Alexey I. Chulichkov

The possibility of estimating the values of a function at given points of the measurement results of a finite number of its linear functionals

The possibility of estimating the values of a function at given points of its domain is investigated. The estimation uses results of the measurements of a finite number of linear functionals; the results of the measurements are distorted by an error. It is shown that only the component of the functions from the linear finite-dimensional subspace can be estimated with a finite error. A method for estimating this component with accuracy control is proposed. The mathematical methods of measurement reduction proposed by Yu.P. Pyt’ev are used. An example of the estimation of an emission spectrum that is measured by a double-slit spectrometer is described.

On cloud bottom boundary determination by digital stereo photography from the Earth’s surface

In this paper, we studied the method for measuring the cloud bottom boundary altitude using the stereo pair of cloud images obtained using two digital photo cameras. We suggested a method for determining the camera orientation parameters using the nighttime images of star sky. The range to the cloud is calculated using the shift of the image of a cloud fragment as a whole. A given fragment on the photographs is identified using the methods of morphological image analysis. When the stereo base is 60 m and images are taken with a resolution of 1200 pixels within a field of view of 60°, the uncertainty does not exceed 10% when cloud altitude is less than 4 km. Optimizing the parameters of photography and increasing the stereo base may substantially improve the accuracy of the cloud base altitude estimation. Examples are presented of cloud bottom boundary determination using a prototype of the experimental setup as compared to data of a laser range finder.

Estimation of cloud height using ground-based stereophotography: methods, error analysis, and validation

Retrieval errors of the atmospheric composition using optical methods (DOAS et al.) are under the determining influence of the cloudiness during the measurements. If there is information about the clouds, the optical model of the atmosphere used to interpret the measurements can be adjusted, and the retrieval errors are reduced. For the reconstruction of the parameters of clouds a method was developed based on taking pictures of the sky by a pair of digital photocameras and subsequent processing of the obtained sequence of stereo frames by a method of morphological analysis of images. Since the directions of the optical axes of the cameras are not exactly known, the calibration of the direction of sight of the cameras was conducted at the first stage using the photographs of the stars in the night sky. At the second stage, the relative shift of the image of the cloud fragment on the second frame of the pair was calculated. Stereo pairs obtained by simultaneous photography, allowed us to estimate the height of cloud. The report describes a mathematical model of measurement, pose and solve the problem of calibration of direction of sight of the cameras, describes methods of combining of image fragments by morphological method, the problem of estimating cloud height and speed of their movement is formulated and solved. The examples of evaluations in a real photo are analyzed and validated by the way of comparing with the results of measurement by laser rangefinder.

Estimation of cloud base height using ground-based stereo photography: method and first results

Retrieval errors of the atmospheric composition using optical methods (DOAS et al.) are under the determining influence of the cloudiness during the measurements. If there is information about the clouds, the optical model of the atmosphere used to interpret the measurements can be adjusted, and the retrieval errors are reduced. For the reconstruction of the parameters of clouds a method was developed based on taking pictures of the sky by a pair of cameras and subsequent processing of the obtained sequence of stereo of frames by a method of morphological analysis of images. Since the directions of the optical axes of the cameras are not exactly known, the graduation of the direction of sight of the cameras was conducted at the first stage using the photographs of the stars in the night sky. As a result, the coefficients of the affine transformation relating own coordinate systems of the cameras were determined. The authors have confined themselves to affine transformations, as the angle between the optical axes was small enough, and the corresponding points on the stereo pair were chosen near the optical axis. At the second stage, the relative shift of the image of the cloud fragment on the second frame of the pair was calculated. Stereo pairs obtained by simultaneous photography, allowed us to estimate the height of cloud. The paper poses and solves the problem of graduation of direction of sight of the cameras, shortly describes the main features of other steps of the method of estimating the height of cloud base. The examples of first evaluations in a real photo are analyzed.

On the estimation of the Maximum possibility for the parameters of a measurement model

The problem of estimating the parameters of the model of a measurement experiment using the results of measurements with an error is considered. The mathematical model of the measurement error is formulated in terms of the theory of possibilities; the distribution of the possibilities on the set of error values determines the order that indicates which error values are preferred (that more probably occur during measurement) and which are less preferred. It is assumed that small error values are preferable to large ones. The mathematical model of the experiment depends on unknown parameters. The problem is to specify the values of these parameters by choosing their estimate for which the difference between the results of experiment and the model prediction is the most possible; this estimate is called the estimate of maximum possibility. An example of estimating the parameters of a Mössbauer spectrometric experiment is given.

An analysis of images of opaque objects in the problem of the optimization of the focusing of optical systems

An urgent problem of the optimization of automated optical microscopy is considered. An approach is suggested during which variations in the brightness of a Gaussian beam in microscope optics are analyzed. A discrete (digital) model, which correlates with the continuous one, is introduced. An analytical approach to finding the maximum of a discrete function that estimates the measure of optical system focusing is suggested; it is based on a discrete model of Gaussian beam behavior and requires a much smaller number of optical microscope images, which speeds up the process of the automated focusing of an optical microscope.

