# Calculating depth of field and defocus

For a thin lens, the defocus and depth of field can be calculated using the Lensmaker's Equation. This is quite a practical calculation these days, because with the advent of high quality plastic lens manufacturing for cell phones, the simple thin lens formulae are increasingly practical.

This script generates a series of graphs that illustrate the defocus in diopters assuming the sensor is either at the focal length of the lens, or further away. The graphs clarify how the depth of field varies as we change the pupil size.

Copyright ImagEval Consultants, LLC, 2003.

## Contents

```ieInit
```

## Use a simple default lens, as one might find in a cell phone camera

```optics  = opticsCreate;
fLength = opticsGet(optics,'focal length','m');
D0      = opticsGet(optics,'power');

% You could try running this with a different focal length
%   optics = opticsSet(optics,'focal length',0.03);
%   fLength = opticsGet(optics,'focal length','m');
%   D0      = opticsGet(optics,'power');

fprintf('Lens power %f and focal length %f (m)\n',D0,fLength);
```
```Lens power 258.882578 and focal length 0.003863 (m)
```

## Defocus of an object at different distances from the thin lens

```% Image distance is the distance from the center of the thin lens to the
% image.  Normally the sensor is placed in this image plane.
%

% Object distance is the distance to the object.
% Here are the object distance
nSteps = 500;
objDist = linspace(fLength*1.5,100*fLength,nSteps);

% Calculate defocus (dioptric error) and plot it relative to total lens
% power  (D0).
D  = opticsDepthDefocus(objDist,optics);

vcNewGraphWin;
semilogx(objDist/fLength,D/D0);
title('Sensor at focal length')
xlabel('Distance to object (units: focal length)');
ylabel('Relative dioptric error')
grid on
t = sprintf('Focal length %.1f (mm)',fLength*ieUnitScaleFactor('mm'));
legend(t);

% The significance of defocus in terms of blur of the image depends on the
% pupil radius, as well as the dioptric defocus.  We calculate this below
% in discussing depth of field and the Hopkins w20.
``` ## The image distance as the object distance increases

```% We use the opticsDepthDefocus to calculate the image distance
% for different object distances
[tmp, imgDist] = opticsDepthDefocus(objDist,optics);

vcNewGraphWin;
o = objDist*ieUnitScaleFactor('mm');
plot(o,imgDist*ieUnitScaleFactor('mm'));
xlabel('Distance to object (mm)'); ylabel('Image dist (mm)')
line([o(1) o(end)],[fLength fLength]*ieUnitScaleFactor('mm'),'Color','k','linestyle','--')
grid on
t = sprintf('Focal length %.1f (mm)',fLength*ieUnitScaleFactor('mm'));
legend(t)
``` ## Plot with respect to focal lengths

```vcNewGraphWin;
plot(objDist/fLength,imgDist/fLength);
xlabel('Distance to object (units: focal length)'); ylabel('Image dist (units: focal length)')
grid on
t = sprintf('Focal length %.1f (mm)',fLength*ieUnitScaleFactor('mm'));
legend(t)
``` ## Defocus with respect to image plane different from focal image plane

```% We arrange the distance between the lens and sensor to be a
% little longer than the focal length. In this case objects in
% good focus are closer than infinity.  We calculate the in-focus
% plane in both meters and focal lengths.
s = 1.1;
D = opticsDepthDefocus(objDist,optics,s*fLength);
[v,ii] = min(abs(D));

fprintf('Sensor plane is located %.3f focal lengths behind thin lens.\n',s)
fprintf('In focus object distance (m) from thin lens: %.3f.\n',objDist(ii));
fprintf('In focus object is %.3f focal lengths from thin lens.\n',objDist(ii)/fLength);

% The dioptric error as a function of object distance (in units of focal
% length
vcNewGraphWin;
semilogx(objDist/fLength,D);
xlabel('Object distance (focal lengths)');
ylabel('Dioptric error (1/m)')
grid on
t = sprintf('Focal length %.1f (mm)',fLength*ieUnitScaleFactor('mm'));
legend(t)
```
```Sensor plane is located 1.100 focal lengths behind thin lens.
In focus object distance (m) from thin lens: 0.042.
In focus object is 10.975 focal lengths from thin lens.
``` ## Calculate the OTF using Hopkins' method

```% The dioptric error depends both on the object distance and the pupil
% aperture.
p = opticsGet(optics,'pupil radius');

% The w20 parameter is used by Hopkins to predict the amount of
% defocus. Notice that the formula depends on both the relative
% defocus (D and D0) and the pupil size.
%
% For discussion about the Hopkins w20 measure used in the depth of field
% calculation, see humanCore.
%
s = [0.5 1.5 3];              % Scale factor times the pupil radius
leg = cell(1,length(s));      % Legend
w20 = zeros(length(D),length(s));
lType = {'k-','r--','b--','g-'};

% Graph of object distance versus defocus for different pupil
% sizes. The point is that as the pupil size changes, the defocus
% parameter, w20, also changes.
vcNewGraphWin;
for ii=1:length(s)
w20(:,ii) = (s(ii)*p)^2/2*(D0.*D)./(D0+D);  % See human Core
semilogx(objDist/fLength,w20(:,ii),lType{ii}); hold on;
leg{ii} = sprintf('%.2f (mm)',p*s(ii)*ieUnitScaleFactor('mm'));
end
hold off; grid on; legend(leg)
title('Depth of field')
xlabel('Object distance (re: focal length)');
ylabel('Hopkins w20 (defocus)');
``` 