Examples with UnivariateOptimizer org.apache.commons.math3.optim.univariate.UnivariateOptimizer used on opensource projects

Example 1 with UnivariateOptimizer

use of org.apache.commons.math3.optim.univariate.UnivariateOptimizer in project GDSC-SMLM by aherbert.

the class FIRE method findMin.

private UnivariatePointValuePair findMin(UnivariatePointValuePair current, UnivariateOptimizer o, UnivariateFunction f, double qValue, double factor) {
    try {
        BracketFinder bracket = new BracketFinder();
        bracket.search(f, GoalType.MINIMIZE, qValue * factor, qValue / factor);
        UnivariatePointValuePair next = o.optimize(GoalType.MINIMIZE, new MaxEval(3000), new SearchInterval(bracket.getLo(), bracket.getHi(), bracket.getMid()), new UnivariateObjectiveFunction(f));
        if (next == null)
            return current;
        //System.out.printf("LineMin [%.1f]  %f = %f\n", factor, next.getPoint(), next.getValue());
        if (current != null)
            return (next.getValue() < current.getValue()) ? next : current;
        return next;
    } catch (Exception e) {
        return current;
    }
}

Example 2 with UnivariateOptimizer

use of org.apache.commons.math3.optim.univariate.UnivariateOptimizer in project imagingbook-common by imagingbook.

the class FourierDescriptor method getStartPointPhase.

/**
 * Calculates the 'canonical' start point. This version uses
 * (a) a coarse search for a global maximum of fp() and subsequently
 * (b) a numerical optimization using Brent's method
 * (implemented with Apache Commons Math).
 *
 * @param Mp number of Fourier coefficient pairs
 * @return start point phase
 */
public double getStartPointPhase(int Mp) {
    Mp = Math.min(Mp, (G.length - 1) / 2);
    UnivariateFunction fp = new TargetFunction(Mp);
    // search for the global maximum in coarse steps
    double cmax = Double.NEGATIVE_INFINITY;
    int kmax = -1;
    // number of steps over 180 degrees
    int K = 25;
    for (int k = 0; k < K; k++) {
        // phase to evaluate
        final double phi = Math.PI * k / K;
        final double c = fp.value(phi);
        if (c > cmax) {
            cmax = c;
            kmax = k;
        }
    }
    // optimize using previous and next point as the bracket.
    double minPhi = Math.PI * (kmax - 1) / K;
    double maxPhi = Math.PI * (kmax + 1) / K;
    UnivariateOptimizer optimizer = new BrentOptimizer(1E-4, 1E-6);
    int maxIter = 20;
    UnivariatePointValuePair result = optimizer.optimize(new MaxEval(maxIter), new UnivariateObjectiveFunction(fp), GoalType.MAXIMIZE, new SearchInterval(minPhi, maxPhi));
    double phi0 = result.getPoint();
    // the canonical start point phase
    return phi0;
}

Example 3 with UnivariateOptimizer

use of org.apache.commons.math3.optim.univariate.UnivariateOptimizer in project GDSC-SMLM by aherbert.

the class Fire method findMin.

private static UnivariatePointValuePair findMin(UnivariatePointValuePair current, UnivariateOptimizer optimiser, UnivariateFunction func, double qvalue, double factor) {
    try {
        final BracketFinder bracket = new BracketFinder();
        bracket.search(func, GoalType.MINIMIZE, qvalue * factor, qvalue / factor);
        final UnivariatePointValuePair next = optimiser.optimize(GoalType.MINIMIZE, new MaxEval(3000), new SearchInterval(bracket.getLo(), bracket.getHi(), bracket.getMid()), new UnivariateObjectiveFunction(func));
        if (next == null) {
            return current;
        }
        if (current != null) {
            return (next.getValue() < current.getValue()) ? next : current;
        }
        return next;
    } catch (final Exception ex) {
        return current;
    }
}

Example 4 with UnivariateOptimizer

use of org.apache.commons.math3.optim.univariate.UnivariateOptimizer in project GDSC-SMLM by aherbert.

the class FIRE method runQEstimation.

