Method and arrangement for outputting residual errors of a function fitted to a set of points

Abstract

Method for outputting residual errors (R) of a function (G) fitted to a set of points, wherein - Data points (P) of the point set are determined from measurement data by calculating a correlation of the measurement data and determining the data points (P) of the point set from support points of the correlation, - the adapted function (G) or the data points (P) of the point set are assigned section-wise or point-wise visual codes depending on the residual errors (R) and - the adapted function (G) is graphically output at an interface, wherein the adapted function (G) is represented section by section or point by point in the form of the associated visual codes, wherein a storage medium, a printer or a display (5) of a laser scanning microscope (2) or an external computer (3) is used as the interface for the output.

Description

[0001] The invention relates to a method and an arrangement for outputting residual errors of a function adapted to a set of points.

[0002] A point set consists of several discrete data points. Each of these points is assigned at least two coordinates from a corresponding multidimensional number space and a one- or multidimensional value. The points can directly represent measurement data or be indirectly derived from measurement data. A fitted function is an implicitly or explicitly defined mapping rule. It can be continuous or discontinuous and defined continuously or only at discrete points. Fitting a function to a point set can be achieved, for example, by varying one or more function parameters within the framework of a least squares calculation or by other function variations.

[0003] Various algorithms are known in the art for fitting a given or specified function to a set of points using least-squares adjustment, for example, the method of least squares. To assess the suitability of a fit in detail, the residual errors of the fitted function are typically used. These residual errors make it easy to identify local deviations in the fitted function. For this purpose, the residual errors are usually displayed graphically.

[0004] Methods for displaying residual errors of a fitted function are already known in the art. For example, Kolin et al. (Biophysical Journal Vol. 90 (2006), 628-639, 638) show graphs with discrete data points and one-dimensional functions fitted to the data points. Below each function graph, a second graph showing the corresponding residuals is displayed, with the graphs aligned along a coordinate axis. From DE 10327531 A1, it is known to represent fluorescence correlations and residual errors in the same graph.

[0005] Digman et al. (Biophysical Journal-Biophysical Letters BioFAST, 105.061788, L01-L03, L03; Biophysical Journal Vol. 89 (2005), 1317-1327, 1321) propose that, for raster image correlation spectroscopy, two-dimensional fitted functions be displayed perspectively in a pseudo-3D representation. The corresponding residuals are shown in a second graph, also in a perspective pseudo-3D representation, above the function graph. The two graphs are aligned with each other along two coordinate axes. The data points are not shown in the graph of the fitted function, as they cannot be interpreted in the perspective view. A color-coded representation of both graphs, depending on the respective vertical coordinate, is intended to increase the visibility of the perspective representation.

[0006] These familiar forms of data output have the disadvantage that it is difficult for a viewer to discern the quality of the function's fit to the data points. In both one-dimensional and two-dimensional fits, the viewer must perceive two graphs simultaneously to obtain information about the fit quality. In the two-dimensional case, the visual correlation of residual errors with the fitted function is particularly difficult and therefore inaccurate due to perspective.

[0007] Schreibmann et al. (Int. J. Radiation Oncology Biol. Phys., Vol. 64 (2006), No. 5, pp. 1537-1550) describe the color coding of residual defects between acquired and interpolated images for the representation of computed tomography data. A similar method for representing surface deviations is known from US 2004 / 0189944 A1. DE 19961631 A1 describes a method for assigning color values ​​to deviations.

[0008] The invention is therefore based on the objective of improving a method and an arrangement of the type mentioned at the outset, so that a visual assessment of the quality of the fit is possible simply and accurately based on a correlation of the measurement data.

[0009] The problem is solved by a method having the features specified in claim 1 and by an arrangement having the features specified in claim 9.

[0010] Advantageous embodiments of the invention are specified in the dependent claims.

[0011] According to the invention, it is provided that visual codes are assigned to the adapted function or to the points of the point set section by section or point by point depending on the residual errors and that the adapted function is graphically output at an interface, wherein the adapted function is represented section by section or point by point in the form of the assigned visual codes.

