the information becomes more precise.”Indeed, this interpretation led Hartley to derive the measure of information associated with a set of equally likely choices, which Shannon later generalised to account for unequally likely choices. Nevertheless, despite being used since the foundation of information theory, there is a surprising lack of a formal characterisation of information in terms of the elimination of choice. Both Fano [X. Then in alignment with Hartley’s description, they consider information to be something which excludes possible choices x, with more eliminations corresponding to greater information; however, this approach does not capture the concept of information in its most general sense since it cannot account for information provided by partial eliminations which merely reduces the likelihood of a choice x from occurring.  Nonetheless, In information theory, there is a duality between the concepts entropy and information: entropy is a measure of uncertainty or freedom of choice, whereas information is a measure of reduction of uncertainty (increase in certainty) or restriction of choice. Interestingly, this description of information as a restriction of choice predates even Shannon , originaoth Fano  and Ash oth Fano  motivatethe same information or just the same amount of information” [Our interest in providing a formal interpretation of information in terms of exclusions is driven by a desire to understand how information is distributed in complex systems ,6. In parmation”  . WitX and Y with discrete sample spaces x from X using an event y from Y which has occurred jointly. Ideally, there is a one-to-one correspondence between the occurrence of events from X and Y such that an event x can be exactly predicted using an event y. However, in most complex systems, the presence of noise or some other such ambiguity means that we typically do not have this ideal correspondence. Nevertheless, when a particular event y is observed, knowledge of the distributions Consider two random variables y is observed, we know that the complementary event Definition 1.A probability mass exclusion induced by the event y from the random variable Y is the probability mass associated with the complementary event (Probability Mass Exclusion)When a particular event y provides about x; however, this is not true in general. To see this, consider the joint event x from the random variable X. Knowing the event x occurred enables us to categorise the probability mass exclusions induced by y into two distinct types: the first is the portion of the probability mass exclusion associated with the complementary event x, i.e., x given y written in terms of these two categories,Echoing Hartley’s description, it is perhaps tempting to think that the greater the probability mass exclusion y induces exclusions which are confined to the probability mass associated with the complementary event probability mass diagram at the top of y provides about x. Hence, we define this type of exclusion as follows:Definition 2.For the joint event (Informative Probability Mass Exclusion)To see why these two types of exclusions are distinct, consider two special cases: The first special case is when the event )=0 into . In thisy induces exclusions which are confined to the probability mass associated with the event x. This means that the portion of exclusion y provides about x. Hence, we define this type of exclusion as follows:Definition 3.For the joint event (Misinformative Probability Mass Exclusion)The second special case is when the event )=0 into . In this)=0 into  is a strNow consider the general case where both informative and misinformative probability mass exclusions are present simultaneously. It is not immediately clear whether the posterior Although it is not immediately clear whether the posterior If Now that a formal relationship between eliminations and information has been established using probability theory, we return to the motivational question—can this understanding of information in terms of exclusions aid in our understanding of how random variables share information?y and z each induce different exclusions, both in terms of size and type, and yet provide the same amount of information about the event x sinceConsider the example in y and z reduce our uncertainty about x in distinct ways and yet, after making the relevant exclusions, we have the same freedom of choice about x. It is our belief that the information provided by y and z should only be deemed to be the same information if they both reduce our uncertainty about x in the same way; we contend that for the events y and z to reduce our uncertainty about x in the same way, they would have to identically restrict our choice, or make the same exclusions with respect to x.The events What this example demonstrates is that the mutual information does not—and indeed cannot—distinguish between how events provide information about other events. By definition, the mutual information only depends on the prior e saw in , there iThe mutual information cannot distinguish between events which induce different exclusions because any given value could arise from a whole continuum of possible informative and misinformative exclusions. Hence, consider decomposing the mutual information into two separate information-theoretic components. Motivated by the strictly positive mutual information observed in the purely informative case and the strictly negative mutual information observed in the purely informative case, let us demand that one of the components be positive while the other component is negative.Postulate 1.The information provided by y about x can be decomposed into two non-negative components, such that (Decomposition)Furthermore, let us demand that the two components preserve the functional dependencies between the mutual information and the informative and misinformative exclusion observed in  for the Postulate 2.The functions 1. For all fixed 2. For all fixed 3. For all fixed (Monotonicity)x could never induce a misinformative exclusion about itself, since the misinformative exclusion x,Before considering the functions which might satisfy Postulates 1 and 2, there are two further observations to be made about probability mass exclusions. The first observation is that an event terms of  yields tPostulate 3.An event cannot misinform about itself, hence (Self-Information)y and z. Firstly, we could consider the information provided by the joint event y which excludes the probability mass in z which excludes the probability mass in z which excludes the probability mass in y which excludes the probability mass in The second observation is that the informative and misinformative exclusions exclusions must individually satisfy the chain rule of probability. As shown in Postulate 4.The functions where the conditional notation denotes the same function only with conditional probability as an argument.(Chain Rule)Theorem 1.The unique functions satisfying Postulates 1–4 areBy rewriting  and 8)  8) in teHence, in order to prove Theorem 1 we must demonstrate that  and 8)  8) are tTheorem 1 answers the question posed at the end of y with respect to x. It is novel that this particular decomposition enables us to uniquely determine the size of the exclusions induced by y with respect to x, rather than y about x since it reflects the change from the prior x with respect to y.It is important to note the directed nature of this decomposition—this equation considers the exclusions induced by only the misinformative exclusion while the positive component both the informative and misinformative exclusions. The positive component depends on the total size of the exclusions induced by y and hence has no functional dependence on x. It quantifies the specificity of the event y: the less likely the outcome y is to occur, the greater the total amount of probability mass excluded by y and therefore the greater the potential for y to inform about x. On the other hand, the negative component quantifies the ambiguity of y given x: the less likely the outcome y is to coincide with the outcome x, the greater the misinformative probability mass exclusion and therefore the greater the potential for y to misinform about x. This asymmetry between the components is apparent when considering the two special cases. In the purely informative case where There is another important asymmetry which can be seen from  and 10)10). The y and z provide the same amount of information about x, and consider this example in terms of the decomposition (Let us now consider how this information-theoretic expression (which has a one-to-one correspondence with exclusion) could be utilised to provide an information decomposition that can distinguish between whether events provide the same information or merely the same amount of information. Recall the example from position ,(12)i+(y and z provide information through differing exclusions even if they provide the same amount of information. As for how to decompose multivariate information using this decomposition? This is not the subject of this paper—those who are interested in seen an operational definition of shared information based on redundant exclusions should see [In contrast to the mutual information in , the decould see .