private void runQEstimation() {
    IJ.showStatus(TITLE + " ...");
    if (!showQEstimationInputDialog())
        return;
    MemoryPeakResults results = ResultsManager.loadInputResults(inputOption, false);
    if (results == null || results.size() == 0) {
        IJ.error(TITLE, "No results could be loaded");
        return;
    }
    if (results.getCalibration() == null) {
        IJ.error(TITLE, "The results are not calibrated");
        return;
    }
    results = cropToRoi(results);
    if (results.size() < 2) {
        IJ.error(TITLE, "No results within the crop region");
        return;
    }
    initialise(results, null);
    // We need localisation precision.
    // Build a histogram of the localisation precision.
    // Get the initial mean and SD and plot as a Gaussian.
    PrecisionHistogram histogram = calculatePrecisionHistogram();
    if (histogram == null) {
        IJ.error(TITLE, "No localisation precision available.\n \nPlease choose " + PrecisionMethod.FIXED + " and enter a precision mean and SD.");
        return;
    }
    StoredDataStatistics precision = histogram.precision;
    //String name = results.getName();
    double fourierImageScale = SCALE_VALUES[imageScaleIndex];
    int imageSize = IMAGE_SIZE_VALUES[imageSizeIndex];
    // Create the image and compute the numerator of FRC. 
    // Do not use the signal so results.size() is the number of localisations.
    IJ.showStatus("Computing FRC curve ...");
    FireImages images = createImages(fourierImageScale, imageSize, false);
    // DEBUGGING - Save the two images to disk. Load the images into the Matlab 
    // code that calculates the Q-estimation and make this plugin match the functionality.
    //IJ.save(new ImagePlus("i1", images.ip1), "/scratch/i1.tif");
    //IJ.save(new ImagePlus("i2", images.ip2), "/scratch/i2.tif");
    FRC frc = new FRC();
    frc.progress = progress;
    frc.setFourierMethod(fourierMethod);
    frc.setSamplingMethod(samplingMethod);
    frc.setPerimeterSamplingFactor(perimeterSamplingFactor);
    FRCCurve frcCurve = frc.calculateFrcCurve(images.ip1, images.ip2, images.nmPerPixel);
    if (frcCurve == null) {
        IJ.error(TITLE, "Failed to compute FRC curve");
        return;
    }
    IJ.showStatus("Running Q-estimation ...");
    // Note:
    // The method implemented here is based on Matlab code provided by Bernd Rieger.
    // The idea is to compute the spurious correlation component of the FRC Numerator
    // using an initial estimate of distribution of the localisation precision (assumed 
    // to be Gaussian). This component is the contribution of repeat localisations of 
    // the same molecule to the numerator and is modelled as an exponential decay
    // (exp_decay). The component is scaled by the Q-value which
    // is the average number of times a molecule is seen in addition to the first time.
    // At large spatial frequencies the scaled component should match the numerator,
    // i.e. at high resolution (low FIRE number) the numerator is made up of repeat 
    // localisations of the same molecule and not actual structure in the image.
    // The best fit is where the numerator equals the scaled component, i.e. num / (q*exp_decay) == 1.
    // The FRC Numerator is plotted and Q can be determined by
    // adjusting Q and the precision mean and SD to maximise the cost function.
    // This can be done interactively by the user with the effect on the FRC curve
    // dynamically updated and displayed.
    // Compute the scaled FRC numerator
    double qNorm = (1 / frcCurve.mean1 + 1 / frcCurve.mean2);
    double[] frcnum = new double[frcCurve.getSize()];
    for (int i = 0; i < frcnum.length; i++) {
        FRCCurveResult r = frcCurve.get(i);
        frcnum[i] = qNorm * r.getNumerator() / r.getNumberOfSamples();
    }
    // Compute the spatial frequency and the region for curve fitting
    double[] q = FRC.computeQ(frcCurve, false);
    int low = 0, high = q.length;
    while (high > 0 && q[high - 1] > maxQ) high--;
    while (low < q.length && q[low] < minQ) low++;
    // Require we fit at least 10% of the curve
    if (high - low < q.length * 0.1) {
        IJ.error(TITLE, "Not enough points for Q estimation");
