# MATLAB: Estimating resolution from a PSF slide image

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20.309: Biological Instrumentation and Measurement

### Measuring resolution

One of the most commonly used definitions of resolution is the distance between two point sources in the sample plane such that the peak of one point source’s blurred image falls on the first zero of the other source. The definition suggests a procedure for measuring resolution: make an image of a point source, measure the peak-to-trough distance in the image plane, and divide by the magnification. In this part of the lab, you will use a procedure inspired by this simple idea to estimate the resolution of your microscope. Instead of measuring the spot sizes with a ruler, you will use nonlinear regression to find best-fit parameters of digital images of point sources.

One practical problem with this method is that true point sources are difficult to come by. If you are an astronomer using a telescope, stars are readily available, and they are very good approximations of point sources. Since there is no natural microscopic sample that is equivalent to the night sky, microscopists have to construct something suitable. One of the most common facsimiles is a microscope slide sprinkled with tiny, fluorescent beads that have diameters in the range of 100-190 nm. These beads are small enough to be considered point sources. Unfortunately, beads small enough for this purpose are not very bright. Imaging them can be challenging. Your microscope must be very well aligned to get good results. (The images you make for this part of the lab will probably remind you of the night sky. If they don't, have an instructor take a look at your setup.)

In this part of the assignment, you will use image processing functions to locate the beads in an image and fit a Gaussian function to them. Gaussians are more amenable to nonlinear regression than Bessel functions, and they are a very good approximation. It is straightforward to convert the Gaussian parameters to Rayleigh resolution. See Converting Gaussian fit to Rayleigh resolution for a discussion of the conversion.

## Measuring resolution

The sequence of steps for the resolution code is:

1. find the connected regions
2. eliminate connected regions that are likely not image of a single microscphere
3. use nonlinear regression to fit a Gaussian model function to the image
4. compute summary statistics
5. convert Gaussian parameter to Rayleigh resolution

### Step 1: find the connected regions

There are many methods for segmenting images. Fortunately, the fluorescent images of microspheres you are making in the lab usually have a high SNR. Because of this, a simple, global threshold works quite well. The bright pixels are the ones where the microspheres were.

The function FindBrightObjectsInImage performs a global threshold and then computes various properties of the connected regions of pixels above the threshold. The MATLAB documentation includes a complete list of the properties that the built-in function regionprops can compute.

```function RegionProperties = FindBrightObjectsInImage( InputImage, GlobalThreshold, DilationRadius, PropertyList )

RegionProperties = regionprops( mask, InputImage, PropertyList );

end
```

The function below, EstimateResolutionFromPsfImage, takes an image of point sources (or approximate point sources) as an input and it returns an estimate of the resolution of an optical system that produced the image. It uses a global threshold to locate bright regions and the built-in MATLAB function regionprops to measure attributes of each connected region of bright pixels. You can add some code to the function that uses the properties of the regions to reject connected regions that are likely not images of single microspheres. The function uses the nonlinear regression function nlinfit to estimate best fit Gaussian parameters for each bright spot.