Type 1 diabetes is a chronic autoimmune disease that affects tens of millions of people. Diabetes mellitus is one of the strongest factors in the development of cerebrovascular diseases. In this study we used NOD.CB17 Prkdcscid mice and the pharmacological model of type 1 diabetes mellitus of different duration to study changes in the cerebral vasculature. We used two combined approaches using magnetic resonance angiography both steady and transient CFD blood flow modeling. We identified the influence of type 1 diabetes on the architectonics and hemodynamics of the large blood vessels of the brain as the disease progresses. For the first time, we detected a statistically significant change in angioarchitectonics  and hemodynamic  in animals with diabetes duration of 2 months, that is manifested by the development of asymmetry of cerebral blood flow. The result shows the negative effect of diabetes on cerebral circulation as well as the practicability of CFD modeling. This may be of extensive interest, in pharmacological and preclinical studies. Long-term course of the disease leads to the development of associated illness2. Thus, diabetes mellitus is one of the strongest factors in the development of cerebrovascular diseases3. Multiple studies suggest that hyperglycemia can cause excess free fatty acids4, loss of endothelial nitric oxide6, prothrombotic state7, endothelial dysfunction9, abnormal release of endothelial vasoconstrictors10, vascular smooth muscle dysfunction11, oxidative stress13, which can lead to damage to blood vessels and the development of pathological blood flow in them. Vascular disorders can also occur in the cerebral vasculature. View of the above it is expected that with an increase in the diabetes duration, the changes in the cerebral vessels hemodynamics can potentially appear or intensify.More than 200 million people have been diagnosed with diabetes. About 10% of them are sick with type 1 diabetes. Type 1 diabetes mellitus is a widespread chronic autoimmune disease14.The cerebral vascular system has a complex structure. The basic component of the cerebral vasculature is a complex of large arteries consisting of carotid and vertebral arteries as well as the circle of Willis formed by anterior cerebral arteries, anterior communicating arteries, posterior cerebral arteries, posterior communicating arteries as well as middle cerebral arteryOn the one hand, the structure of these vessels should be sufficiently conservative. On the other hand, it is natural to assume that with the course of the disease leading to changes in the functioning of the circulatory system and the need for its adaptation there may be a change in the geometry and hemodynamic regime of this arterial complex.These arteries are the largest in the brain and therefore their architectonics is available for study by Magnetic Resonance Imaging (MRI) methods.17, are often used to solve the problem of moving blood through the vessels.Blood vessel geometry data obtained by magnetic resonance angiography (MRA) can be used for subsequent CFD simulation. Since the hemodynamics study is associated with a complex flow geometry and complicated flow, application packages, ANSYS, COMSOL, OpenFOAM, etc.In this work, the influence of type 1 diabetes lasting 1 and 2 months on the architectonics and hemodynamics of cerebral blood vessels was studied using NODSCID mice as model objects. We used a combined approach to obtain the actual configuration of the mice' cerebral vessels using MRI, then both steady and transient numerical analysis in the ANSYS CFX software and subsequent statistical analysis of geometric and hemodynamic characteristics using PLS-DA. The data supporting the conclusions of this article is included within the article in the Supplementary files.All procedures were performed in accordance with the European Convention for the Protection of Vertebrate Animals used for Experimental and other Scientific Purposes. The experimental protocol was approved by the Bioethical Committees at the Institute of Cytology and Genetics of the Siberian Branch of the Russian Academy of Sciences. All animal experiments described below were in compliance with the ARRIVE  guidelines.scid mice (NOD SCID) from the Centre for Laboratory Animal Genetic Resources, Institute of Cytology and Genetics, Siberian Branch of the Russian Academy of Sciences (RFMEFI62119X0023). This line of mice is chosen because of their increased sensitivity in the pharmacological model of diabetes mellitus (streptozotocin injection).In this study we use NOD.CB17 PrkdcAt the beginning of the experiment animals of SPF (Specific Pathogen Free) status were at the age of 8 weeks. The diabetes group was injected intraperitoneally with streptozotocin  dissolved in 0.01 M citrate buffer at pH 4.2. A total of 150 mg streptozotocin/kg body weight was administered to induce diabetes. The control group was injected with the same volume of the citrate buffer.Four groups of animals were formed: 1c—7 males; 1d—5 males; 2c—10 males; 2d—9 males. Here: 1, 2—the duration of the experiment in months, c—control, d—diabetes.Initially, the number of animals in group 1d was 7. However, one animal died during the experiment . Another one from this group was excluded from the analysis due to the substandard quality of the MRA data.18 using Diacont® glucometer  and individual test strips of the same company compatible with the glucometer. Measurements were carried out in animals before the experiment, 7 days after the introduction of streptozotocin and at the end of the experiment. Blood sampling was made from the tip of the animal's tail 4 h after removal of feed, so the blood was mixed type, taken on an empty stomach.To confirm the presence of type 1 diabetes in all animals used in the experiment a multiple measurement of blood glucose level (mmol/l) was carried out by electrochemical method19 with pulse sequence parameters Echo Time (TE) = 3.2 ms, and Repetition Time (TR) = 15 ms . The blood flow rate was determined in single slice (the common carotid arteries) using 2D phase-contrast magnetic resonance imaging method with pulse sequence parameters TE = 6 ms, TR = 20 ms (1- mm thick slice with a field of view of 2 cm × 2 cm and a matrix size of 256 × 256). The data obtained have a voxel size of 78 × 78 × 156 microns.MRA data were obtained on an ultrahigh-field tomograph with a magnetic field strength of 11.7 T  with a volume RF coil for head of mouse MRA data. Images and the blood flow velocity in blood vessels of the mouse were recorded with a transmitter and receiver volume (T11232V3) 1H radiofrequency coil. The vessel’s architectonics was determined using 3D-TOF (Time of Flow) method20 using the Cubic (Catmull-Rom) interpolation method. This allowed us to obtain data of sufficient quality for CFD simulation without excessive use of interpolation.The small size of the mice cerebral vessels required an increase in image resolution. To do this, MRA data were interpolated to the matrix 512 * 512 * 256 in the Seg3D program21. After segmentation, the images of the vessels were cleaned from noise, the cerebral arterial network was recognized, and the models smoothing was performed. Figure 22.Non-fragmented three-dimensional models of vascular networks of all animals have been constructed using the software ITK-Snap for segmentationAt the obtained configurations of all mice, vascular networks geometrical parameters were measured. Namely, the angles entclass1pt{minima24.Software ANSYS CFX is used for numerical analysis of hemodynamics. Blood flow CFD calculations in the circle of Willis and adjoining arteries were carried out based on the obtained vascular geometry by MRA velocity data in the common carotid arteries. Verification of the reliability of the simulation by comparing high-quality experimental MRA data with the results of CFD modeling was carried out in our early works26. In our case, the Womersley’s number is quite large and on the order of tens. Therefore, it is impossible to say in advance whether unsteady effects can be ignored. In this regard, we carry out both stationary and non-stationary CFD modeling.In hemodynamic modeling, it is widely used both with and without flow pulsationsThe incompressible viscous liquid model is used for numerical modeling. Walls of blood vessels are considered to be the rigid and no-slip condition is used. Flow considered being laminar. Thus, the mathematical model of blood flow has the form:Here For steady calculation at the inlets of vertebral arteries and common carotid arteries we set individual values of the average blood flow velocity obtained during the MRA measurements. These velocities are shown in the Supplementary Table . At the 27. The outlets volumetric flow rates are proportional cross sectional areas and inlet velocity value in each timepoint. Lengthened outlets 7 and 8 are used. The transient calculation time = 0.137 s, which corresponds to one cardiac cycle, time step = 0.005 s.For transient calculation at the inlets we used the same average velocity. The varying-time velocity pattern was used based on data fromVelocity and pressure values were obtained as a result of numerical calculations in the vasculature. In the transient case, their average values for the cardiac cycle are used. An example of pressure and velocity distribution within the circulatory system is shown in Fig. These CFD modeling results were used to obtain the following blood flow parameters:Mass flow entclass1pt{minimaMaximum blood flow velocity Hydraulic resistance Here Blood flow velocity characteristics, angioarchitectonics, and CFD simulation were carried out in accordance with the requirements of the blind study.As parameters for further statistical analysis were chosen: the angles between the vessels of the circle of Willis mean ± SE. Comparison between groups was also made using parametric statistics methods—Student's t-test .To establish the normal sample distribution of the estimated parameters, we used the Kolmogorov–Smirnov criterion. Further, the data are shown as 28. The PLS-DA builds a linear regression model by projecting the variables to a new space, with Y as a categorical variable29. Additionally, PLS-DA model was tested using a stepwise variable elimination procedure. Then two-way ANOVA (on pathology and disease duration) was performed, with Y values as the dependent variable, in order to assess the distribution of experimental groups in