        return;
    }
    // Obtain initial estimate of Q plateau height and decay.
    // This can be done by fitting the precision histogram and then fixing the mean and sigma.
    // Or it can be done by allowing the precision to be sampled and the mean and sigma
    // become parameters for fitting.
    // Check if we can sample precision values
    boolean sampleDecay = precision != null && FIRE.sampleDecay;
    double[] exp_decay;
    if (sampleDecay) {
        // Random sample of precision values from the distribution is used to 
        // construct the decay curve
        int[] sample = Random.sample(10000, precision.getN(), new Well19937c());
        final double four_pi2 = 4 * Math.PI * Math.PI;
        double[] pre = new double[q.length];
        for (int i = 1; i < q.length; i++) pre[i] = -four_pi2 * q[i] * q[i];
        // Sample
        final int n = sample.length;
        double[] hq = new double[n];
        for (int j = 0; j < n; j++) {
            // Scale to SR pixels
            double s2 = precision.getValue(sample[j]) / images.nmPerPixel;
            s2 *= s2;
            for (int i = 1; i < q.length; i++) hq[i] += FastMath.exp(pre[i] * s2);
        }
        for (int i = 1; i < q.length; i++) hq[i] /= n;
        exp_decay = new double[q.length];
        exp_decay[0] = 1;
        for (int i = 1; i < q.length; i++) {
            double sinc_q = sinc(Math.PI * q[i]);
            exp_decay[i] = sinc_q * sinc_q * hq[i];
        }
    } else {
        // Note: The sigma mean and std should be in the units of super-resolution 
        // pixels so scale to SR pixels
        exp_decay = computeExpDecay(histogram.mean / images.nmPerPixel, histogram.sigma / images.nmPerPixel, q);
    }
    // Smoothing
    double[] smooth;
    if (loessSmoothing) {
        // Note: This computes the log then smooths it 
        double bandwidth = 0.1;
        int robustness = 0;
        double[] l = new double[exp_decay.length];
        for (int i = 0; i < l.length; i++) {
            // Original Matlab code computes the log for each array.
            // This is equivalent to a single log on the fraction of the two.
            // Perhaps the two log method is more numerically stable.
            //l[i] = Math.log(Math.abs(frcnum[i])) - Math.log(exp_decay[i]);
            l[i] = Math.log(Math.abs(frcnum[i] / exp_decay[i]));
        }
        try {
            LoessInterpolator loess = new LoessInterpolator(bandwidth, robustness);
            smooth = loess.smooth(q, l);
        } catch (Exception e) {
            IJ.error(TITLE, "LOESS smoothing failed");
            return;
        }
    } else {
        // Note: This smooths the curve before computing the log 
        double[] norm = new double[exp_decay.length];
        for (int i = 0; i < norm.length; i++) {
            norm[i] = frcnum[i] / exp_decay[i];
        }
        // Median window of 5 == radius of 2
        MedianWindow mw = new MedianWindow(norm, 2);
        smooth = new double[exp_decay.length];
        for (int i = 0; i < norm.length; i++) {
            smooth[i] = Math.log(Math.abs(mw.getMedian()));
            mw.increment();
        }
    }
    // Fit with quadratic to find the initial guess.
    // Note: example Matlab code frc_Qcorrection7.m identifies regions of the 
    // smoothed log curve with low derivative and only fits those. The fit is 
    // used for the final estimate. Fitting a subset with low derivative is not 
    // implemented here since the initial estimate is subsequently optimised 
    // to maximise a cost function. 
    Quadratic curve = new Quadratic();
    SimpleCurveFitter fit = SimpleCurveFitter.create(curve, new double[2]);
    WeightedObservedPoints points = new WeightedObservedPoints();
    for (int i = low; i < high; i++) points.add(q[i], smooth[i]);
    double[] estimate = fit.fit(points.toList());
    double qValue = FastMath.exp(estimate[0]);
    //System.out.printf("Initial q-estimate = %s => %.3f\n", Arrays.toString(estimate), qValue);
    // This could be made an option. Just use for debugging
    boolean debug = false;
    if (debug) {
        // Plot the initial fit and the fit curve
        double[] qScaled = FRC.computeQ(frcCurve, true);