[0012] The assignment of visual codes can be done using a lookup table or a functional relationship. Ideally, in both cases, each deviation should correspond to exactly one visual code. With a limited set of values, a single visual code can be assigned to multiple deviations or a range of deviations. A functional relationship can, for example, be linear. Using a table has the advantage that arbitrary codes can be easily defined. This allows the output to be tailored to specific requirements. For example, the contrast can be increased to make smaller deviations clearly visible or decreased to suppress statistical noise more effectively. Using a functional relationship has the advantage of enabling a quantitative evaluation of the graphical representation when a legend explaining the code is also included.Mixed forms of assignment are also possible, whereby, for example, different functional relationships across value ranges can be stored in one table.

[0013] Visual codes can be assigned to sections or individual points of the customized function. Additionally or alternatively, visual codes can be assigned point-by-point to the data points within the point set. The output of the customized function is then modulated based on these visual codes. The output can be in vector graphics, raster graphics, or solid models.

[0014] This type of representation allows the viewer to identify the residuals in the graph of the fitted function and thus assess the quality of the fit. The viewer does not need to perceive two graphs simultaneously, but only a single graph. This also has the advantage that the graphical representation requires less space, because the separate residual graph can be omitted. Alternatively, the graph of the fitted function can be displayed larger so that it is easier to see.

[0015] Preferably, optical densities and / or color intensities and / or hues and / or color saturations are used as visual codes. Such visual codes are easy to grasp and allow the visualization of a wide range of values ​​for high reading accuracy. When using optical densities, color coding can be applied to a multidimensional, adapted function depending on one of the coordinates.

[0016] In particularly preferred embodiments, a first color tone is assigned to visual codes for positive residual defects and a second color tone to visual codes for negative residual defects. This makes it easy to visually distinguish areas of the adapted function with positive residual defects from those with negative residual defects.

[0017] Advantageously, the intensity or saturation of the respective visual code is determined depending on the magnitude of the residual error. In this way, the degree of deviation of the fit from the data points can be easily visualized.

[0018] The invention is particularly advantageous in the case of two- or three-dimensional adapted functions, because a perspective assignment of the residual errors from a separate graph is not necessary due to the representation of the visual codes within the function itself.

[0019] The points of the point set are determined from measurement data by calculating a correlation of the measurement data and determining the points of the point set from the support points of the correlation. Particularly preferred are configurations in which the measurement data are acquired using fluorescence scanning spectroscopy (FCS) and / or raster image correlation spectroscopy (RICS). Advantageously, the measurement data are acquired using a laser scanning microscope (LSM). An interface of a laser scanning microscope can also advantageously be used for output. For output, a storage medium, a printer, or a display of a laser scanning microscope or an external computer is used as the interface.

[0020] In the case of one-dimensional FCS measurement series, especially time series, the correlation functions of the individual measurements can be displayed in a single graph in pseudo-3D representation. This allows the temporal progression of data to be visualized.

[0021] The invention also includes, in particular, an arrangement and a computer program for carrying out the various embodiments of the method.

[0022] The invention will now be explained in more detail using exemplary embodiments.

[0023] The drawings show: Fig. 1. A function graph and a residue graph with only statistical errors, Fig. 2. A function graph and a residue graph with a systematic error of 10%. Fig. 3. A function graph and a residue graph with a systematic error of 30%. Fig. 4. A flowchart of the process flow and Fig. 5 an order to carry out the procedure.

[0024] The Fig. Figures 1A to 3A each show a graph of a function G(x,y) in a perspective pseudo-3D grid representation. The respective function G(x,y) is fitted to an identical set of data points (not shown) for all figures using a least-squares adjustment, exemplified by the method of least squares. The data points originate from an autocorrelation analysis of an LSM-RICS measurement simulation superimposed with artificial noise. In the following, the... Fig. 2 and Fig. In addition to the three adjustments shown, a systematic error of 10% or 30% is artificially generated.