the space of Y axes.The number of initial variables was reduced with partial least-squares discriminant analysis (PLS-DA), which is widely used in biological and medical research because it allows maximization of the separation between groups of observationData on glucose levels for all animal groups are given in Supplementary Table .Regarding the MRA data, in group 1d the blood flow rate in both the right and left common carotid arteries is significantly lower than in group 1c. In addition, there is no significant asymmetry of blood flow in both groups. There are no differences in the blood flow rate in the left common carotid artery in animals of group 2d compared to group 2c, but there is a significant decrease in blood flow in the right common carotid artery. At the same time, a significant asymmetry of blood flow is observed only in group 2d. Data about all four groups are shown in Supplementary  and in transient calculation , see Supplementary  and Sect. 6 area of right internal carotid artery . Also difference in the base of the Sect. 1 area basilar artery  was detected. Also, significant difference was found in the angle between right ACA and MCA .In steady calculation a significant difference in hemodynamics between groups 2c and 2d is observed in: hydraulic resistance between planes 5 and 6 ; the maximum velocity in cross Sect. 1 , Sect. 15 . Also there were several of parameters closely approaches the brink of significance: mass flow rate between Sect. 15 and 16 ; the maximum velocity in cross Sect. 13 , Sect. 16 .Transient calculation shows corresponding results. Hydraulic resistance between planes 5 and 6 ; the maximum velocity in cross Sect. 1 , Sect. 15 . Also there were several of parameters are closely approaches the brink of significance: mass flow rate between Sect. 15 and 16 ; the maximum velocity in cross Sect. 13 , Sect. 16 . Comprehensive data set included in Supplementary  and the disease  have a significant influence on the Y1 axis. Only the duration of the experiment has a significant effect on the Y2 axis . The transient case analysis showed that both the duration of the experiment  and the disease  have a significant influence on the Y1 axis. Only the duration of the experiment has a significant effect on the Y2 axis .Partial least square discriminant analysis (PLS-DA) was performed on the basis of the obtained vessel geometry data and CFD simulation results. The similarity of the results of steady and transient calculations is reflected in the similar results of PLS-DA Fig. . AccordiIn a study on a model object—the NODSCID mice, we used two combined approach using MRA together with CFD steady or transient blood flow modeling. We identified the influence of type 1 diabetes on the architectonics and hemodynamics of the major blood vessels of the brain as the disease progresses. It was confirmed statistically, in particular by PLS-DA analysis. Steady and transient modeling showed similar results.31. At the same time, if these pathologies are not sudden in nature, for example, a stroke or trauma, then such changes are formed gradually32. Diabetes mellitus as a chronic disease has a long-term effect on blood vessels with the first time microvascular and then macrovascular changes33.The geometry changing of large vessels is possible in a number of pathologies34. An important role in the implementation of such compensation is assigned to the circle of Willis. Its configuration, although conservative, is still capable of significant variations35. No less important factor for the compensation can be a change in the lumen of blood vessels. According to the results of angiography, we did not observe acute occlusion in animals which could lead to a brain infarction. But we detected a decrease in the lumen of the internal carotid arteries only in animals with disease duration of 2 months. The reason that led to decrease in the lumen of vessels was not examined in this work. Perhaps one of the reasons that led to a decrease in the lumen of vessels could be atherosclerotic vascular changes caused by prolonged hyperlipidemia37. Metabolic disorder can also lead to the development of hypertension38. According to the European Society of Cardiology-EASC and the European Association for the Study of diabetes-EASD in Europe, the frequency of detection of arterial hypertension in type 1 diabetes mellitus is 10–30%, and in type 2 diabetes mellitus, this frequency reaches 70–80%. Since hypertension can affect blood flow characteristics, we set special boundary conditions when performing CFD calculations: the values of fluxes obtained from high-field MRA data individually for each animal were set as boundary conditions. We detected a decrease in blood flow velocity in the internal carotid arteries both for animals with diabetes duration 1 and 2 months, but only in animals with a duration of 2 months, this led to a significant change in hemodynamic characteristics. Apparently, such changes were the result of a combination of a decrease in blood flow velocity and a narrowing of the vessel lumen, which was observed only in animals with diabetes duration of 2 months. Since the change in blood vessels occurs gradually, the duration of diabetes for 2 months is sufficient for the influence of factors such as the loss of endothelial nitric oxide, endothelial dysfunction, oxidative stress and abnormal release of endothelial vasoconstrictors to become significant.The vascular network structure of the brain has the ability of significant flow compensation39. Systolic heart failure with high glucose and no treatment can occur in a short time40. Another result is a lower blood flow velocity tendency in the right internal carotid artery compared with the left one, regardless of the animal's group. This result is a physiological norm caused by anatomical features of the blood supply to the common carotid arteries from the aortic arch basin41. Note that, only in one (a diabetes duration of 2 months) of the four groups the differences in blood flow velocity between the internal carotid arteries were statistically significant. In this group, the blood flow rate in the right internal carotid artery was significantly lower than in the left one. Some diseases at a certain stage of development can lead to the formation of temporary or stable fluctuating asymmetry42. The asymmetry of a wide range of organism’s characteristics may indicate disorders43. What has been demonstrated recently, including in relation to the disease of diabetes mellitus45 and diabetes-related vascular disorders46. In a recent paper, an approach to analyzing the fluctuating asymmetry of fingerprints was used as an element of a prognostic tool for detecting diabetes mellitus type 1 and 247. This can be based on epigenetic changes on the platform of emerging genetic noise when asymmetric gene expression occurs in the left and right parts of the body. Thus, we can assume that the genetic manifestation of asymmetric changes in hemodynamics is observed only in a group of animals with duration of diabetes of 2 months. This is manifested in an increase in hydraulic resistance between Sects. 5 and 6, a decrease in the maximum velocity of blood flow in Sect. 15, as well as several of parameters are closely approaches the brink of significance (See Fig. 48 carried out on the posthumous material of people it is noted that against the background of the large variability of the circle of Willis organization, the posterior communicating artery is the most variable. Unfortunately, there is no information about what diseases people had during their lifetime, and the study was carried out on one ethnic group, although researchers point to similarities with data obtained by other researchers50. An interesting fact is that according to our data, no hemodynamic changes have been detected on the downstream right side while there was a change in the angle between the ACA and MCA. Is it related to the fact that the normalization of hemodynamics occurs on the same segment where the divergence angle of the middle cerebral artery changes? In a number of papers, it is noted that with an increase in blood pressure on the vessel walls, it is possible to change it geometry as a result of the change of the expression level of the genes responsible for the endothelium growth factor, cell proliferation change and migration52. In our case, it is possible that this change in angle has a compensatory nature since a change in the divergence angle has a significant effect on the hydraulic resistance53.The decrease in the blood flow rate in animals with diabetes can be explained by changes in the volume characteristics of the left ventricular ejection fraction of the heart. Systolic heart failure is a common comorbidity in diabetes mellitus while the pathogenesis, in this case, may be due to myocardial nutritional disordersThe work does not concern a number of important issues, the consideration of which would be interesting and which can be considered in future studies. Our research was focused on modeling the situation when endogenous insulin ceases to be produced at a young age and insulin therapy is not provided for a long time or is insufficient, which occurs, for example, in patients if the disease was not detected in time. This study did not concern the change in angioarchitectonics in type 1 diabetes on the background of insulin therapy that requires another one. We also do not consider the issue of type 2 diabetes mellitus, and, although it is also accompanied by hyperglycemia, nevertheless arises in an older age, has another pathogenesis and therefore requires another study. Our model is implemented in young animals, and therefore not quite correct to study vascular changes in type 2 diabetes mellitus.Thus, in our work, it is demonstrated that more strongly statistically significant differences in angioarchitectonics and hemodynamics are observed in animals with diabetes duration of 2 months, than in 1 month one. This was valid for both steady and transient simulations. The result shows the negative effect of diabetes on cerebral circulation as well as practicability of CFD modeling. This may be of extensive interest, for example, for pharmacological and preclinical studies. To sum up, with an increase in the duration of the disease, type 1 diabetes begins to influence the angioarchitectonics and hemodynamics of the large cerebral vessels of mice.Supplementary Information.