There are four subfunctions that should be included in the same m-file as EstimateResolutionFromPsfImage.

```function [ Resolution, StandardError, BestFitData ] = EstimateResolutionFromPsfImage( ImageData )

figure(1);
imshow( ImageData );
hold on;
title( 'PSF Image' );

% create bilevel image mapping bright regions
thresholdLevel = 0.5 * ( max( ImageData(:) ) + median( ImageData(:) ) ); % this may not work in all cases
binaryImage = im2bw( ImageData, thresholdLevel );
% increase the size of each connected region using dilation
dilatedImage = imdilate( binaryImage, strel( 'disk', 7, 0 ) );
% remove objects touching the edges
dilatedImage = imclearborder( dilatedImage );

% assign a unique object number to each connected region of pixels
labeledImage = bwlabel( dilatedImage );

% draw a red circle around labeled objects
CircleObjectsInImage( labeledImage, [ 1 0 0 ] );

% compute the parameters of each object identified in the image
objectProperties = regionprops( dilatedImage, ImageData, ...
'Area', 'Centroid', 'PixelList', 'PixelValues', 'MaxIntensity' );

% eliminate outliers -- PSF beads that are touching, aggregates, etc...
% begin by eliminating objects whose areas are outliers, as occurs when
medianArea = median( [ objectProperties.Area ] );
outliers = [ objectProperties.Area ] > OutlierTolerancePercentage(2) * medianArea | [ objectProperties.Area ] < OutlierTolerancePercentage(1) * medianArea;

if( sum( outliers ) > 0 )
dilatedImage = RemoveObjectsFromImage( dilatedImage, objectProperties( outliers ) );
objectProperties( outliers ) = [];  % remove outliers
end

CircleObjectsInImage( dilatedImage, [ 1 1 0 ] );

% next eliminate intensity outliers
medianMaxIntensity = median( [ objectProperties.MaxIntensity ] );
outliers = [ objectProperties.MaxIntensity ] > OutlierTolerancePercentage(2) * medianMaxIntensity | [ objectProperties.MaxIntensity ] < OutlierTolerancePercentage(1) * medianMaxIntensity;
outliers = outliers | [ objectProperties.MaxIntensity ] > 0.995;

if( sum( outliers ) > 0 )
dilatedImage = RemoveObjectsFromImage( dilatedImage, objectProperties( outliers ) );
objectProperties( outliers ) = [];
end

% circle all the remaining objects in green
CircleObjectsInImage( dilatedImage, [ 0 1 0 ] );
LabelObjectsInImage( objectProperties );
hold off;

BestFitData = cell(1, length(objectProperties));

% use nlinfit to fit a Gaussian to each object
for ii = 1:length(objectProperties)
% initial guess for sigma based on area of bright spot
maximumPixelValue = max( objectProperties(ii).PixelValues );
darkPixelValue = median( objectProperties(ii).PixelValues );
pixelCountAboveHalf = sum( objectProperties(ii).PixelValues > .5 *  ( maximumPixelValue + darkPixelValue ) );
sigmaInitialGuess = 0.8 * sqrt( pixelCountAboveHalf / 2 / pi / log(2) );

initialGuesses = [ ...
objectProperties(ii).Centroid(1), ... % yCenter
objectProperties(ii).Centroid(2), ... % xCenter
max(objectProperties(ii).PixelValues) - min(objectProperties(ii).PixelValues), ... % amplitude
sigmaInitialGuess, ... % (objectProperties(ii).BoundingBox(3) - 6) / 4, ... % sigma
min(objectProperties(ii).PixelValues) ];

BestFitData{ii} = nlinfit( objectProperties(ii).PixelList, objectProperties(ii).PixelValues, @Gaussian2DFitFunction, initialGuesses );

% plot data, initial guess, and fit for each peak
figure(2)
clf

% generate a triangle mesh from the best fit solution found by
% nlinfit and plot it
gd = delaunay( objectProperties(ii).PixelList(:,1), ...
objectProperties(ii).PixelList(:,2) );
trimesh( gd, objectProperties(ii).PixelList(:,1), ...
objectProperties(ii).PixelList(:,2), ...
Gaussian2DFitFunction(BestFitData{ii}, ...
objectProperties(ii).PixelList ) )
hold on

% plot initial guesses -- commented out to make plots less
% cluttered. put this back in to debug initial guesses
% plot3( objectProperties(ii).PixelList(:,1), ...
%   objectProperties(ii).PixelList(:,2), ...
%   Gaussian2DFitFunction(initialGuesses, ...
%   objectProperties(ii).PixelList ), 'rx' )

% plot image data
plot3( objectProperties(ii).PixelList(:,1), ...
objectProperties(ii).PixelList(:,2), ...
objectProperties(ii).PixelValues, 'gx', 'LineWidth', 3)
title(['Image data vs. Best Fit for Object Number ' num2str(ii)]);
end

allPeakData = vertcat( BestFitData{:} );
Resolution = mean( allPeakData(:,4) ) ./ 0.336;
StandardError = std( allPeakData(:,4) ./ 0.336 ) ./ sqrt( length( BestFitData ) );
end

function out = Gaussian2DFitFunction( Parameters, Coordinates )
yCenter = Parameters(1);
xCenter = Parameters(2);
amplitude = Parameters(3);
sigma = Parameters(4);
offset = Parameters(5);

out = amplitude * ...
exp( -(( Coordinates(:, 1) - yCenter ).^2 + ( Coordinates(:, 2) - xCenter ).^2 ) ...
./ (2 * sigma .^ 2 )) + offset;

end

function CircleObjectsInImage( LabelImage, BorderColor )
boundaries = bwboundaries( LabelImage );
numberOfBoundaries = size( boundaries );
for k = 1 : numberOfBoundaries
thisBoundary = boundaries{k};
plot(thisBoundary(:,2), thisBoundary(:,1), 'Color', BorderColor, 'LineWidth', 2);
end
end

function LabelObjectsInImage( ObjectProperties )
labelShift = -9;
fontSize = 10;

for ii = 1:length(ObjectProperties)
unweightedCentroid = ObjectProperties(ii).Centroid;		% Get centroid.
text(unweightedCentroid(1) + labelShift, unweightedCentroid(2), ...
num2str(ii), 'FontSize', fontSize, 'HorizontalAlignment', ...
'Right', 'Color', [0 1 0]);
end
end

function OutputBinaryImage = RemoveObjectsFromImage( InputBinaryImage, ObjectProperties )
OutputBinaryImage = InputBinaryImage;
eliminatedPixels = vertcat( ObjectProperties.PixelList );
allObjectIndexes = sub2ind( size( InputBinaryImage ), ...
eliminatedPixels(:, 2), eliminatedPixels(:,1) );
OutputBinaryImage( allObjectIndexes ) = 0;
end

% initial version 9/23/2013 by SCW
```

## Testing the code

It is an excellent idea to test functions before using them. One way to test EstimateResolutionFromPsfImage is to generate a synthetic point-source image of known resolution.