        double[] line = new double[q.length];
        for (int i = 0; i < q.length; i++) line[i] = curve.value(q[i], estimate);
        String title = TITLE + " Initial fit";
        Plot2 plot = new Plot2(title, "Spatial Frequency (nm^-1)", "FRC Numerator");
        String label = String.format("Q = %.3f", qValue);
        plot.addPoints(qScaled, smooth, Plot.LINE);
        plot.setColor(Color.red);
        plot.addPoints(qScaled, line, Plot.LINE);
        plot.setColor(Color.black);
        plot.addLabel(0, 0, label);
        Utils.display(title, plot, Utils.NO_TO_FRONT);
    }
    if (fitPrecision) {
        // Q - Should this be optional?
        if (sampleDecay) {
            // If a sample of the precision was used to construct the data for the initial fit 
            // then update the estimate using the fit result since it will be a better start point. 
            histogram.sigma = precision.getStandardDeviation();
            // Normalise sum-of-squares to the SR pixel size
            double meanSumOfSquares = (precision.getSumOfSquares() / (images.nmPerPixel * images.nmPerPixel)) / precision.getN();
            histogram.mean = images.nmPerPixel * Math.sqrt(meanSumOfSquares - estimate[1] / (4 * Math.PI * Math.PI));
        }
        // Do a multivariate fit ...
        SimplexOptimizer opt = new SimplexOptimizer(1e-6, 1e-10);
        PointValuePair p = null;
        MultiPlateauness f = new MultiPlateauness(frcnum, q, low, high);
        double[] initial = new double[] { histogram.mean / images.nmPerPixel, histogram.sigma / images.nmPerPixel, qValue };
        p = findMin(p, opt, f, scale(initial, 0.1));
        p = findMin(p, opt, f, scale(initial, 0.5));
        p = findMin(p, opt, f, initial);
        p = findMin(p, opt, f, scale(initial, 2));
        p = findMin(p, opt, f, scale(initial, 10));
        if (p != null) {
            double[] point = p.getPointRef();
            histogram.mean = point[0] * images.nmPerPixel;
            histogram.sigma = point[1] * images.nmPerPixel;
            qValue = point[2];
        }
    } else {
        // If so then this should be optional.
        if (sampleDecay) {
            if (precisionMethod != PrecisionMethod.FIXED) {
                histogram.sigma = precision.getStandardDeviation();
                // Normalise sum-of-squares to the SR pixel size
                double meanSumOfSquares = (precision.getSumOfSquares() / (images.nmPerPixel * images.nmPerPixel)) / precision.getN();
                histogram.mean = images.nmPerPixel * Math.sqrt(meanSumOfSquares - estimate[1] / (4 * Math.PI * Math.PI));
            }
            exp_decay = computeExpDecay(histogram.mean / images.nmPerPixel, histogram.sigma / images.nmPerPixel, q);
        }
        // Estimate spurious component by promoting plateauness.
        // The Matlab code used random initial points for a Simplex optimiser.
        // A Brent line search should be pretty deterministic so do simple repeats.
        // However it will proceed downhill so if the initial point is wrong then 
        // it will find a sub-optimal result.
        UnivariateOptimizer o = new BrentOptimizer(1e-3, 1e-6);
        Plateauness f = new Plateauness(frcnum, exp_decay, low, high);
        UnivariatePointValuePair p = null;
        p = findMin(p, o, f, qValue, 0.1);
        p = findMin(p, o, f, qValue, 0.2);
        p = findMin(p, o, f, qValue, 0.333);
        p = findMin(p, o, f, qValue, 0.5);
        // Do some Simplex repeats as well
        SimplexOptimizer opt = new SimplexOptimizer(1e-6, 1e-10);
        p = findMin(p, opt, f, qValue * 0.1);
        p = findMin(p, opt, f, qValue * 0.5);
        p = findMin(p, opt, f, qValue);
        p = findMin(p, opt, f, qValue * 2);
        p = findMin(p, opt, f, qValue * 10);
        if (p != null)
            qValue = p.getPoint();
    }
    QPlot qplot = new QPlot(frcCurve, qValue, low, high);
    // Interactive dialog to estimate Q (blinking events per flourophore) using 
    // sliders for the mean and standard deviation of the localisation precision.
    showQEstimationDialog(histogram, qplot, frcCurve, images.nmPerPixel);
    IJ.showStatus(TITLE + " complete");
}