New resonant emission of dispersive waves by oscillating solitary structures in optical fiber cavities is considered analytically and numerically. The pulse propagation is described in the framework of the Lugiato-Lefever equation when a Hopf-bifurcation can result in the formation of oscillating dissipative solitons. The resonance condition for the radiation of the dissipative oscillating solitons is derived and it is demonstrated that the predicted resonances match the spectral lines observed in numerical simulations perfectly. The complex recoil of the radiation on the soliton dynamics is discussed. The reported effect can have importance for the generation of frequency combs in nonlinear microring resonators. In the pioneering work by Akhmediev and Karlsson3 it was discovered that optical solitons propagating in conservative fibers with focusing Kerr nonlinearity and high order dispersion can resonantly emit dispersive waves. The condition of the resonance is the equality of the soliton velocity to the phase velocity of a dispersive wave. This effect is similar to the effect of Cherenkov radiation of charged particles moving at a superluminal velocity. Later it was shown that Bragg solitons moving in a periodic medium emit resonant radiation which is analogous to the transitional radiation of charges flying in a periodic system4. Periodical variations of the soliton parameters can lead to resonant radiation, too. It has been shown that resonant radiation can be excited by solitons in periodical systems where the motion of the solitons results in their oscillations. For the case of optical nonlinear Schrödinger solitons this effect was reported in the literature6 where it was shown that the radiation appears if phase matching conditions are satisfied. It should be mentioned here that the effect of resonant radiation of oscillating solitons is of general nature and can be observed in different physical systems, for example in long Josephson junctions7.In recent time resonant radiation of solitons propagating in fibers with high order dispersion attracts much of attention because of both the fundamental interest and practical importance8. It should be noted here that from the point of view of the field theory the synchrotron radiation can be interpreted as an excitation of resonant waves by an oscillating localized source moving at a relativistic velocity. The periodical variations of the velocity of a relativistic charge is just a way to obtain the right driving force in the equation for the electromagnetic waves. Aiming to develop the analogy between synchrotron radiations of the charges and solitons it is much simpler and more instructive to consider one-dimensional oscillating solitons rather than two-dimensional nonlinear localized structures relativistically moving along one direction and wobbling in the transverse direction .The aim of the present paper is to address the resonant radiation of oscillating optical solitons in the presence of high-order dispersion and to reveal the analogy of this radiation to the synchrotron radiation of charges appearing when the velocity of relativistic charges varies periodically in time2. Recently, studies with focus on Cherenkov emission shed by shock waves9 and ultra-short pulses13 were published. It was discovered that the effect of Cherenkov radiation affects not only the intensity but also the frequency of the resonantly emitted waves15.The effects of Cherenkov and synchrotron radiation are widely used in vacuum electronics, in particular in forward- and backward waves oscillators. The optical counterparts of Cherenkov effects can play an important role in the generation of new optical frequencies by the solitons propagating in fibers with high order dispersion. Indeed, since 1995 when Cherenkov radiation of optical solitons was reported for the first time this effect has been actively studied both experimentally and theoretically, especially in the context of the generation of optical supercontinuum spectra16. The possibility of frequency comb generation in optic ring microresonators21 is another motivation to study these systems. The effect of high-order dispersion on the dynamics of the solitons in such systems was investigated and it was shown that high-order dispersion can be crucial for stabilization or destabilization of the soliton clusters28. The steering of solitary structures by dispersive waves is an important effect discussed in the literature32. An important problem of the formation of stable bound states of dissipative solitons interacting through the radiation field was considered in detail in terms of a theoretical approach33 and the experimental observation of the dispersive waves mediated interaction between solitons in fiber cavities34.All these effects are important for the explanation of the dynamics of solitons and domain walls in wave-guiding systems. The effect of synchrotron emission of oscillating solitons can also affect the dynamics of the solitons strongly and thus is of interest from both fundamental and practical points of view. For example, it can possibly be used for the design of optical analogues of gyrotrons and free-electron lasers. Optical solitons in fibers seem to be the best candidates for experimental observation of the discussed effect of synchrotron emission, however similar effects can take place in polariton condensates, plasmas, hydrodynamics and other systems of different physical origins. The resonant emission is possible not only in the conservative but also in dissipative systems providing that the dissipation is not too strong and slowly decaying waves can propagate in the medium. The advantage of weakly dissipative systems is that the resonant radiation can occur there in a stationary regime when it is not disguised by any transitional processes. For this reason, the present study is focused on weakly dissipative annular optical waveguides with high order dispersion, resonantly pumped by external coherent light. Our choice is also motivated by the fact that these systems are not only a convenient test bench for investigation of synchrotron radiation but are of interest on their own and have been actively investigated in recent years theoretically and experimentally. For instance, it was shown that clusters of solitons forming in such systems can be used for information storage and processing38. Cherenkov radiation of oscillating cavity solitons was recently considered39 but the effect of the synchrotron radiation was out of the scope of that paper. In the present article we consider synchrotron radiation of oscillating solitons propagating in an optical fiber cavity with third order dispersion.Propagation of oscillating solitons in the presence of high order dispersion was studied experimentally and numerically and very rich dynamics of the solitons was reported40 in the form27θ is the detuning between the cavity resonance and the frequency of the pump, P is the amplitude of the pump and d3 is the third order dispersion. To make the resonance spectral lines well pronounced we choose the detuning to be equal to a realistic value θ = 15, just slightly larger than the value reported by a numerical study reported in the literature27. For fiber cavities the boundary conditions are periodic and the spectrum of the eigenmodes is discrete. In this paper we restrict ourselves to the case of cavities so long that the spectrum of the eigenmodes can be considered quasi-continuous. However, we remark that in the case of shorter cavities the boundary conditions can play an important role41. In the literature, propagation models represented by equations of the kind of Eq. |2 of the field for the small value d3 = 0.02 is shown in Fig. 39. A close up view of the frequency range k ≈ 47–54 in terms of a spectrogram PS is shown in Fig. x domain exhibit a polychotomous, i.e. multi-peaked, structure in the k domain. The intensity of the resonant radiation strongly depends on the strength of high order dispersion and for larger dispersion the radiation spectrum becomes more intense, see Fig. d3 = 0.04. While the behavior is qualitatively similar, the periodic recoil of the soliton becomes more apparent and the resonant radiation is shifted towards smaller values of k to k ≈ 28. The positions of the spectral lines observed in numerical experiment match the predictions of the resonant condition if we account for the change of the soliton velocity appearing because of the recoil of the resonant radiation. The dependence of the spectral characteristics of the resonant radiation on the strength of third-order dispersion in the range d3 = 0.02–0.04 is shown in the Supplementary Information  of Fig. k = 0 corresponding to the background. Comparing spectra of non-oscillating and oscillating solitons one can notice that in the latter case several additional maxima appear on the spectral curve.For l ≤ 4 are shown in Fig. l  = 0 in Eq.  with the dispersion characteristics gives the position of Cherenkov resonance and it is seen that it matches the position of the one of the spectral lines observed in numerical experiment perfectly. The other resonances (l ≠ 0) fit the other spectral lines. So we can explain the spectral structure of soliton radiation as a number of lines produced by different temporal harmonics of the oscillating soliton. It is obvious that the emitted waves have phase velocities close to the velocity of the soliton  and thus these radiations can be interpreted as relativistic radiation of the oscillating soliton, i.e. synchrotron radiation. Looking at the dispersion characteristic in the vicinity of the resonances one can conclude that in the considered case Cherenkov and synchrotron radiations always have group velocities higher than the velocity of the soliton and so the radiation must appear in front of the soliton. It is instructive to compare the radiation of oscillating and not oscillating solitons. The blue dashed curves in Fig. P = 6. The velocities of the solitons are practically the same for P = 6 and P = 8 and so the Cherenkov spectral lines of the oscillating and non-oscillating solitons practically coincide. However it is clearly seen that the non-oscillating soliton emits neither cyclotron no synchrotron radiation. In the case of third order dispersion the synchrotron resonances are always situated on the left and on the right of the Cherenkov spectral line, but it is important to remark here that there are dispersion characteristics allowing for synchrotron resonances for l ≠ 0 in the absence of the Cherenkov one.To explain the observed maxima on the spectrum we solve the resonance conditions given by Eq.  graphica0 in Eq. ] does noP = 6 when third order dispersion is set to d3 = 0.02. The resonantly emitted radiation contains only a single mode, see the blue dashed line in Fig. vg − v]. An account of the temporal evolution of the corresponding intensity, spectral intensity as well as a spectrogram at propagation time t = 10 is given in the Supplementary. For stronger pump P = 8 the soliton develops oscillations and, as discussed above, the cyclotron radiation appears in front and behind the soliton. This radiation is clearly seen in Fig. k has small group velocity and decays quite quickly in x. Consequently, this radiation is seen only in the immediate vicinity of the soliton. Cherenkov and synchrotron radiation at large k have higher group velocities and thus decay slower in x than the radiation at low k. In Fig. x ≥ 60 –plane is referred to as a 2-cycle44. This comprises the simplest periodic motion in which the soliton periodically oscillates between a unique intensity minimum and maximum. Such a behavior is given at very small values of third order dispersion d3 < 0.044. For increasing values of d3 the system passes through a sequence of period doublings. E.g., for d3 = 0.047 (Fig. d3 = 0.06 (Fig. t = 16 is found for d3 = 0.063 (Fig. d3 = 0.07–0.11 the homogeneous stationary state can become the only stable solution, portraying an extinction event in which the oscillating soliton can vanish completely. An example of this behavior is shown in Fig. d3 = 0.09. For larger values of d3, remerging bifurcation events occur46. E.g., at d3 = 0.175 (Fig. To summarize the propagation dynamics of oscillating solitons in the parameter range ion Fig. . In case.02 Fig. , these p047 Fig. , the tra.06 Fig. , the tra063 Fig. . Here, t175 Fig.  the traj175 Fig. .Figure 5d3 ≈ 0.44–0.17, whereas nearly periodic oscillations are observed outside this intermediate range. Possibly this explains the complicated dynamics of cavity solitons discussed in the literature35.Overall, this behavior can be explained by the fact that the pulsations of the synchrotron radiation causes a pulsating recoil of the radiation on the soliton. Such a recoil leads to the oscillating variations of the soliton velocity and intensity. In its turn, the change of the soliton parameters affects the instantaneous frequencies of the radiation. The interplay of these effects can result in a very complex dynamics of the soliton. As evident from Fig. In summary, it is shown that oscillating solitons can emit resonant radiation analogous to the synchrotron and cyclotron radiations of moving charges. The resonance condition for the radiation is derived and it is demonstrated that the predicted resonances match the observed spectral lines very precisely. The structure of the field of the emitted radiation and the effect of the recoil of the radiation on the soliton dynamics were also addressed in the presented study. The reported results are of very general physical nature and thus can be observed not only in fiber cavities but in a wide class of physical systems supporting oscillating solitons.Supplementary Information.Supplementary Information2.