### Generating a synthetic PSF image

SimulatePsfSlide repeatedly calls subfunction SimulateAiryDiskImage to generate a synthetic PSF slide image.

```function SimulatedImage = SimulatePsfSlide( ...
NumericalAperture, Magnification, Wavelength, ...
ProbabilityOfAggregation, TechnicalNoise, ElectronsPerCount )

resolution = 0.61 * Wavelength / NumericalAperture;
airyDiskRadiusPixels = resolution * Magnification / PixelSize;

SimulatedImage = zeros( ImageSize );

beadCenter = rand( 1, 2 ) .* ImageSize;
intensity = ( 1 + 0.1 * randn() ) * Intensity * ( 1 + poissrnd(ProbabilityOfAggregation) );
SimulatedImage = SimulatedImage + ...
end

SimulatedImage = SimulatedImage + TechnicalNoise .* randn( ImageSize );
SimulatedImage = uint16( round( SimulatedImage ./ ElectronsPerCount ) );
end

function out = SimulateAiryDiskImage( AiryDiskCenter, AiryDiskRadius, ImageSize, Intensity )

yAxis = (1:ImageSize(1)) - AiryDiskCenter(1);
xAxis = (1:ImageSize(2)) - AiryDiskCenter(2);

[ xCoordinate, yCoordinate ] = meshgrid( xAxis, yAxis );

firstBesselZero = 3.8317;

out = Intensity .* ( 2 * besselj(1, normalizedRadius ) ./ ( normalizedRadius ) ) .^ 2;

% remove NaN from result if center is an integer
if( all(AiryDiskCenter == fix( AiryDiskCenter )) )
out( AiryDiskCenter(1), AiryDiskCenter(2) ) = Intensity;
end
end

```

### Testing the code

The script below generates synthetic images over a range of resolutions and compares them to the expected results.

```
naValues = 0.65:.025:1.25;
figureHandle = figure();

measuredResolution = zeros( 1, length(naValues) );
theoreticalResolution = zeros( 1, length(naValues) );
standardError = zeros( 1, length(naValues) );

for ii=1:length(naValues)
% generate synthetic PSF image
NumericalAperture = naValues(ii); %0.5; %65;
Magnification = 40;
Wavelength = 590e-9;    % m
ProbabilityOfAggregation = 0; %0.2;
PixelSize = 7.4e-6;     % m
ImageSize = [500 500];  % pixels x pixels
TechnicalNoise = 0;    % electrons per exposure
ElectronsPerCount = 2;
Intensity = 4095 * ElectronsPerCount * 0.8; % photons per exposure at peak

simulatedPsfImage = SimulatePsfSlide( ...
NumericalAperture, Magnification, Wavelength, ...
ProbabilityOfAggregation, TechnicalNoise, ElectronsPerCount );

% detect and simulate clipping due to overexposure
if( max( simulatedPsfImage(:) ) > 4095 )
disp('*** Overexposed Image ***');
simulatedPsfImage( simulatedPsfImage > 4095 ) = 4095;
end

simulatedPsfImage = im2double( simulatedPsfImage * 16 );

theoreticalResolution(ii) = 0.61 * Wavelength / NumericalAperture;

[ measuredResolution(ii), standardError(ii), bestFitData ] = ...
EstimateResolutionFromPsfImage( simulatedPsfImage );

measuredResolution(ii) = measuredResolution(ii) .* PixelSize / Magnification;

==Effect of finite bead size ==
Here is some code you can use to investigate the effect of finite bead size:
<pre>
close all
xAxis = -10:0.001:10;

% generate an Airy disk profile with a normalized resolution R=1
firstBesselZero = 3.8317;
airyFunction = (2 * besselj(1, firstBesselZero * xAxis)./ (firstBesselZero * xAxis)).^2;
airyFunction(xAxis == 0) = 1;

% use nonlinear regression to fit a Gaussian function to the Airy disk
gaussianFunction = @(parameters, xdata)  parameters(2) * exp( -xdata .^ 2 / (2 * parameters(1) .^ 2) );

bestFitGaussianParameters = nlinfit(xAxis, airyFunction, gaussianFunction, [1 1]);
lsqcurvefit(gaussianFunction, [1 1], xAxis, airyFunction);
bestFitGaussian = gaussianFunction(bestFitGaussianParameters, xAxis);
sigma = bestFitGaussianParameters(1);
sigmaToFwhmConversionFactor = 2 * sqrt(2*log(2));
fwhm = sigmaToFwhmConversionFactor * sigma;

% plot the results
plot(xAxis, airyFunction);
hold on;
plot(xAxis, bestFitGaussian, 'r');
title(texlabel(['Airy Function (R=1) versus Gaussian (sigma = ' num2str(sigma) ' FWHM = ' num2str(fwhm) ')']));

% generate a simulated bead with diameter = 0.33 * R

figure

hold on
plot(xAxis, airyFunction, 'r');