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Generalised cross-correlation functions for engineering applications. Application to experimental data
Abstract
A recent generalisation of cross-correlation (GC-C) makes it possible to model transformations between data such as pairs
of images by sets of parameterised functions as opposed to constant shifts, rotations etc. as employed in conventional cross-correlation.
Typical applications of GC-C are in areas such as particle image velocimetry (PIV), or two-dimensional or three-dimensional
surface strain field determinations. As the flow, strain etc. descriptions developed by GC-C are global or zonal, the parameters
required are estimated using all or a large fraction of the information in the images typically used to provide the basic
data in such techniques. This is in complete contrast to traditional cross-correlation methods used in PIV, where the image
domains are segmented into small sub-regions and a constant shift, rotation etc. is determined separately in each local cell.
Such local cellular methods inevitably introduce a compromise between spatial resolution and the statistical confidence that
can be placed in the estimates of the shifts, rotations etc. GC-C removes the need for such compromises. This paper examines
the application of the small perturbation form of GC-C to real experimental data sets with special emphasis on showing the
effects of the analytical approximations employed in the perturbation scheme. In particular, the key issue of the effects
of the bandwidth of the images used are explored and a very simple procedure is described for checking that optimal results
are being obtained.
... The shape of the interest and the shape of the search areas can assume many types [20]. Depending of the fluid flow, a specific shape can be more suitable because it can decrease the uncertainty in the measured velocity vector. ...
The fluid dynamics presents a complex behavior and in certain regimes it has not been theoretically understood by the physics laws yet. For this reason, the experimental techniques of flow visualization have been used since the 70's with the first methods introduced in laser applications like LDV (Laser Doppler Velocimetry). In these last ten years, due to the digital CCD camera improvements, the development of frame grabbers and microcomputers with better processing performance and large memory, image processing techniques have been applied to investigate fluid dynamics. However, the scientific works in this field of velocimetry through image processing have been always focused in the physical basis behind the measurement, in the improvements or developments of specific algorithms, and in the use of different equipments for instrumentation, not considering the engineering aspects involved in the methodology for the conception and the design of the entire Velocimetry System. In this way, this work presents the design and the implementation of an image processing algorithm for the measurement of the two-dimensional velocity field of fluid flows. This image processing algorithm is a part of a whole Velocimetry System developed by us and that has been successfully applied to different physical systems as our recent publications shown.
S t r e s z c z e n i e . W artykule przedstawiono wyniki badań doświadczalnych zmian zachodzących w materiale sypkim podczas opróŜniania silosu. Pomiarów dokonano dla przyściennej warstwy piasku przemieszczającej się w silosie z przepływem masowym oraz kominowym. Doświadczenia prze-prowadzono dla zróŜnicowanego zagęszczenia początkowego materiału sypkiego oraz zmiennej szorstkości ścian. Zwrócono uwagę na moŜliwość obserwacji lokalizacji odkształceń powstających w materiale sypkim w czasie opróŜniania silosu. S ło wa k lu c zo we : P I V , silos, korelacja, materiał sypki, zagęszczenie WSTĘP Popularną techniką, szczególnie przydatną dla badań nad przepływem materiałów w stanie ciekłym lub przepływu materiałów sypkich prowadzoną z wykorzystaniem kamery CCD jest tzw. metoda laserowego noŜa świetlnego (Barker i Fourney 1976, Adrian 1991). W technice tej pomiar dokonywany jest najczęściej przez oświetlenie strumienia materiału wąskim pasmem światła lasera i rejestrację rozproszonego bądź absorbowanego światła przez aparat czy kamerę cyfrową. Światło lasera jest rozpraszane przez cząstki przepływającego medium, co pozwala na zarejestrowanie obrazów rozkładu natęŜenia światła rozproszonego przez poruszający się materiał. Technika ta wykorzystywana jest do generowania obrazów, których analiza pozwala obliczyć między innymi pole rozkładu prędkość przepływu. Metoda ta nazywana jest anemometrią obrazową i znana pod techniczna nazwą PIV (Particle Image Velocimetry) (Kadambi i in. 2001, Scarano i Riethmuller 1999).
Traditional cross-correlation considers situations where two functions or data sets are linked by a constant shift either in time or space. Correlation provides estimates of such shifts even in the presence of considerable noise corruption. This makes the technique valuable in applications like sonar, displacement or velocity determination and pattern recognition. When regions are decomposed into patches in applications such as Particle Image Velocimetry it also allows estimates to be made of whole displacement/flow fields. The fundamental problem with traditional correlation is that patch size and hence statistical reliability must be compromised with resolution. This article develops a natural generalization of cross-correlation which removes the need for such compromises by replacing the constant shift with a function of time or space. This permits correlation to be applied globally to a whole domain retaining any long-range coherences present and dramatically improves statistical reliability by using all the data present in the domain for each estimate.