In this paper, a mobile sound localization setup is described that can be used to measure a persons’ localization performance in a sophisticated way. With this mobile setup, researchers can travel to subjects, and studies are not limited by the willingness of participants to visit the clinic. In the setup, sounds are presented within a partial sphere in both the horizontal (−70° to 70° azimuth) and vertical (−35° to 40° elevation) plane. Participants are asked to indicate the perceived sound origin by pointing with a head-mounted LED. Head movements are recorded and instantly visualized (i.e. online target response plots). Depending on the research question, the setup can be adjusted for more advanced or simplified measurements, making the setup suitable for a wide range of research questions. The rationale for building this mobile setup was to test horizontal sound localization abilities  and vertical sound localization abilities  of children and patients who were otherwise not accessible for testing. In this setup loudspeakers are not visible and subjects are asked to indicate the perceived sound direction by a natural head-pointing response towards the perceived location. An advantage of the implemented pointing-method is the playful manner in which children are tested. They are ‘shooting’ at the perceived sound target location with a head-mounted LED and have fun while performing the test.•We present a mobile sound localization setup suitable for measuring horizontal and vertical sound localization in children and adult patients in the convenience of their own environment.  Specifications TableThe ability to localize sounds is essential in daily life, for example, in traffic situations and for overall feelings of comfort i)There is a preference to test the participants’ localization abilities close to their home or school.ii)A setup in a specific clinic is not suitable for measuring sound localization.As sound localization is an important aspect of hearing, it has been investigated extensively in the past for normal hearing subjects and patients The initial reason for developing a mobile setup was to test normal-hearing children at schools. In the mobile setup, these children can be tested in a playful manner. The children are having fun while ‘shooting’ with their head-mounted pointer in the perceived sound direction. Testing at schools enables researchers to measure groups of children in a convenient way, without asking them and their parents to make a trip to the research institute, which could be a barrier to participate in studies.An additional benefit of the setup's mobility is the opportunity to evaluate different treatment options for specific groups of patients at nearby clinics. This is of particular interest in Europe because due to the differences in reimbursement systems among countries a variety of different treatments are available for similar patient populations.Because the mobile setup has proven to be a successful tool T60<0.09 s). The available lighting in the lab is dimmed during experiments to prevent any remaining visual cues. Performing the test under dimmed light conditions instead of complete darkness provides the ability to observe the participant, and is an advantage for testing patients with disturbed vestibular functioning Twenty-four speakers  are mounted on an iron framework built in a sound-isolated anechoic trailer . Walls, The participant is seated comfortably, while wearing a pair of custom-made glasses, which consists of a frame on which a rod with six infrared reflectors  is mounted. On the frame, also a green LED is centrally fixated in front of the participant's nose . The parMatlab  is utilized to control the experiments via custom-made scripts in which the test settings can be adjusted. Stimuli are delivered by a sound card with 24 analog output channels  and an electronic board , which triggers the fixation LED located at the center of the speaker array. For experiments that require more than twenty-four speakers, an additional sound card (MotuMini), and 8 Genelec speakers can be added. These additional speakers can also be used to widen the azimuth span. The costs for 24 active speakers is about $6000. The soundcard $1000 and the cables about $500. In addition, there are costs for the trailer and software licenses.The glasses are placed on the participant's face like normal glasses and stabilized with cotton straps to prevent movement of the glasses relative to the head. Before the start of the experiment, participants are instructed to move their head to test whether the glasses do not slip off. The glasses are constructed in a way that there is enough space around the ear to wear hearing aids or auditory implants, and can be used by persons with aural atresia. Head movements are recorded online via infrared cameras  tracking the position of the reflectors . A red Lsee 1.6 Data analysis and representation and Video 1). The participant receives no visual reinforcement at the sound source. The researcher monitors whether head movements are made well within the acquisition time of 1.5 s. The acquisition time can be extended if needed. After the head movement, the fixation LED is turned on again (within 2 s), indicating the beginning of a new trial.The participant controls presentation of stimuli by a button box. After pressing the button, the central fixation LED is turned off, and a stimulus with 150 millisecond duration is presented after a variable 200–300 millisecond delay. The participant is instructed to point as fast and accurate as possible to the perceived sound origin. Each individual response (a completed head movement) is directly visualized online on a laptop in front of the researcher :‘you are doing excellent’ or ‘perfect’ is provided. When the task is performed incorrectly the participant is instructed again. For instance, when the participant is still not pointing with the fixated LED to the target, but primarily looking to the target, the participant is asked to point to the experimenter's hand, and the difference between looking and pointing with the head-mounted LED is explained. This instruction helps especially when stimuli are presented about 70° opposite of the ‘frontal 0° direction’ of the participant. When needed, the participant is repeatedly instructed to look straight-ahead at the fixation LED before each trial.Usually, after 2 or 3 trials participants are instructed to initiate the next sound presentation by pressing the button themselves. During the training block, positive feedback like: A standard experiment includes sound stimuli of broadband (0.1–20 kHz), low-pass (0.1–1.5 kHz) and high-pass (3 - 20 kHz) Gaussian noise bursts (duration of 150 milliseconds with 10 milliseconds on- and offset ramping). The low-pass and high-pass stimuli are applied to specifically measure the use of ITD's and ILD's. Stimuli are presented interleaved in a pseudorandom order with sound levels ranging from 45 to 65 dB  to limit the possibility that monaural level cues facilitate monaural localization abilities The stimulus duration is kept short to ensure the participant's head remains stationary during stimulus presentation, resulting in a true response to a single location. Sound presentations are evenly distributed over the two-dimensional frontal hemifield. Usually the minimum number of stimuli per condition is 75  and can be changed depending on the research question. If it is preferable to present a low number of trials, for example because of a limited attention span, one can choose to only measure broadband stimuli to reduce testing time. Especially when working with children, the researcher should keep the number of stimuli to a minimum in order to reduce the risk of insufficient concentration and, consequently, less reliable responses. A trial refers to a single measurement and a block  refers to all trials that make up a complete experimental condition. The time required for one block varies from 5 to 10 min, depending on the number of stimuli  and the time the participant takes to initiate a new trial.For each trial, the stimulus settings  and the raw data containing the recorded head movement  are stored in a data structure in the Matlab Workspace. The data structure is saved after each trial and can be used for offline analysis.During measurements, every individual response is visualized online by plotting the head movement over time (see video 1). During each block of trials, preliminary target-response plots for azimuth and elevation are created online showing all completed trials. In Afterward, all measurements are checked and erroneously processed start- and end-points of the head movement are manually corrected. Subsequently, a final target-response plot is created per block and a linear fit of the data is made based on below equation:a and c, for the horizontal and vertical plane respectively. Any bias in the responses is quantified by parameters b and d. In case of perfect localization the response of the listener is equal to the target, b and d in In The deviation from perfect localization is expressed by the mean absolute error (MAE), which is computed using below equationThe target-response plots are the standard way of presenting the data, of course, further analysis of the data is possible. For instance, in case of asymmetric hearing loss or unilaterally aided patients, Vogt et al. The rationale for developing the mobile setup was to enable researchers to investigate sound localization in children and patients who were otherwise not accessible for testing. Vogt et al. Recently studies have been published with normal hearing children tested in Kleef, Germany One of the outcomes of rehabilitation that can be tested is whether binaural hearing is restored. For example, in case of unilateral hearing loss or a large hearing asymmetry, accurate localization testing can provide information about the effectiveness of treatment. In the study presented by Vogt et al. In addition to assessing the effects of rehabilitation with hearing aids or auditory implants, the setup is also suitable to study different specifications of hearing aids or auditory implants. For example, properties such as sound processing strategies, directionality of microphones, pre-processing (and time delay) in hearing aids or the effect of synchronization of bilateral worn devices can be investigated.Another advantage of the described mobile lab is the possibility to investigate sound localization in the vertical plane. There is an increasing interest for testing sound localization abilities in single-sided deafness (SSD) patients using a cochlear implant Finally, there are a couple of drawbacks of the setup that should be mentioned. Firstly, a small but acceptable drawback of equipping a trailer with this setup is the limited sound isolation. For example, high-intensity external sounds, such as emergency helicopters or heavy rain, are audible. Secondly, the limitation to the size of the trailer (dimension 4.03 × 2.32 × 2.25 m) means a full circle for measuring in the horizontal plane does not fit. Therefore, front-back confusions cannot be tested. Thirdly, the range of elevation is reduced because of height constraints of the vehicle.In summary, the presented mobile lab provides a versatile test environment to investigate horizontal and vertical sound localization abilities  of both children and adults, normal hearing and hearing impaired, away from the clinic. Depending on the research question, adjustments can be made for more advanced or simplified measurements. The better our measurement methods become, the better we can assess the effectiveness of treatment.