Example 5 with UnivariateOptimizer

use of org.apache.commons.math3.optim.univariate.UnivariateOptimizer in project GDSC-SMLM by aherbert.

the class Fire method runQEstimation.

@SuppressWarnings("null")
private void runQEstimation() {
    IJ.showStatus(pluginTitle + " ...");
    if (!showQEstimationInputDialog()) {
        return;
    }
    MemoryPeakResults inputResults = ResultsManager.loadInputResults(settings.inputOption, false, null, null);
    if (MemoryPeakResults.isEmpty(inputResults)) {
        IJ.error(pluginTitle, "No results could be loaded");
        return;
    }
    if (inputResults.getCalibration() == null) {
        IJ.error(pluginTitle, "The results are not calibrated");
        return;
    }
    inputResults = cropToRoi(inputResults);
    if (inputResults.size() < 2) {
        IJ.error(pluginTitle, "No results within the crop region");
        return;
    }
    initialise(inputResults, null);
    // We need localisation precision.
    // Build a histogram of the localisation precision.
    // Get the initial mean and SD and plot as a Gaussian.
    final PrecisionHistogram histogram = calculatePrecisionHistogram();
    if (histogram == null) {
        IJ.error(pluginTitle, "No localisation precision available.\n \nPlease choose " + PrecisionMethod.FIXED + " and enter a precision mean and SD.");
        return;
    }
    final StoredDataStatistics precision = histogram.precision;
    final double fourierImageScale = Settings.scaleValues[settings.imageScaleIndex];
    final int imageSize = Settings.imageSizeValues[settings.imageSizeIndex];
    // Create the image and compute the numerator of FRC.
    // Do not use the signal so results.size() is the number of localisations.
    IJ.showStatus("Computing FRC curve ...");
    final FireImages images = createImages(fourierImageScale, imageSize, false);
    // DEBUGGING - Save the two images to disk. Load the images into the Matlab
    // code that calculates the Q-estimation and make this plugin match the functionality.
    // IJ.save(new ImagePlus("i1", images.ip1), "/scratch/i1.tif");
    // IJ.save(new ImagePlus("i2", images.ip2), "/scratch/i2.tif");
    final Frc frc = new Frc();
    frc.setTrackProgress(progress);
    frc.setFourierMethod(fourierMethod);
    frc.setSamplingMethod(samplingMethod);
    frc.setPerimeterSamplingFactor(settings.perimeterSamplingFactor);
    final FrcCurve frcCurve = frc.calculateFrcCurve(images.ip1, images.ip2, images.nmPerPixel);
    if (frcCurve == null) {
        IJ.error(pluginTitle, "Failed to compute FRC curve");
        return;
    }
    IJ.showStatus("Running Q-estimation ...");
    // Note:
    // The method implemented here is based on Matlab code provided by Bernd Rieger.
    // The idea is to compute the spurious correlation component of the FRC Numerator
    // using an initial estimate of distribution of the localisation precision (assumed
    // to be Gaussian). This component is the contribution of repeat localisations of
    // the same molecule to the numerator and is modelled as an exponential decay
    // (exp_decay). The component is scaled by the Q-value which
    // is the average number of times a molecule is seen in addition to the first time.
    // At large spatial frequencies the scaled component should match the numerator,
    // i.e. at high resolution (low FIRE number) the numerator is made up of repeat
    // localisations of the same molecule and not actual structure in the image.
    // The best fit is where the numerator equals the scaled component, i.e. num / (q*exp_decay) ==
    // 1.
    // The FRC Numerator is plotted and Q can be determined by
    // adjusting Q and the precision mean and SD to maximise the cost function.
    // This can be done interactively by the user with the effect on the FRC curve
    // dynamically updated and displayed.
    // Compute the scaled FRC numerator
    final double qNorm = (1 / frcCurve.mean1 + 1 / frcCurve.mean2);
    final double[] frcnum = new double[frcCurve.getSize()];
    for (int i = 0; i < frcnum.length; i++) {
        final FrcCurveResult r = frcCurve.get(i);
        frcnum[i] = qNorm * r.getNumerator() / r.getNumberOfSamples();
    }
    // Compute the spatial frequency and the region for curve fitting
    final double[] q = Frc.computeQ(frcCurve, false);
    int low = 0;
    int high = q.length;
    while (high > 0 && q[high - 1] > settings.maxQ) {
        high--;
    }
    while (low < q.length && q[low] < settings.minQ) {
        low++;
