package gdsc.smlm.function.gaussian;
/*-----------------------------------------------------------------------------
* GDSC SMLM Software
*
* Copyright (C) 2017 Alex Herbert
* Genome Damage and Stability Centre
* University of Sussex, UK
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*---------------------------------------------------------------------------*/
/**
* Implements a astigmatism model of a 2D Gaussian function, where z-depth determines the x and y width.
* <p>
* Ref: Smith et al, (2010). Fast, single-molecule localisation that achieves theoretically minimum uncertainty.
* Nature Methods 7, 373-375 (supplementary note).
* <p>
* Ref: Holtzer, L., Meckel, T. & Schmidt, T. Nanometric three-dimensional tracking of individual quantum dots in cells.
* Applied Physics Letters 90, 1–3 (2007).
*/
public class HoltzerAstimatismZModel implements AstimatismZModel
{
public final double gamma, one_d2, Ax, Bx, Ay, By;
/**
* Static constructor.
*
* @param gamma
* the gamma parameter (half the distance between the focal planes)
* @param d
* the depth of focus
* @param Ax
* Empirical constant A for the x-astigmatism of the PSF
* @param Bx
* Empirical constant B for the x-astigmatism of the PSF
* @param Ay
* Empirical constant A for the y-astigmatism of the PSF
* @param By
* Empirical constant B for the y-astigmatism of the PSF
* @return the holtzer astimatism Z model
*/
public static HoltzerAstimatismZModel create(double gamma, double d, double Ax, double Bx, double Ay, double By)
{
final double d2 = d * d;
return new HoltzerAstimatismZModel(gamma, 1.0 / d2, Ax, Bx, Ay, By);
}
/**
* Constructor.
*
* @param gamma
* the gamma parameter (half the distance between the focal planes)
* @param one_d2
* one over the depth of focus squared (1/d^2)
* @param Ax
* Empirical constant A for the x-astigmatism of the PSF
* @param Bx
* Empirical constant B for the x-astigmatism of the PSF
* @param Ay
* Empirical constant A for the y-astigmatism of the PSF
* @param By
* Empirical constant B for the y-astigmatism of the PSF
*/
public HoltzerAstimatismZModel(double gamma, double one_d2, double Ax, double Bx, double Ay, double By)
{
this.gamma = gamma;
this.one_d2 = one_d2;
this.Ax = Ax;
this.Bx = Bx;
this.Ay = Ay;
this.By = By;
}
/**
* Gets the standard deviation, first and second derivatives for the z-depth.
*
* @param z
* the z
* @param one_d2
* one over the depth of focus squared (1/d^2)
* @param A
* Empirical constant A for the astigmatism of the PSF
* @param B
* Empirical constant B for the astigmatism of the PSF
* @param ds_dz
* the first and second derivative of s given z
* @return the standard deviation
*/
public static double getS2(double z, double one_d2, double A, double B, double[] ds_dz)
{
final double z2 = z * z;
final double z3 = z2 * z;
final double z4 = z2 * z2;
// Eq. 17a
final double s = Math.sqrt(1 + one_d2 * (z2 + A * z3 + B * z4));
// Eq. 19a
ds_dz[0] = (one_d2 * (2 * z + A * 3 * z2 + B * 4 * z3)) / (2 * s);
// Eq. 19b
ds_dz[1] = (one_d2 * (2 + A * 6 * z + B * 12 * z2)) / (2 * s) -
pow2(one_d2 * (2 * z + A * 3 * z2 + B * 4 * z3)) / (4 * s * s * s);
return s;
}
private static double pow2(final double d)
{
return d * d;
}
/**
* Gets the standard deviation and first derivative for the z-depth.
*
* @param z
* the z
* @param one_d2
* one over the depth of focus squared (1/d^2)
* @param A
* Empirical constant A for the astigmatism of the PSF
* @param B
* Empirical constant B for the astigmatism of the PSF
* @param ds_dz
* the first derivative of s given z
* @return the standard deviation
*/
public static double getS1(double z, double one_d2, double A, double B, double[] ds_dz)
{
final double z2 = z * z;
final double z3 = z2 * z;
final double z4 = z2 * z2;
// Eq. 17a
final double s = Math.sqrt(1 + one_d2 * (z2 + A * z3 + B * z4));
// Eq. 19a
ds_dz[0] = (one_d2 * (2 * z + A * 3 * z2 + B * 4 * z3)) / (2 * s);
return s;
}
/**
* Gets the standard deviation for the z-depth.
*
* @param z
* the z
* @param one_d2
* one over the depth of focus squared (1/d^2)
* @param A
* Empirical constant A for the astigmatism of the PSF
* @param B
* Empirical constant B for the astigmatism of the PSF
* @return the standard deviation
*/
public static double getS(double z, double one_d2, double A, double B)
{
final double z2 = z * z;
final double z3 = z2 * z;
final double z4 = z2 * z2;
// Eq. 17a
return Math.sqrt(1 + one_d2 * (z2 + A * z3 + B * z4));
}
/*
* (non-Javadoc)
*
* @see gdsc.smlm.function.gaussian.AstimatismZModel#getSx(double)
*/
public double getSx(double z)
{
return getS(z - gamma, one_d2, Ax, Bx);
}
/*
* (non-Javadoc)
*
* @see gdsc.smlm.function.gaussian.AstimatismZModel#getSx(double, double[])
*/
public double getSx(double z, double[] ds_dz)
{
return getS1(z - gamma, one_d2, Ax, Bx, ds_dz);
}
/*
* (non-Javadoc)
*
* @see gdsc.smlm.function.gaussian.AstimatismZModel#getSx2(double, double[])
*/
public double getSx2(double z, double[] ds_dz)
{
return getS2(z - gamma, one_d2, Ax, Bx, ds_dz);
}
/*
* (non-Javadoc)
*
* @see gdsc.smlm.function.gaussian.AstimatismZModel#getSy(double)
*/
public double getSy(double z)
{
return getS(z + gamma, one_d2, Ay, By);
}
/*
* (non-Javadoc)
*
* @see gdsc.smlm.function.gaussian.AstimatismZModel#getSy(double, double[])
*/
public double getSy(double z, double[] ds_dz)
{
return getS1(z + gamma, one_d2, Ay, By, ds_dz);
}
/*
* (non-Javadoc)
*
* @see gdsc.smlm.function.gaussian.AstimatismZModel#getSy2(double, double[])
*/
public double getSy2(double z, double[] ds_dz)
{
return getS2(z + gamma, one_d2, Ay, By, ds_dz);
}
}