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


The paper presents the concept of planning the optimal trajectory of fixed-wing unmanned aerial vehicle (UAV) of a short-range tactical class, whose task is to recognize a set of ground objects as a part of a reconnaissance mission. Tasks carried out by such systems are mainly associated with an aerial reconnaissance using Electro-Optical/Infrared (EO/IR) systems and Synthetic Aperture Radars (SARs) to support military operations. Execution of a professional reconnaissance of the indicated objects requires determining the UAV flight trajectory in the close neighborhood of the target, in order to collect as much interesting information as possible. The paper describes the algorithm for determining UAV flight trajectories, which is tasked with identifying the indicated objectives using the sensors specified in the order. The presence of UAV threatening objects is taken into account. The task of determining the UAV flight trajectory for recognition of the target is a component of the planning process of the tactical class UAV mission, which is also presented in the article. The problem of determining the optimal UAV trajectory has been decomposed into several subproblems: determining the reconnaissance flight method in the vicinity of the currently recognized target depending on the sensor used and the required parameters of the recognition product , determining the initial possible flight trajectory that takes into account potential UAV threats, and planning detailed flight trajectory considering the parameters of the air platform based on the maneuver planning algorithm designed for tactical class platforms. UAV route planning algorithms with time constraints imposed on the implementation of individual tasks were used to solve the task of determining UAV flight trajectories. The problem was formulated in the form of a Mixed Integer Linear Problem (MILP) model. For determining the flight path in the neighborhood of the target, the optimal control algorithm was also presented in the form of a MILP model. The determined trajectory is then corrected based on the construction algorithm for determining real UAV flight segments based on Dubin curves. In the literature describing the problems of planning unmanned systems missions, there are theoretical algorithms for determining unmanned aerial vehicle (UAV) flight routes for individual UAVs or UAV swarms that cooperate together. A detailed introduction to the issues of route planning along with the presentation of the taxonomy of the problem is given in the paper by Coutinhoa et al. . Yang etThe general plans include the allocation of targets for each of the UAVs available during the mission, but they do not specify how to fly to and between destinations. Examples of articles dealing with this type of a problem are given in the paper by Stecz, Gromada . The worA specific approach to route planning is a Computational-Intelligence-based (CI) UAV path planning presented in the paper by Zhao et al. . They prAnother type of a route planning problem, where a sensor use is not usually considered, is route planning in real-time. An example is the paper by Schellenberg et al. , in whicDetailed schedules determine the exact flight routes along which the UAV should fly, and additionally they take into account the parameters of the sensor assigned to the target ,7. This Preparation of a detailed mission plan for tactical class UAVs was discussed in the article written by Stecz, Gromada . The artFrom the optimization theory point of view, when planning a mission based on recognition orders, the global task Vehicle Routing Planning with Time Windows (VRPTW) ,10,11 isThis article presents an in-depth analysis of the mission planning method for tactical class fixed-wing UAV and introduces new algorithms related to determining the UAV flight trajectory in the neighborhood of the recognized target. The results of our experience, as part of the construction work on tactical class fixed-wing UAV, are described. The article presents the algorithms for planning the detailed flight trajectory of a fixed-wing UAV, which uses EO/IR and SAR for recognition. As the work concerns tactical class UAVs, the trajectories for the SAR must be particularly carefully defined. Even slight deviations from the designated traffic section may make the prepared SAR scan unreadable. Thus, the task of determining the correct trajectory including terrain obstacles becomes critical. So far, the assumption is that a short-range tactical class system, that operates on a ceiling of 1000–5000 m above the ground level, is considered. By default, its flight length is no more than 6–8 h. This class of systems includes, for example, Hermes 450, which is a tactical class system project.In the first part of the article, in The second part of the article  presentsThe third part of the article in When determining the flight trajectory for UAV target recognition, a set of constraints related to battlefield recognition is defined that should be taken into account when planning the flight trajectory. The most important constraints taken into account include the type of a sensor to be used, the quality of the recognition material to be obtained (set on the NIIRS), and the parameters of the UAV itself. In the case of UAV parameters, the most important are operational speed, operational flight altitude, available payload useful for recognition, and time to reach the optimum altitude for the recognition with a given type of a sensor. For these systems, meteorological conditions prevailing in the recognition area (mainly wind speed and cloud cover) are extremely important. This applies also to SAR.There are several approaches in the literature to solve the local task of UAV flight trajectory optimization. One of them was presented in the work by Stecz, Gromada , when thFor an advanced overview of dynamic optimization rules, see Pytlak . An alteThe rest of the article shows the methods of smoothing the flight trajectory to reflect the real flight trajectory of a given class platform. Due to the size of the platform , turning radius and turning time are not negligible. The article presents a new algorithm for the construction of real trajectories of tactical platforms. Based on the definitions introduced by Coutinhoa et al.  and 5000 [m] above ground level, respectively.For each of the presented tasks, exemplary results are provided along with examples of recognition materials collected at the designated trajectories.UAV moving along a route uses a sensor to recognize a predefined target within a range of the sensor. The method of determining the route segments during a reconnaissance, the UAV should follow, depends on a type of a sensor and on the required data quality. In this section the route planning model for a UAV that uses EO/IR and SAR is presented. At the beginning, however, it should be defined how the analyst can select the expected quality of recognition material in both SAR and EO/IR recognition.NIIRS is a scale used for rating the quality of imagery acquired from various types of imaging systems. It defines the levels of image quality (interpretability) based on the types of tasks an analyst can perform with these images. The higher the NIIRS level, the more details are visible in the image, so the analysts are able to perform more sophisticated interpretations.The NIIRS consists of 10 levels, but taking into account the parameters of UAV, presented in the paper, one can get the images with a scale not higher than 7 . detect a medium-sized port facility; distinguish between runways and taxiways at a large airport [ airport .7—Confirm the movement of a car-door, as it is opened or closed, while car is not moving [t moving .9—Consistently track the movement of a pedestrian’s body and arms in a busy public area [lic area .Examples of some requirements for selected VNIIRS levels (NIIRS for visible light):Optical images are the most popular and best described data collection method, which is why most of mathematical approximations of the NIIRS relate to this type of imaging. The most popular approximation pattern for NIIRS is the General Image-Quality Equation (GIQE).GIQE is an empirical equation used to approximate the NIIRS level of a given image. Currently, the version widely used is GIQE 4, developed in 1996. It considers both raw images and images with processing performed to improve their quality . Processing operation parameters must be constant and known. The definition of GIQE 4 is presented in Equation (The newest version of GIQE (version 5) is similar, but it uses less parameters and does not consider image