Part I of this article, ibid. 64, No. 2, 321-326 (1997; Zbl 0890.62045), introduced a generalization of cross-correlation in which the constant shift used in traditional cross-correlation is replaced by a function of time or space. This allows correlation to be applied globally to the whole domain of interest avoiding the need to compromise spatial resolution with statistical reliability. The development in Part I was entirely in terms of continuous variables. This article examines the issues that arise when generalized cross-correlation is applied to discrete variable situations. Topics discussed include sampling rate requirements, noise rejection, and efficient approximate algorithms, with special attention being paid to the condition number for the system.
An objective technique is developed to calculate advective surface velocities from sequential images. A pattern matching method is used with the identification of maximum cross correlation (MCC) between a template window in the first image and search areas in the second image. We examined the limitations of the MCC method in two cases: (1) eddy size (LE) much smaller than the radius of deformation (Lp), and (2) LE>=Ls. In the first case an eddy is regarded as a particle. Maximum detectable time period is estimated to be about 1 day. For the second case we developed the MCC method to detect a rotational motion. We compared velocity fields drived from the method with a real velocity field in a numerical analysis of tracer and quasi-geostrophic eddy fields. The optimal template size is about the eddy diameter which spans between the maximum velocity points. We also applied the MCC method to infrared images in the Kuroshio-Oyashio confluence zone.
This paper is concerned with the implementation of a new image processing technique called Generalised Cross-Correlation designed to detect a two-dimensional displacement field between consecutive pictures of some object. The particular problem considered here is the determination of a flow field from experimental tank data. The method consists of maximizing a Generalised Cross-Correlation (GC-C) function in order to compute the parameters in a global mathematical description of the displacement field. This is in complete contrast to traditional approaches such Cellular Correlation, where an array of constant vectors are estimated using spatially local data. The focus here is on the case where the displacements are too large to employ perturbation techniques and thus numerical optimization is required to calculate the parameters in the flow/ displacement model used. A combination of a Genetic Algorithm (GA) and a Hillclimbing Method (HC) is applied to recover the global maximum of the GC-C function in the presence of many large local maxima. The paper also briefly reviews the basic background of GC-C and develops those GA issues relevant for the implementation of the method.
The accurate description of material response consists of a material constitutive formulation and material parameter values. The reduction of material test specimen load versus displacement data to constitutive parameters is often precluded by inelastic material response and deformation inhomogeneity within the specimen. For ductile engineering alloys, these effects are influenced by specimen geometry and must be uncoupled from specimen geometry to characterize the large strain material response. The accuracy of material parameters at such strains should be demonstrated for subsequent applications to design and analysis. Iterative solution for material constitutive parameters is discussed in the investigation. The use of video processing of laboratory tensile test specimens is combined with successive computational simulations of the specimen responses. The solution for HSLA-80 steel constitutive parameters, in the context of incremental plasticity theory, is presented. The material response is treated as the unknown in the computational simulations. It is iteratively modified to achieve correlation with the laboratory experiments. Two different specimen length-to-diameter aspect ratios are utilized to ensure the geometry independence of the material solution and to facilitate efficient solution. The constitutive iteration sequence illustrates the sensitivity of specimen response to material nonlinearity.
A newly developed correlation interpolation method to measure the regional velocity of moving tissue is evaluated in an experimental setup. Pulsed ultrasound echo signals (center frequency 3.5 MHz) are received from a rotating Agar disk containing scattering particles. When averaging over a depth range of 2.2 mm at a pulse repetition frequency (PRF) of 930 Hz, the standard deviation of the measured displacement between 2 successive pulses was found to be +/- 6 microns. In a second series of experiments, the angular velocity of the disk is estimated from the displacement, as measured simultaneously in two different regions located on separate echo lines (PRF = 465 Hz per line). The exact position of both regions in respect to the center of rotation was found to be irrelevant. The accuracy of the calculated angular velocity was found to be better for large angles between the two lines of observation than for small angles.
We have previously reported initial clinical results of a novel blood velocity imaging technique utilizing a two-dimensional correlation search applied to consecutively acquired echoes. In this paper, we describe both the physical principles underlying this technique and test tank experiments which define its performance under a variety of conditions. The results indicate that, unlike Doppler flow imaging systems, this technique defines the flow velocity vector in two dimensions and is not subject to aliasing.