Selection of optical model of stereophotography experiment for determination the cloud base height as a problem of testing of statistical hypotheses

Earlier, we developed a method for estimating the height and speed of clouds from cloud images obtained by a pair of digital cameras. The shift of a fragment of the cloud in the right frame relative to its position in the left frame is used to estimate the height of the cloud and its velocity. This shift is estimated by the method of the morphological analysis of images. However, this method requires that the axes of the cameras are parallel. Instead of real adjustment of the axes, we use virtual camera adjustment, namely, a transformation of a real frame, the result of which could be obtained if all the axes were perfectly adjusted. For such adjustment, images of stars as infinitely distant objects were used: on perfectly aligned cameras, images on both the right and left frames should be identical. In this paper, we investigate in more detail possible mathematical models of cloud image deformations caused by the misalignment of the axes of two cameras, as well as their lens aberration. The simplest model follows the paraxial approximation of lens (without lens aberrations) and reduces to an affine transformation of the coordinates of one of the frames. The other two models take into account the lens distortion of the 3rd and 3rd and 5th orders respectively. It is shown that the models differ significantly when converting coordinates near the edges of the frame. Strict statistical criteria allow choosing the most reliable model, which is as much as possible consistent with the measurement data. Further, each of these three models was used to determine parameters of the image deformations. These parameters are used to provide cloud images to mean what they would have when measured using an ideal setup, and then the distance to cloud is calculated. The results were compared with data of a laser range finder.

Stereoscopic ground-based determination of the cloud base height: theory of camera position calibration with account for lens distortion

For the reconstruction of some geometrical characteristics of clouds a method was developed based on taking pictures of the sky by a pair of digital photo cameras and subsequent processing of the obtained sequence of stereo frames to obtain the height of the cloud base. Since the directions of the optical axes of the stereo cameras are not exactly known, a procedure of adjusting of obtained frames was developed which use photographs of the night starry sky. In the second step, the method of the morphological analysis of images is used to determine the relative shift of the coordinates of some fragment of cloud. The shift is used to estimate the searched cloud base height. The proposed method can be used for automatic processing of stereo data and getting the cloud base height. The earlier paper described a mathematical model of stereophotography measurement, poses and solves the problem of adjusting of optical axes of the cameras in paraxial (first-order geometric optics) approximation and was applied for the central part of the sky frames. This paper describes the model of experiment which takes into account lens distortion in Seidel approximation (depending on the third order of the distance from optical axis). We developed procedure of simultaneous camera position calibration and estimation of parameters of lens distortion in Seidel approximation.

Stereoscopic ground-based determination of the cloud base height: camera position adjusting with account for lens distortion

For the reconstruction of the cloud base height a method was developed based on taking pictures of the sky by a pair of digital photo cameras from the ground and subsequent processing of the obtained sequence of stereo frames. Since the directions of the optical axes of the stereo cameras are not exactly known, a procedure of adjusting of obtained frames was developed which use photographs of the night starry sky. In the second step, the method of the morphological analysis of images is used to determine the relative shift of the coordinates of some fragment of cloud. The shift is used to estimate the searched cloud base height. The proposed method can be used for automatic processing of stereo data and getting the cloud base height. The earlier paper described a mathematical model of stereophotography measurement, poses and solves the problem of adjusting of optical axes of the cameras in paraxial (first-order geometric optics) approximation and was applied for the central part of the sky frames. This paper describes the model of experiment which takes into account lens distortion in Seidel approximation (depending on the third order of the distance from optical axis). Based on this model a procedure of simultaneous camera position adjusting and estimation of parameters of lens distortion in Seidel approximation was developed. The first experimental results of its application are shown.

Method of estimation of cloud base height using ground-based digital stereophotography

Errors of the retrieval of the atmospheric composition using optical methods (DOAS et al.) are under the determining influence of the cloudiness during the measurements. Information on cloud characteristics helps to adjust the optical model of the atmosphere used to interpret the measurements and to reduce the retrieval errors are. For the reconstruction of some geometrical characteristics of clouds a method was developed based on taking pictures of the sky by a pair of digital photo cameras and subsequent processing of the obtained sequence of stereo frames to obtain the height of the cloud base. Since the directions of the optical axes of the stereo cameras are not exactly known, a procedure of adjusting of obtained frames was developed which use photographs of the night starry sky. In the second step, the method of the morphological analysis of images is used to determine the relative shift of the coordinates of some fragment of cloud. The shift is used to estimate the searched cloud base height. The proposed method can be used for automatic processing of stereo data and getting the cloud base height. The report describes a mathematical model of stereophotography measurement, poses and solves the problem of adjusting of optical axes of the cameras, describes method of searching of cloud fragments at another frame by the morphological image analysis; the problem of estimating the cloud base height is formulated and solved. Theoretical investigation shows that for the stereo base of 60 m and shooting with a resolution of 1600x1200 pixels in field of view of 60° the errors do not exceed 10% for the cloud base height up to 4 km. Optimization of camera settings can farther improve the accuracy. Available for authors experimental setup with the stereo base of 17 m and a resolution of 640x480 pixels preliminary confirmed theoretical estimations of the accuracy in comparison with laser rangefinder.