processing. GIQE 5 is not considered in this paper due to the lack of support for IR imaging.ST (Sensor Type)—the value determined by the sensor type is 10.25 for EO and 10.75 for IR.GSD (Ground Sampled Distance)—field distance corresponding to the distance between two adjacent image pixels (expressed in inches) (see hes) see .f—focal length, and h—flight height above ground level.GSD in the GIQE equation is determined as a Ground Plane projection. In the case of orthogonal projection, the formula for GSD is as follows,For high viewing angles (close to 90°), one can approximate this value with Equation :(3)GSDGh angles :(5)GSDCRelative Edge Response (RER)—A fixed value that indicates the camera’s ability to image sharp edges. This parameter is usually determined empirically based on the observation of a known test pattern. Two more coefficients in Equation —The signal-to-noise ratio of the matrix can be determined experimentally or on the basis of formulas taking into account the detailed parameters of the matrix.SNR results from photon noise, dark current, FPN (Fixed Pattern Noise), and reading noise. However, it is often also given in sensors datasheets.In the paper byVelenzuela, Reyes , the autG (Noise Gain)—A parameter resulting from the operation of edge sharpening, which increases the effective RER, but also increases the noise in the image. For matrix sharpening filters, the gain is determined as the root of the sum of squares of the individual values of the filter mask. An example considering a typical edge detecting filter matrix (r matrix  is shownr matrix :(7)MfilGain equals:For images without edge sharpening performed G = 1.H (Height of Overshoot)—value of overshoot of intensity resulting from edge sharpening. Due to that operation, MTFC (Modulation Transfer Function Compensation) is introduced. It causes additional distortions in the image resulting from the amplitude overshoot.The parameters RER, H, and G are defined after image postprocessing, by contrast SNR is calculated for an unprocessed image. GIQE applies to uncompressed images. In case of the usage of a compression additional loss of NIIRS has to be assessed for the used algorithm with its degree of compression.Algorithm 1 Reverse NIIRS calculation—requirements of flight geometry to achieve given NIIRS1:Input expected NIIRS,2:Calculate G, H values for currently used video processing filters,3:Read constant NSR, RER, A and B coefficients,4:5:A < 0)Based on the presented equation, it is possible to define technical (semi-objective) requirements for the given optical or SAR image required to provide sufficient quality. The procedure for determining maximum Ground Sample Distance for given NIIRS value is presented in Algorithm 1. A calculated value is approximated maximum GSD image to maintain the demanded NIIRS level.When designing the flight path associated with observing a specific area, the Field of View (FOV) of the selected sensor must be specified. This information is also useful for recordings playbacks. It is worth noting that the calculations estimating the FOV depend not only on the sensor type, parameters, but also on the method of its installation and manipulation on the platform.An example of a typical camera with a manipulator with two rotary axes will be discussed: C (around Z axis) and A . This implies that the upper and lower edges of the image will be marked by horizontal lines in the camera projected on a plane.Algorithm 2 Field of Vision calculation . SARs generally have much larger scanning angles than optical cameras, and therefore the algorithm accuracy will be much more influential in their case.A very detailed introduction to remote sensing can be found in the paper by Colomina, Molina . The autIn case of SAR, an image resolution is defined by digital parameters set before data acquisition. Short description of these systems can be found in Gromek et al. . For exaAnalog/digital converters precision or digital processing noises can be treated as a constant factor reducing the NIIRS level. In case of external disturbances, it is difficult to predict and consider their influence on image quality. Atmospheric conditions have little direct influence over the SAR signal, but they affect the UAV flight, thus the closer the flight is to the scanned area (the shorter the path is) the lesser is the weathers influence.NIIRS applies for SAR images as well as for EO images. Due to a difference in the characteristics of sensors, different requirements are set for each of them to be accounted to the respective NIIRS level . In case of detectable objects, interesting for military and security services (vehicles and buildings), the differences are insignificant.The SARs beam angles spread wider than a size of an output image. Due to the higher signal power density, radar should be imaging an area as close to the center of the beam as possible. Nonetheless in appliances, where movement limitations can apply (due to threat zones or optimization of a flight trajectory), algorithms should include the options related to scanning with a squint angle.R is yjr slant range and There is no single, most popular GIQE equivalent for SAR/NIIRS estimations. One algorithm (presented in Equation ) was proAnother alternative approximation was introduced in Nolan et al. . Output In case of SARs, there is a major problem in creation of an universal equation due to many parameters affecting image quality including algorithms and the used principle of data processing. Thus, NIIRS should be researched for every device.For applications, where tactical UAVs are used, it is important to enable a scanning area under the angle from a perpendicular (default) direction. In military appliances, it might be related to speed flight optimization, but it also needs to avoid threat zones. In case of civil applications, like border guarding, it is convenient to gather scans of few separate areas with a single flight segment. Especially while monitoring the whole space with the Moving Target Indicator (MTI) , which mAn example geometry with the squint angle w→=Equation .(12)Tα=For an optical camera coordinate system, a calculation is presented in Equation .(13)w→=The presented algorithm enables to calculate what a sensor could see under the given angles. Often it is required to determine where (under what angles) the sensor should scan to view an area of interest. This requires a solution of inverse kinematics, which outputs an angular position for servomechanisms or electronic correction systems for SARs. Simplified (omitting displacements) equations for SAR coordinate reversion are presented in Equations –16):(14:(1416):. This lEquation .R is the slant range, SARs enable to conduct single scans or series of scans. A sequential scan can be conducted as a single long, continuously expanded strip scan or as a mosaic pattern of separate scans, depending on the radars’ processing software. In case of mosaic output images, depending on the used algorithms and parameters, it can overlay or not. A minimal distance of flight to execute a single scan is determined by the required synthetic aperture length (Equation .(17)LA= Gromada .n is the coefficient depending on requirements of a size of the image  multiplEquation . In caseIn case, where the created scan image is longer than the aperture length  task in Algorithm 3 Route planning algorithm1:Import data: 2:Select the route segments assigned to all targets e target 3:Find feasible route plan for UAV between airfields 4:Update distances among vertices of ented in  or algor5:Find optimal route—VRPTW model –33) (Se (Se33) 6:Transform generated path into a feasible trajectories7:  Generate trajectories—MILP model –46) (Se (Se46) 8:  Smooth generated trajectories The presented procedure for preparing the mission plan has three main steps and it is presented in Algorithm 3.At the first step all data is imported. When planning a real mission, obstacles that threaten UAVs must be imported. All targets that UAVs must recognize are also loaded into the model.Next, for all the targets, the sensors used for recognition of each target are assigned with predefined quality of images that must be collected as described in In the third step all the possible route segments between all pairs of vertices are selected. The vertices model the points lying near the targets to be recognized. For determining the shortest paths between pair of vertices, the Dijkstra algorithm can be used.Next, at the fourth step, the distances between all pairs of waypoints are updated. The reason is that the network  Gromada , a versi Gromada . There a Gromada ). One of Gromada  and Kim  Gromada . Xin et  Gromada . In thisIt is assumed that a robot can communicate. As the sensor network built by one robot meets the network built by another robot, the robots exchange the information with each other. The method presented by Zhang et al.  is also As a result of determining the shortest paths between the pairs of vertices, the square matrix At the fifth step, the routing problem VRPTW is solved. If exists the feasible solution, this solution is also optimal. The details of VRPTW were shown in the paper by Stecz, Gromada . In thisIn the last step, the optimal trajectories are calculated. This process is divided into two parts. The section presents the modified version of the VRPTW model for determining UAV flight routes which was originally presented in Stecz, Gromada . The modModel parameters:e—earliest date when an operation or task can start, d—date when a planned task should be completed, p—priority.Each waypoint Model variables:An optimization task based on the minimization of the travel time of the UAVs:The optimization function is presented in . This fuConstraint  ensures Constraint  means thConstraint  means thIn the article, there is an assumption that every UAV must fly on a mission. This is a technical constraint. In a real situation, there is no need to begin a mission for each UAV. The presented situation is modeled by a constraint .A classical flow preservation requirement for any net is presented as a constraint . When thEach UAV, that has begun its mission, must return to the landing