    }
    // Require we fit at least 10% of the curve
    if (high - low < q.length * 0.1) {
        IJ.error(pluginTitle, "Not enough points for Q estimation");
        return;
    }
    // Obtain initial estimate of Q plateau height and decay.
    // This can be done by fitting the precision histogram and then fixing the mean and sigma.
    // Or it can be done by allowing the precision to be sampled and the mean and sigma
    // become parameters for fitting.
    // Check if we can sample precision values
    final boolean sampleDecay = precision != null && settings.sampleDecay;
    double[] expDecay;
    if (sampleDecay) {
        // Random sample of precision values from the distribution is used to
        // construct the decay curve
        final int[] sample = RandomUtils.sample(10000, precision.getN(), UniformRandomProviders.create());
        final double four_pi2 = 4 * Math.PI * Math.PI;
        final double[] pre = new double[q.length];
        for (int i = 1; i < q.length; i++) {
            pre[i] = -four_pi2 * q[i] * q[i];
        }
        // Sample
        final int n = sample.length;
        final double[] hq = new double[n];
        for (int j = 0; j < n; j++) {
            // Scale to SR pixels
            double s2 = precision.getValue(sample[j]) / images.nmPerPixel;
            s2 *= s2;
            for (int i = 1; i < q.length; i++) {
                hq[i] += StdMath.exp(pre[i] * s2);
            }
        }
        for (int i = 1; i < q.length; i++) {
            hq[i] /= n;
        }
        expDecay = new double[q.length];
        expDecay[0] = 1;
        for (int i = 1; i < q.length; i++) {
            final double sinc_q = sinc(Math.PI * q[i]);
            expDecay[i] = sinc_q * sinc_q * hq[i];
        }
    } else {
        // Note: The sigma mean and std should be in the units of super-resolution
        // pixels so scale to SR pixels
        expDecay = computeExpDecay(histogram.mean / images.nmPerPixel, histogram.sigma / images.nmPerPixel, q);
    }
    // Smoothing
    double[] smooth;
    if (settings.loessSmoothing) {
        // Note: This computes the log then smooths it
        final double bandwidth = 0.1;
        final int robustness = 0;
        final double[] l = new double[expDecay.length];
        for (int i = 0; i < l.length; i++) {
            // Original Matlab code computes the log for each array.
            // This is equivalent to a single log on the fraction of the two.
            // Perhaps the two log method is more numerically stable.
            // l[i] = Math.log(Math.abs(frcnum[i])) - Math.log(exp_decay[i]);
            l[i] = Math.log(Math.abs(frcnum[i] / expDecay[i]));
        }
        try {
            final LoessInterpolator loess = new LoessInterpolator(bandwidth, robustness);
            smooth = loess.smooth(q, l);
        } catch (final Exception ex) {
            IJ.error(pluginTitle, "LOESS smoothing failed");
            return;
        }
    } else {
        // Note: This smooths the curve before computing the log
        final double[] norm = new double[expDecay.length];
        for (int i = 0; i < norm.length; i++) {
            norm[i] = frcnum[i] / expDecay[i];
        }
        // Median window of 5 == radius of 2
        final DoubleMedianWindow mw = DoubleMedianWindow.wrap(norm, 2);
        smooth = new double[expDecay.length];
        for (int i = 0; i < norm.length; i++) {
            smooth[i] = Math.log(Math.abs(mw.getMedian()));
            mw.increment();
        }
    }
    // Fit with quadratic to find the initial guess.
    // Note: example Matlab code frc_Qcorrection7.m identifies regions of the
    // smoothed log curve with low derivative and only fits those. The fit is
    // used for the final estimate. Fitting a subset with low derivative is not
    // implemented here since the initial estimate is subsequently optimised
    // to maximise a cost function.
    final Quadratic curve = new Quadratic();
    final SimpleCurveFitter fit = SimpleCurveFitter.create(curve, new double[2]);
    final WeightedObservedPoints points = new WeightedObservedPoints();
    for (int i = low; i < high; i++) {
        points.add(q[i], smooth[i]);
    }
    final double[] estimate = fit.fit(points.toList());
    double qvalue = StdMath.exp(estimate[0]);
    // This could be made an option. Just use for debugging
    final boolean debug = false;
    if (debug) {
        // Plot the initial fit and the fit curve
