Digitizing and approximating graphs of functions using Wolfram Mathematica and Graph Digitizer

The task of digitizing graphs of functions and curves has to face almost every engineer and student. The traditional "manual" method is very inconvenient and also introduces large errors in the data. For a one-time task, this method is not so bad, but if there are more than one graphs and each one contains not one curve, but a family of curves?

In the process of carrying out laboratory practical work in physics, I often have the task of determining the value of a function from its graph presented on paper, for further calculations. Since the processing of such graphs on a computer significantly increases the speed and accuracy of this process, it was decided to explore the possibilities for digitizing the graph and building a mathematical model of the curve presented on the graph.



As an example, I took a graph of the efficiency of the generator from its power from a laboratory workshop on electrical engineering. During the execution of the work, I performed a scan of the graph, image processing of the graph, digitization of coordinates and the construction of a mathematical model of the curve.
')

1. Image preparation

After scanning, the first thing to do is to bring the resulting image to full contrast and align one of the axes of the graph. Next, you need to increase the sharpness and resize the image. If the size and resolution are too large, difficulties arise at the subsequent stages of work.

Image processing, I recommend the program Adobe Photoshop. Using the Curves tool, we achieve a full-fledged contrast, then use the Smart Sharpen filter to increase sharpness. The undoubted advantage of Photoshop is the ability to process a large number of images by recording the action (Action) and applying it in conjunction with batch processing (File - batch processing).

To speed up the process, the processing can be performed in the scanning program using pre-prepared presets or automatic algorithms.


Figure 1.1 - Graphic image Before processing and After processing

2. Digitization of coordinates

To digitize coordinates, I used the shareware program GetData Graph Digitize version 2.26. After starting the program, open our processed image "File - Open Image". After opening, we will see a standard workspace.


Figure 2.1 - Standard Graph Digitize Interface

2.1. Setting the coordinate system (SC)

The first thing we need to do is to establish a coordinate system, i.e. mark the lines of the axes. To do this, go to "Commands - Set the coordinate system." Further, holding the LMB, we find the point of origin and click on it. In the window that appears, enter the value of the origin (Xmin). Next, set the values ​​Xmax, Ymin and Ymax in the same way. For convenient installation of points it is necessary to open the “View - Magnifier” magnifying glass window. After the control points are set, the axis lines appear and the "Coordinate System Parameters" window opens in which you can reassign the values ​​of the control points and set the logarithmic axis scale.

For visual quality control of the SC installation, you can display the grid with the specified step “View - Show grid”. In case of correct installation of the IC, the grid lines should be strictly parallel to the lines in the image of the graph. It should be noted that when scanning turns, the graph often turns out to be in the bend area, and one of the axes turns out to be bent. In this case, it is not possible to set the CS correctly, therefore at the scanning stage it is necessary to press the turn to the glass more tightly.


Figure 2.2 - View with the established coordinate system and grid

2.3. Curve digitization

Let's start installing points on the chart. To do this, go to the point setting mode (Ctrl + P). In this mode, click LMB to set a new point. To display the table of coordinates of the selected points, go to the “View - Information Window”. To delete points, use the data point eraser “Commands - Data Eraser” (Ctrl + E)

In my experience, a greater number of points must be installed in the vicinity of the inflection points of the curve; on the linear sections of the curve, it can be limited to a small number of them.

If there is more than 1 curve or a family of curves on the graph, then after setting the points on the first one, you need to add a new line “Commands - Add Line”. Then it will be possible to set points on the second curve, etc.

If there is no grid on the graph image, then you can use the automatic curve trace algorithm (Ctrl + T). In the presence of a grid, the algorithm gives a lot of errors.


Figure 2.3 - View with set points on the curve

2.4. Data export

For further processing of the data, it is necessary to export the coordinates of the points to a .txt file or to the clipboard (conveniently if we have only one curve). In the GetData Graph Digitize program, export to .txt is performed by calling the “File - Data Export” command (Ctrl + Alt + E). After clicking in the window that opens, you are prompted to set the save path and file name.


In the “Settings - Parameters” menu the output format is set. You can also enable sorting points by the value of the X coordinate, if a unique Y exists on your curve for each X, to eliminate random errors in the sequence of setting points.


Figure 2.4 - Export Settings

3. Construction of a mathematical model of the curve

In the final, we will perform an approximation of the obtained data and verify the correctness of the obtained mathematical model. For this, I propose to use the computer algebra system Wolfram Mathematica.

To quickly import data into Wolfram Mathematica, copy the coordinates of points from the exported file and paste it into an empty Excel cell. As a result, 2 columns of data X and Y will appear on the sheet, respectively.


Figure 3.1 - Data in Excel

The next step is to create a new Wolfram Mathematica document and drag the Excel file into it. The result is a list of lists containing the coordinates of the points. Give it the variable data.


Figure 3.2 - Imported data in Wolfram Mathematica

Display the imported data using the ListPlot [] function.


Figure 3.3 - Graphic display of points in the form of a scatter diagram

Approximate the points by a polynomial of degree 5. For this we use the function LinearModelFit []. As a result, we get an object of class FittedModel []. Give it the fit variable.

We calculate the coefficient of determination R ^ 2, which shows how much of the variation (spread) of the variable explains the resulting equation. The closer this coefficient is to unity, the greater the proportion of variation that explains the equation. To do this, we specify “RSquared” as the argument of the fit function. In this case, R ^ 2 = 0.99, which means that our model explains 99.9% of the variation of the variable.

To calculate the Y value, it is necessary to specify the required X value as an argument to the fit function.


Figure 3.4 - Approximation of points, calculation of the coefficient of determination and calculation of the function value

In addition to calculating the coefficient of determination, we will conduct a regression analysis. This time, the argument of the fit function is “ANOVATable”. According to the obtained result, it can be argued that the use of each term of the approximating polynomial is justified. Let's display the resulting equation explicitly, for this we apply the function Normal [] to the fit variable.


Figure 3.5 - Regression analysis and the polynom explicitly

Next, we construct a graph of the polynomial and display the initial points on it. Using the standard syntax, set the style of the chart and add labels to the axes and the name of the chart.


Figure 3.6 - Final Schedule


Figure 3.7 - Comparison of the final schedule with the original data

The possibilities for analyzing the mathematical model in Wolfram Mathematica are truly enormous, but we will limit ourselves to those presented above. Interested parties can learn more by calculating the fit ["Properties"] function.

Conclusion:

As a result, we studied the possibility of using Wolfram Mathematica and Graph Digitizer for digitizing graphs and selecting a mathematical model of the curve. The software used allows you to complete the task with minimal effort and with high quality.

PS: In the comments you can briefly tell how often you have to deal with a similar task in your activity? I will be glad to answer questions.