// Compute 2D Polynomial Fit function. This function fits an polynomial to map the relationship between // two sets of 2D data. // D. R. G. Mitchell, adminnospam@dmscripting.com (remove the nospam to make this address work) // version:20190905, v1.0, September 2019, www.dmscripting.com // The function takes as the following arguments // Input data - an array of data points in the following format: // [x1, y1, x2, y2 . . xn, yn // xn+1, yn+1 etc. // . . . . . . . . .xfinal, yfinal] // These are the data being input into the system. In the example here, they are a series of coordinates from a STEM // scan - being symmetric about the zero position. // Ouputdata - this is an array of similar format to that above. It contains the corresponding outputs from the system. // In this case the positions of the STEM (000) beam produced by the aforementioned STEM scan. Due to lens aberrations // there is not a linear correlation between the two - hence the need for a polynomial fit. // Inputdata and Outputdata must have the same dimensions. // This function fits a polynomial to map the relationship between the two data sets. // i'= a0+a1*i+a2*j+a3*i^2+a4*i*j+a5*j^2 // j'= b0+b1*i+b2*j+b3*i^2+b4*i*j+b5*j^2 // Here i' and j' are the x and y coordinates respectively of the Outputdata, i and j are the are the x and y // coordinates of the Inputdata and a0 - a5, and b0 - b5 are the polynomial coefficients of the fit. Higher order // polynomial fits can produce erratic results with noisy data and here you will see the agreement is OK near the // the centre of the plots, but in the corners, where the aberrations are most strongly felt, the deviations are significant. // This type of analysis might be useful for calibrations, where you need to understand/predict how an input signal // will affect an output signal. // The function uses a Pseudo-inverse matrix method to compute the polynomial coefficients producing the best // fit to the data. These coefficients are returned (by reference) to the two passed in images acoefficients and // bcoefficients. These images are one pixel wide and 6 tall, and contain the a0 - a5 and b0 - b5 values,, top to bottom. // The function returns 1 if the calculation was successful and 0 if a problem was encountered. // Use the returned coefficient values in the above expression to calculate Outputdata from input data. // Function definition number Fit2DPolynomialFunction(image inputdata, image outputdata, image &acoefficients, image &bcoefficients) { // Get the size of the input and output images passed in and make sure they are the same size number inxsize, inysize, outxsize, outysize getsize(inputdata, inxsize, inysize) getsize(outputdata, outxsize, outysize) if(inxsize!=outxsize && inysize!=outysize) { showalert("inputdata and outputdata images are not the same size.",2) return 0 } number nopoints=(inxsize/2)*inysize // Note half of inputxsize, due to x1,y1, x2, y2 . . // Place the input and output data into a single data image for convenience. There are four columns // inputx, inputy, outputx, outputy and these run vertically, first point to last image alldataincolumns=realimage("", 4, 4, nopoints) number x, y, counter=0 for(y=0; y