[0025] The residual errors R(x,y) of the fitted function G(x,y), i.e., the differences between the respective function values ​​G(x,y) and the data points contained in the point set, are in the Fig. 1B to 3B, as is customary in the prior art, are also shown as independent graphs in perspective.

[0026] Also in the Fig. Figures 1A to 3A represent the residual errors R(x,y) of the fitted function G(x,y), advantageously integrated into the function graph by means of a visual coding of the grid faces of the fitted function G(x,y). Various black-and-white patterns with different optical densities are used as visual codes. The optical densities are proportional to the magnitude of the respective residual error for each value range. Residual errors with a positive sign, for example in grid facet F1, are encoded with a low optical density with a predominant white component. Residual errors with a negative sign, for example in grid facet F2, are encoded with a high optical density with a predominant black component. Residual errors close to zero, for example in grid facet F3, are encoded with a medium optical density with approximately equal black and white components.By viewing the visually encoded fit in this way, the quality of the fit can be assessed immediately. This contrasts with the conventional separate representation of the residual errors R(x,y) (. Fig. 2B) There are already small systematic deviations in the visually coded graph of the fitted function G(x,y) ( Fig. 2A) easily recognizable. In an alternative embodiment (not shown), the signs of the residual errors can be encoded by two different black-and-white basic patterns, for example, hatching. The first pattern then represents positive residual errors, the second pattern represents negative residual errors. The magnitudes of the residual errors can, in turn, be encoded by different optical densities. For this partial encoding and assignment, either a discrete table, a functional relationship, or a hybrid of these can be used.

[0027] According to the invention, the residual error graphs can be omitted from the output because their information content is contained in the adapted functions. This frees up more space for the output of the function graphs.

[0028] In other configurations (not shown), grayscale or color gradations can be used for visual coding. Visual coding with color tones is particularly advantageous because it makes areas with positive or negative residual errors easy to identify. For example, a red hue can be assigned to the fitted function G(x,y) as the visual code for positive residual errors and a blue hue as the visual code for negative residual errors. Beyond these primary colors, the assignment of visual codes can be further refined by assigning a color saturation or intensity depending on the magnitude of the respective residual errors R(x,y). This allows the magnitudes of deviations across entire regions of the fitted function G(x,y) to be easily observed.

[0029] The invention can also be used with adapted three-dimensional functions. The output can then be, for example, in pseudo-3D representation or in true 3D representation, for example holographically or stereoscopically, in particular as a color-coded point cloud.

[0030] In Fig. Figure 4 schematically illustrates the process according to the invention in the form of a flowchart. The starting point is a set of discrete data points P(x). i ,y i ) from an LSM measurement in a memory table as well as a parameterized function G(x,y). First, the parameters of the function G(x,y) are assigned to the point set P(x) using the method of least squares. i ,y i ) adjusted. For each point P(x i ,y i ) of the set of points stored in a memory table, a residual error R(x) is calculated by taking the difference R=GP. i ,y i) determined. For each point P(x i ,y i Visual codes are assigned piecewise to the adapted function G(x,y). The sections extend around the support points (x i ,y i ) and correspond, for example, to a single grid facet of a pseudo-3D grid. In this example, each visual code consists of a hue and saturation depending on the sign and magnitude of the residual errors. The function G(x,y) is then displayed as an example on a monitor of an LSM control computer. The function G(x,y) is represented as a color-coded grid surface in the form of the visual codes. The grid lines can be displayed in black or also colored using the visual codes. Alternatively or additionally, the visually coded, customized function can be output to a file, a clipboard on a storage medium, or a printer.

[0031] In an alternative implementation (not shown), the visual codes are not assigned to the adapted function section by section, but only point by point. The output can still be modulated section by section within the visual codes. Each section then symmetrically surrounds, for example, one of the support points (x). i ,y i Alternatively, the visual codes can also be assigned to the data points P(x). i ,y i ) occur, for example, when the support points (x k ,y k ) the adapted function G(x,y) does not match the data points (x i ,y i ) agree.

