Following last week's impulse, here is a tutorial about the subtleties of multiscale particle localisation from confocal 3D images.

For reference, this has been published as

- A novel particle tracking method with individual particle size measurement and its application to ordering in glassy hard sphere colloids.
- Leocmach, M., & Tanaka, H.
- Soft Matter (2013), 9, 1447–1457.
- doi:10.1039/c2sm27107a
- ArXiV

# Particle localisation in a confocal image¶

First, import necessary library to load and display images

```
import numpy as np
from scipy.ndimage import gaussian_filter
from matplotlib import pyplot as plt
from colloids import track
%matplotlib inline
```

We also need to define functions to quickly display tracking results.

```
def draw_circles(xs, ys, rs, **kwargs):
for x,y,r in zip(xs,ys,rs):
circle = plt.Circle((x,y), radius=r, **kwargs)
plt.gca().add_patch(circle)
def display_cuts(imf, centers, X=30, Y=25, Z=30):
"""Draw three orthogonal cuts with corresponding centers"""
plt.subplot(1,3,1);
draw_circles(centers[:,0], centers[:,1], centers[:,-2], facecolor='none', edgecolor='g')
plt.imshow(imf[Z], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,2);
draw_circles(centers[:,0], centers[:,2], centers[:,-2], facecolor='none', edgecolor='g')
plt.imshow(imf[:,Y], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,3);
draw_circles(centers[:,1], centers[:,2], centers[:,-2], facecolor='none', edgecolor='g')
plt.imshow(imf[:,:,X], 'hot',vmin=0,vmax=255);
```

## Raw confocal image¶

Load the test data, which is only a detail of a full 3D stack. Display different cuts.

Note that raw confocal images are heavily distorted in the \(z\) direction. We actually have 4 particles in the image: two small on top of each other, one on the side and a slightly larger below and on the side.

```
imf = np.fromfile('../../../multiscale/test_input/Z_elong.raw', dtype=np.uint8).reshape((55,50,50))
plt.subplot(1,3,1).imshow(imf[30], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,2).imshow(imf[:,25], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,3).imshow(imf[:,:,30], 'hot',vmin=0,vmax=255);
```

Create a finder object of the same shape as the image. Seamlessly work with 3D images.

```
finder = track.MultiscaleBlobFinder(imf.shape, Octave0=False, nbOctaves=4)
```

Feed the image in the tracker with default tracking parameters. The output has four columns : \((x,y,z,r,i)\) where \(r\) is a first approximation of the radius of the particle (see below) and \(i\) is a measure of the brightness of the particle.

```
centers = finder(imf)
print centers.shape
print "smallest particle detected has a radius of %g px"%(centers[:,-2].min())
display_cuts(imf, centers)
```

This is a pretty bad job. Only two particle detected correctly. The top and side small particles are not detected.

Let us try not to remove overlapping.

```
centers = finder(imf, removeOverlap=False)
print centers.shape
print "smallest particle detected has a radius of %g px"%(centers[:,-2].min())
display_cuts(imf, centers)
```

Now we have all the particles.

We have two effects at play here that fool the overlap removal: - The radii guessed by the algorithm are too large. - The particles aligned along \(z\) are localised too close together.

Both effects are due to the elongation in \(z\).

## Simple deconvolution of a confocal image¶

The image \(y\) acquired by a microscope can be expressed as

\(y = x \star h + \epsilon,\)

where \(\star\) is the convolution operator, \(x\) is the perfect image, \(h\) is the point spread function (PSF) of the microscope and \(\epsilon\) is the noise independent of both \(x\) and \(h\). The process of estimating \(x\) from \(y\) and some theoretical or measured expression of \(h\) is called deconvolution. Deconvolution in the presence of noise is a difficult problem. Hopefully here we do not need to reconstruct the original image, but only our first Gaussian blurred version of it. Indeed, after a reasonable amount of blur in three dimensions, the noise can be neglected and we thus obtain:

\(y_0 \approx G_0 \star h,\)

or in Fourier space

\(\mathcal{F}[y_0] = \mathcal{F}[G_0] \times \mathcal{F}[h].\)

Once \(\mathcal{F}[h]\) is known the deconvolution reduces to a simple division in Fourier space.

### Measuring the kernel¶

\(\mathcal{F}[h]\) should be measured in an isotropic system and in the same conditions (magnification, number of pixels, indices of refraction, etc.) as the experiment of interest.

Since a 3D image large enough to be isotropic is too big to share in an example, I just use the image of interest. Don't do this at home.

```
imiso = np.fromfile('../../../multiscale/test_input/Z_elong.raw', dtype=np.uint8).reshape((55,50,50))
kernel = track.get_deconv_kernel(imiso)
plt.plot(kernel);
```

The `kernel`

variable here is actually \(1/\mathcal{F}[h]\)

### Kernel test¶

What not to do: deconvolve the unblurred image.

```
imd = track.deconvolve(imf, kernel)
plt.subplot(1,3,1).imshow(imd[30], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,2).imshow(imd[:,25], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,3).imshow(imd[:,:,30], 'hot',vmin=0,vmax=255);
```

Noisy isn't it? Yet we have saturated the color scale between 0 and 255, otherwise artifacts would be even more apparent.

What the program will actually do: deconvolved the blurred image.

```
imb = gaussian_filter(imf.astype(float), 1.6)
plt.subplot(1,3,1).imshow(imb[30], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,2).imshow(imb[:,25], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,3).imshow(imb[:,:,30], 'hot',vmin=0,vmax=255);
```

```
imbd = track.deconvolve(imb, kernel)
plt.subplot(1,3,1).imshow(imbd[30], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,2).imshow(imbd[:,25], 'hot',vmin=0,vmax=255);
plt.subplot(1,3,3).imshow(imbd[:,:,30], 'hot',vmin=0,vmax=255);
```

The way the noise is amplified by deconvolution is obvious when we compare the histograms of the 4 versions of the image.

```
for i, (l, im) in enumerate(zip(['raw', 'blurred', 'deconv', 'blur deconv'], [imf, imb, imd, imbd])):
plt.plot(
np.arange(-129,363,8),
np.histogram(im, np.arange(-129,371,8))[0]*10**i,
label=l+' (%d,%d)'%(im.min(), im.max()));
plt.xlim(-130,360);
plt.yscale('log');
plt.legend(loc='upper right');
```

Obviously negative values should be discarded, and will be discarded by the algorithm.

### Localisation using deconvolved image¶

Not removing the overlaping particles, we use the deconvolution kernel.

```
reload(track)
centers = finder(imf, removeOverlap=False, deconvKernel=kernel)
print centers.shape
print "smallest particle detected has a radius of %g px"%(centers[:,-2].min())
display_cuts(imf, centers)
```

All particles are found and actually they are not overlaping. However we got a spurious detection, probably some noise that got amplified by the deconvolution. It is easily removed by a threshold in intensity.

```
print centers
```

```
centers = centers[centers[:,-1]<-1]
display_cuts(imf, centers)
```

We can also display the results with respect to the blurred and deconvolved version of the image that we can find by introspection.

```
display_cuts(finder.octaves[1].layersG[0], centers)
```

## From scales to sizes¶

This is pretty much like in 2D.

```
from colloids.particles import get_bonds
s = track.radius2sigma(centers[:,-2], dim=3)
bonds, dists = get_bonds(positions=centers[:,:-2], radii=centers[:,-2], maxdist=3.0)
radii1 = track.global_rescale_intensity(s, bonds, dists, centers[:,-1])
centers2 = np.copy(centers)
centers2[:,-2] = radii1
display_cuts(imf, centers2)
```