base b∈W .In the case of the VRPTW problem, additional time window constraints should be added. The time window for recognition is the time it takes for the target to be recognized in order to obtain reliable and useful information. Time window constraints are presented in Equations –28)..28).A time of UAV arrival to the waypoint nstraint . This tinstraint  preventsnstraint  preventsstraints  and 31)∈E, It should be remembered that the task of VRPTW will be solved correctly if for the indicated network arcs, which have been designated as route segments of the potential UAV flight over the target, the higher priority will be set than for other sections of the route.Next two constraints are technical constraints that forces solver to choose the proper vertices and edges. Constraint  ensures nstraint ). These nstraint –31) are arei∈W tThe constraints commonly used to eliminate subtours in VRP were omitted. They were presented in Stecz, Gromada  and theyThe essential part of determining the optimal trajectory for the UAV recognizing the target is to determine the possible trajectory based on the mathematical model presented below. The model takes into account the movement of UAVs in the area where there is probably a threat that is modeled in the form of hemispheres in 3D or circles in 2D. The 2D model is used when it is assumed that the UAV moves at a given height, which is not being changed. These types of threats are most common in practice and they are usually modeled in this way.The UAV path planning problem is an optimization problem to obtain an optimal cost value under limited constraints, and it is usually modeled as a nonlinear optimal control problem. A good introduction to solving this type of task is the book written by Pytlak .However, some of these tasks can be discretized and solved by classical linear integer methods, as shown in the article in As a result of solving the above task, one get two initial UAV flight trajectories. In order to accurately determine the real UAV flight trajectory along predetermined routes, these trajectories should be smoothed, which is presented in In the described problem, we assume that UAV may encounter obstacles modeled in the form of hemispheres on their flight path. To simplify the description in the article, obstacles are modelled in the form of circles, which means that changes in a flight altitude are not included. This means that the task has a constraint in the form: This type of a constraint is a non-linear one and it requires linearization, as described in Parikshit et al.  and Lin i-th coordinate of the point t-th and the variable o-th obstacle center in 2D.In the further part of the paper, the waypoints ).).gton=1 Optimization task based on the minimization of the travel time of the UAVs:Formula  is the cConstraints –36) are are36) anstraint  guarantenstraint –44) wil wilbtol=R is at least two times smaller than minimal travel distance between two consecutive waypoints. For In order to prepare an efficient path plan, optimal turning approaches must be researched. Every proposed flight order has to be modeled with regard to dynamic limitations of the platform. Otherwise, the quality (cost function) of a path would base on The Euclidean distance. This is very coarse assumption viable only for trajectories, where the turn radius There are analytical algorithms for path planning like Fast Marching —fastest presented by Anderson et al. , but doeIt should be noted that the waypoints generated in the process of determining the flight trajectory, show the optimal UAV flight path. Therefore, after calculations, the number of the route points W increases significantly. But we must keep in mind the UAV maneuverability, so in the process of smoothing the flight path, the number of waypoints is minimized. In this section, for the sake of simplicity and without loss of generality, we assume that the waypoint In the further part, for ease of understanding, an acute or obtuse angle (w) and the lines (L) (as their form depends only on the used metrics). The line segment L is defined by a pair of the waypoints In this paragraph, a set of geometry shapes is described. For the higher description precision will be defined using a symbol and an explicit vector of attributes, except for the points  and segment targets (Algorithms 5 and 6). This requirement induces the existence of a post-maneuver arc. That allows for creation of a single, universal rule to adjust movement to the next target.An algorithm for through-line-segment trajectory generation is described in Algorithm 5 and shown in R is the radius for maneuver with current speed and It is important to note that in case of long distances between waypoints, UAV maneuverability and atmospheric conditions can differ considerably. That is why in this algorithm two separate turn radiuses’ symbols are used. For this algorithm, also the direction of an arc has to be defined. Direction defines right (clockwise) and left (counterclockwise) turn type using a classical vector Equation :(49)dir=z axis value of vector multiplication) is lower than 0. Turn is counter clockwise for Algorithm 5 Ingress arc calculation for line segment target1:if last maneuver is line segment thengment Lm  then2:  Delete last maneuver , an argument distance is compared to Algorithm 6 Egress arc calculation for line segment target1:Find circle int wi+4 ,2:ifthen3: Replace circle  wil wil33) wIf the UAV is to recognize a target located in an area where a threat to the UAV is located, a flight trajectory should be determined that minimizes the risk of platform destruction according to the algorithm –46)..46).The last example presented in The example presented is very often used in practice when the analysts want to check some part of the terrain comparing them with larger photos.This section discusses the important elements to pay attention to when implementing the UAV flight trajectory generation algorithm. The performance aspects of the algorithm for determining real UAV trajectories are also presented. The goal function was used to minimize controls, see . It can It is also worth discussing the conclusions resulting from the analysis of changes in the flight path depending on the optimization function used and the number of obstacles. Many experiments were conducted to check trajectory changes in a situation of changes in the number of threat areas for each of the examined objective functions. It should be noted that the greatest impact on the performance of the flight trajectory generation algorithm –46) has has46) harea see . OtherwiThe procedure of determining the flight trajectory for a short-range tactical UAV, described in the article, was actually implemented in one of the projects in which the authors participated. The procedure is a part of the construction of the so-called Mission Plan, which consists of a set of routes that UAV can follow during the mission. In the mission plan, methods of using payload used for reconnaissance are also being developed. From the point of view of the optimization task, which is the preparation of the mission plan, in the case of determining the flight trajectory to recognize a target, one can talk about solving the local task. The global task is to determine a flight plan that ensures recognition of all major targets.The article discussed an algorithm used for detailed flight route planning, which takes into account the payload installed on the UAV. The optimization algorithm was presented, used to determine the exact flight trajectory. Detailed algorithm for determining the turns in flight was presented also. The latter algorithm belongs to the group of algorithms for plotting a trajectory based on Dubin curves, but it is very easy to implement, unlike many others presented in the literature. The set of algorithms presented allows for the implementation of the flight routing subsystem for one or several UAVs that use payload for target recognition. Therefore, the article contains a complete set of algorithms that allow detailed mission planning on the air platform using available commercial or non-commercial solvers.Further work is associated with the implementation of methods for dynamic change of a UAV flight plan, which is a difficult task because tactical class systems are not equipped with appropriate number of sensors. In addition, it should be noted that UAV that performs reconnaissance tasks, is not usually integrated with other sources of reconnaissance data. In terms of optimization, especially the graph and network theory, to perform this task, the presented model should be extended with mechanisms for efficient modification of the connection network, i.e., the construction of the so-called ad hoc networks.The need to expand the presented model with elements of network modification in the ad hoc mode forces the use of video tracking mechanisms in practical operational activities. In this case, the algorithms of the EO/IR alone are insufficient  and the mechanisms of flight route planning should be modified due to the mutual positioning of the UAV and the tracked target.Considering the dynamics of changes in the battlefield situation and the fact that tactical class UAVs are the outermost operational element, it will be required to increase their recognition capabilities through the use of ELINT electronic reconnaissance subsystems on board UAVs.It is also worth emphasizing the important issue mentioned in the paper. The issue involves the need to build models for specific classes of unmanned systems. Each optimization model presented in the paper for solving the route planning task and determining flight trajectories concerned a tactical class system. Detachment from the UAV class and its equipment is too much simplification of the problem and usually makes models purely theoretical.It is only when the flight trajectories are planned in detail for the UAV mission that it is possible to start further tasks related to the location of the unmanned platform in the field in the event of loss of GPS signal. It is currently one of the most strongly developed directions of work in the field of building intelligent flying systems.