[0032] In another alternative configuration (not shown), the output in the form of visual codes can be limited to individual points of the function. In particular, these can be points whose coordinates correspond to those of the support points (x). i ,y i ) of the data points P(x i ,yi ) are equivalent to.

[0033] Fig. Figure 5 shows a block diagram of an arrangement 1 with a laser scanning microscope 2 connected to a control unit 3. The control unit 3 is equipped with an interface 4 for a display 5. The control unit 3 is programmed to acquire RICS measurement data using the LSM 2, to calculate correlation functions, and to perform the procedure described above. After a triggering operation, it outputs the adapted, visually coded function G(x,y) on the display 5. Reference symbol list 1. Arrangement 2 Laser scanning microscope 3 Control unit 4 Interface 5 ads G(x,y) Fitted Function R(x i ,y i Residual errors (residuals) F 1,2,3 Lattice facets

Claims

[1] Method for outputting residual errors (R) of a function (G) fitted to a set of points, wherein - Data points (P) of the point set are determined from measurement data by calculating a correlation of the measurement data and determining the data points (P) of the point set from support points of the correlation, - the adapted function (G) or the data points (P) of the point set are assigned section-wise or point-wise visual codes depending on the residual errors (R) and - the adapted function (G) is graphically output at an interface, wherein the adapted function (G) is represented section by section or point by point in the form of the associated visual codes, wherein a storage medium, a printer or a display (5) of a laser scanning microscope (2) or an external computer (3) is used as the interface for the output. [2] Method according to claim 1, wherein optical densities and / or color intensities and / or hues and / or color saturations are used as visual codes. [3] Method according to claim 2, wherein a first hue is assigned to visual codes for positive residual defects (R) and a second hue is assigned to visual codes for negative residual defects (R). [4] Method according to claim 2 or 3 wherein an intensity or a saturation of the respective visual code is determined as a function of the amount of the residual error (R) in question. [5] Method according to any of the preceding claims, wherein a two- or three-dimensional adapted function (G) is used. [6] Method according to one of the preceding claims, wherein the measurement data are determined by means of fluorescence scanning spectroscopy and / or by means of raster image correlation spectroscopy. [7] Method according to one of the preceding claims, wherein the measurement data are determined using a laser scanning microscope (2). [8] Computer program configured to carry out a method according to any one of claims 1 to 7. [9] Arrangement (1) for outputting residual errors (R) of a function (G) adapted to a set of points, comprising a control unit (3) and an output interface (4), in particular for carrying out a method according to one of claims 1 to 7, characterized by , that the control unit (3) - Data points (P) of the point set are determined from measurement data by calculating a correlation of the measurement data and determining the data points (P) of the point set from support points of the correlation, - assigns section-wise or point-wise visual codes to the adapted function (G) or the data points (P) of the point set depending on the residual errors (R) and - graphically outputs the adapted function (G) at an interface (4), representing the adapted function (G) section by section or point by point in the form of the associated visual codes. [10] Arrangement (1) according to claim 9, characterized by that the visual codes are optical densities and / or color intensities and / or hues and / or color saturations. [11] Arrangement (1) according to claim 10, characterized by , that the control unit (3) assigns a first hue to visual codes for positive residual errors (R) and a second hue to visual codes for negative residual errors (R). [12] Arrangement (1) according to claim 10 or 11, characterized by , that the control unit (3) determines an intensity or a saturation of a respective visual code depending on the amount of the relevant residual error (R). [13] Arrangement (1) according to any one of claims 9 to 12, wherein the adapted function (G) is two- or three-dimensional. [14] Arrangement (1) according to any one of claims 9 to 13, characterized by , that the control unit (3) determines the measurement data by means of fluorescence scanning spectroscopy and / or by means of raster image correlation spectroscopy. [15] Arrangement (1) according to any one of claims 9 to 14, characterized by , that the arrangement (1) is a laser scanning microscope (2) or a control computer (3) for a laser scanning microscope (2). [16] Arrangement (1) according to any one of claims 9 to 15, wherein the interface (4) is a storage medium, a printer or a display (5) of a laser scanning microscope (2) or an external computer (3).

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