Cubic spline in excel

Spline fitting or spline interpolation is a way to draw a smooth curve through n+1 points (x0, y0), …, (xn,yn). Thus, we seek a smooth function f(x) so that f(xi) = yi for all i. In particular, we seek n cubic polynomials p0, …, pn-1 so that f(x) = pi(x) for all x in the interval [xi, xi+1].Key PropertyProperty 1: The cubic polynomials that we are seeking can be defined bypi(x) = ai(x–xi)3 + bi(x–xi)2 + ci(x–xi) + diwhere for i = 0, …, n-1based on hi = xi+1 – xi and ki = yi+1 – yi. The bi coefficients are defined via matrix operations as follows. First define for i = 1, …, nNow define U = [uij] to be an n+1 × n+1 matrix (where i, j = 0, …, n) whereand define B = [bi] and V = [vi] to be n+1 × 1 matrices where the vi are as defined above and B is defined byB = U-1VProofThe proof uses calculus.We define n cubic polynomials p0, …, pn-1 with the following properties:pi(x) = ai(x–xi)3 + bi(x–xi)2 + ci(x–xi) + di for all i = 0, …, n–1if xi-1 ≤ x ≤ xi then f(x) = pi(x) for all i = 0, …, n–1 (f defined piecewise by the pi)f(xi) = yi for all i = 0, …, n (points (xi, yi) lie on the y = f(x) curve)pi-1 (xi) = pi (xi) for all i = 0, …, n–1 (f is continuous)p′i-1 (xi) = p′i (xi) for all i = 0, …, n–1 (f′ exists and is continuous)p′′i-1 (xi) = p′′i (xi) for all i = 0, …, n–1 (f′′ exists and is continuous)p′′0 (x0) = 0 and p′′n-1 (xn) = 0 (initial conditions)We also define hi = xi+1 – xi and ki = yi+1 – yi for i = 0, …, n–1.By 1, 2, and 3, for i = 0, …, n–1and so by 1, 3, and 4, for i = 0, …, n–1or equivalentlyBy 1and soIt now follows by 6 thatand so for i = 0, …, n–1It now follows thatSolving for ci it follows thatAs we have seen previouslyand soIt now follows by 5 thator equivalently for i = 0, …, n–1This can be re-expressed asWe now have n+1 linear equations in n+1 unknowns as follows for i = 1, …, n–1Dividing both sides of this last equation by hi-1 + hi, we obtain the equivalent formwhere ri, si, and vi are defined as in the statement of the property. With B, U, and V defined as in the statement of the property, the n+1 linear equations can be re-expressed asUB = Vfrom which it follows that B = U-1V, which completes the proof.ReferencesChen, M-Q (2013) Cubic spline interpolationThis paper has been removed from the InternetJameson, A. (2019) Cubic splines K. (2013) A study of cubic spline interpolation

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Pixel values histograms are presented for the same aligned movie from the EMPIAR entry 10,288: (A) Cryosparc; (B) FlexAlign; (C) Motioncor2; (D) Relion MotionCor; (E) Warp. CV stands for coefficients of variation (standard deviation as a percentage of the mean). For representation, we removed the outliers by means of the interquartile range (IQR of 80%) method. Figure 9. Scalability of parallel GPU processing. The plots represent the mean processing time in seconds (y-axis) required to process a single movie on one GPU. The x-axis represents the number of GPUs, which is increased in parallel with the number of movies to process. This figure demonstrates the scalability of the algorithm with GPU parallel processing. The scalability analysis was performed on three different movie sizes commonly encountered in single-particle analysis (SPA) experiments. These movie sizes are as follows: (A) corresponds to the 4096 × 4096 × 70 experiment, which represents a movie size typically observed in lower-resolution SPA experiments; (B) represents the 7676 × 7420 × 70 experiment; (C) corresponds to the 11,520 × 8184 × 70 experiment, which is a movie size commonly used in super-resolution acquisitions. To ensure reliable results, each algorithm was executed 10 times per movie size, thereby avoiding unstable runs and obtaining more accurate measurements. Figure 9. Scalability of parallel GPU processing. The plots represent the mean processing time in seconds (y-axis) required to process a single movie on one GPU. The x-axis represents the number of GPUs, which is increased in parallel with the number of movies to process. This figure demonstrates the scalability of the algorithm with GPU parallel processing. The scalability analysis was performed on three different movie sizes commonly encountered in single-particle analysis (SPA) experiments. These movie sizes are as follows: (A) corresponds to the 4096 × 4096 × 70 experiment, which represents a movie size typically observed in lower-resolution SPA experiments; (B) represents the 7676 × 7420 × 70 experiment; (C) corresponds to the 11,520 × 8184 × 70 experiment, which is a movie size commonly used in super-resolution acquisitions. To ensure reliable results, each algorithm was executed 10 times per movie size, thereby avoiding unstable runs and obtaining more accurate measurements. Table 1. Comparison of various movie alignment programs. Table 1. Comparison of various movie alignment programs. ProgramHWMethod + InterpolationCryoSPARCGPUProprietary codeFlexAlignGPUCC + cubic B-spline in space and timeMotionCor2GPUCC + quadratic (space), cubic (time) polynomialsRelion MotionCorCPUCC + quadratic (space), cubic (time) polynomialsWarpGPUCC + cubic B-spline in space and time Table 2. CTF Resolution limit (Å) comparison. The following table presents the means and standard deviations of the CTF criteria for the CTF estimation using two different methods, Gctf and Xmipp. The data in the table correspond to 30 image samples per EMPIAR entry, divided into three datasets: 10,196, 10,288, and 10,314. Table 2. CTF Resolution limit (Å) comparison. The following table presents the means and standard deviations of the CTF criteria for the CTF estimation using two different methods, Gctf and Xmipp. The data in the table correspond to 30 image samples

Various corners.tap on where you'd like to drop control points.It's a lot less intuitive to use than spline mode Cubic Bezier. It's like a combination of spline mode and Quadratic bezier.click tap, then hold where you'd like the curve to go along.while holding you can control how the curve behaves.This allows you to sharp corners in your curves. If you only tap instead of holding, you get a sharp corner. Bezier Curve It's basically the Continous curve, but on a separate preset tool. The Figure Tools - The Rectangle The Rectangle is used by tapping and dragging to create the rectangle and then lifting pen where you want to other corner to sit.The rectangle is a member of the figure subtool category, but it only allows you to draw rectangles, similarly the Ellipse tool is also a variant of the figure subtool that only allows to draw rounded shapes.Like continuous curves, figures can fill with or without outline on Raster layers IF you check Adjust Angle after Fixed, it will allow you to rotate your shape along the center after your initial tap, drag, and pen lift. Just drag across the canvas to rotate. This is also true for all figures. You can also opt to draw out rounded corners or not. as well as specify how rounded you'd like the corners to be.All the figure tools allow you to change the brush shapes, like the continuous curves and unit curves.I'll discuss aspect types settings when I discuss the shift