        final double[] qScaled = Frc.computeQ(frcCurve, true);
        final double[] line = new double[q.length];
        for (int i = 0; i < q.length; i++) {
            line[i] = curve.value(q[i], estimate);
        }
        final String title = pluginTitle + " Initial fit";
        final Plot plot = new Plot(title, "Spatial Frequency (nm^-1)", "FRC Numerator");
        final String label = String.format("Q = %.3f", qvalue);
        plot.addPoints(qScaled, smooth, Plot.LINE);
        plot.setColor(Color.red);
        plot.addPoints(qScaled, line, Plot.LINE);
        plot.setColor(Color.black);
        plot.addLabel(0, 0, label);
        ImageJUtils.display(title, plot, ImageJUtils.NO_TO_FRONT);
    }
    if (settings.fitPrecision) {
        // Q - Should this be optional?
        if (sampleDecay) {
            // If a sample of the precision was used to construct the data for the initial fit
            // then update the estimate using the fit result since it will be a better start point.
            histogram.sigma = precision.getStandardDeviation();
            // Normalise sum-of-squares to the SR pixel size
            final double meanSumOfSquares = (precision.getSumOfSquares() / (images.nmPerPixel * images.nmPerPixel)) / precision.getN();
            histogram.mean = images.nmPerPixel * Math.sqrt(meanSumOfSquares - estimate[1] / (4 * Math.PI * Math.PI));
        }
        // Do a multivariate fit ...
        final SimplexOptimizer opt = new SimplexOptimizer(1e-6, 1e-10);
        PointValuePair pair = null;
        final MultiPlateauness f = new MultiPlateauness(frcnum, q, low, high);
        final double[] initial = new double[] { histogram.mean / images.nmPerPixel, histogram.sigma / images.nmPerPixel, qvalue };
        pair = findMin(pair, opt, f, scale(initial, 0.1));
        pair = findMin(pair, opt, f, scale(initial, 0.5));
        pair = findMin(pair, opt, f, initial);
        pair = findMin(pair, opt, f, scale(initial, 2));
        pair = findMin(pair, opt, f, scale(initial, 10));
        if (pair != null) {
            final double[] point = pair.getPointRef();
            histogram.mean = point[0] * images.nmPerPixel;
            histogram.sigma = point[1] * images.nmPerPixel;
            qvalue = point[2];
        }
    } else {
        // If so then this should be optional.
        if (sampleDecay) {
            if (precisionMethod != PrecisionMethod.FIXED) {
                histogram.sigma = precision.getStandardDeviation();
                // Normalise sum-of-squares to the SR pixel size
                final double meanSumOfSquares = (precision.getSumOfSquares() / (images.nmPerPixel * images.nmPerPixel)) / precision.getN();
                histogram.mean = images.nmPerPixel * Math.sqrt(meanSumOfSquares - estimate[1] / (4 * Math.PI * Math.PI));
            }
            expDecay = computeExpDecay(histogram.mean / images.nmPerPixel, histogram.sigma / images.nmPerPixel, q);
        }
        // Estimate spurious component by promoting plateauness.
        // The Matlab code used random initial points for a Simplex optimiser.
        // A Brent line search should be pretty deterministic so do simple repeats.
        // However it will proceed downhill so if the initial point is wrong then
        // it will find a sub-optimal result.
        final UnivariateOptimizer o = new BrentOptimizer(1e-3, 1e-6);
        final Plateauness f = new Plateauness(frcnum, expDecay, low, high);
        UnivariatePointValuePair result = null;
        result = findMin(result, o, f, qvalue, 0.1);
        result = findMin(result, o, f, qvalue, 0.2);
        result = findMin(result, o, f, qvalue, 0.333);
        result = findMin(result, o, f, qvalue, 0.5);
        // Do some Simplex repeats as well
        final SimplexOptimizer opt = new SimplexOptimizer(1e-6, 1e-10);
        result = findMin(result, opt, f, qvalue * 0.1);
        result = findMin(result, opt, f, qvalue * 0.5);
        result = findMin(result, opt, f, qvalue);
        result = findMin(result, opt, f, qvalue * 2);
        result = findMin(result, opt, f, qvalue * 10);
        if (result != null) {
            qvalue = result.getPoint();
        }
    }
    final QPlot qplot = new QPlot(frcCurve, qvalue, low, high);
    // Interactive dialog to estimate Q (blinking events per flourophore) using
    // sliders for the mean and standard deviation of the localisation precision.
    showQEstimationDialog(histogram, qplot, images.nmPerPixel);
    IJ.showStatus(pluginTitle + " complete");
}