Spot detection and measurements

Note

New for version 0.4.4. More documentation planned.

Deconwolf includes a few option to detect diffraction limited dots (or spots, or signals) via the module dw dots. This documentation gives and overview of what the program does. Please see the man page for up to date usage and exact command line arguments.

A tiny light emitter on a microscope slide will be seen as a blurry spot under a microscope. The optical parameters decide how small it can be. If it is as small as possible it will be denotes as a diffraction limited spot.

The dot extraction pipeline consists of:

1. Detection spot candidates, using DoG or just by intensity combined with a local maxima finder. A big part of this is selecting a suitable filter to use.

2. Sub pixel localization using 3D gaussian blobs as a template.

3. Extra measurements. Most of the features that are measured are side-results of the fitting routine.

4. Exports to disk.

5. Filtering. This step is up to you to implement in you favorite programming language.

Spot size

For wide field microscopy the optical parameters gives the spot size. The Abbe resolution provides the full width half max (fwhm) in the lateral plane (l) and along the axial direction (a). If $\mathrm{\Delta }x$ is the axial pixel sizes and $\mathrm{\Delta }z$ is the lateral pixel size, then:

$$\begin{array}{r}\begin{array}{c}{\text{fwhm}}_l=\frac{1}{\mathrm{\Delta }x}\frac{\lambda }{2\text{NA}}\\ {\text{fwhm}}_a=\frac{1}{\mathrm{\Delta }z}\frac{2\lambda }{(\text{NA}{)}^2}\end{array}\end{array}$$

And the equivalent size of a Gaussian bell is:

$$\begin{array}{r}\begin{array}{c}{\sigma }_l=\frac{fwh{m}_l}{2\sqrt{2\log (2)}}\\ {\sigma }_a=\frac{fwh{m}_a}{2\sqrt{2\log (2)}}\end{array}\end{array}$$

which is used for the fitting. For the LoG filter the optimal size is given by:

$$\begin{array}{r}\begin{array}{c}{\sigma }_{l,LoG}={\sigma }_l\sqrt{2}\\ {\sigma }_{a,LoG}={\sigma }_a\sqrt{2}\end{array}\end{array}$$

These values are calculated automatically by dw dots if the relevant optical parameters are supplied. Alternatively they can also be set manually.

Dot candidates

In an fictional world where optics is perfect and there are no noise sources, we could use the pixel values (after deconvolution) as direct measurements on the number of photons.

In practice one could detect dots before deconvolution or after deconvolution. If the images are not deconvolved, or only lightly deconvolved there will be blur left in the image (due to the PSF). In that case the pixel values are not very good estimators of the signal brightness.

A common proxy for signal strength is the Laplacian of Gaussian (LoG) feature. Intuitively the features is calculated by the difference between what is in the centre of the filter (signal) to what is in the surrounding pixels (background). I.e., it is a measure of the elevation of the spot relative to the surroundings.

If there is a spot with a size that somewhat matches the LoG filter it can be located as an extrema in the LoG filtered image.

Please note that spot finding method LoG+M will detect spots of any size (although with decreasing amplitude as the spots size deviates from the design size) and that it will also pick up noise and non-dot structures in the image. Hence it is more or less mandatory to filter the candidate spots by a few image features later on.

Fitting dots

Fitting is performed using a multivariate Gaussian as the model, restricting the shape to covariance matrices of the form:

$$\begin{array}{r}\mathrm{\Sigma }=\left(\begin{array}{ccc}{\sigma }_l^2& 0& 0\\ 0& {\sigma }_l^2& 0\\ 0& 0& {\sigma }_a^2\end{array}\right)\end{array}$$

The spot model is written as:

$$M(x)=c_0+c_{1}G(x;\mu ,\mathrm{\Sigma })$$

If we collect all the parameters that we optimize over into $\mathrm{\Theta }$, then we have:

$$\mathrm{\Theta }=\arg \min -\sum (I(x)\log M(x)-M(x))$$

Using the assumption of Poisson noise only.

Measurements

The possible column in the output are as follows:

• x, y, z the coordinates of the spot before fitting. Note that this is 0-indexed and given in the units of pixels.

• value the filter value, i.e. the LoG value or the pixel value depending on what feature that was used to identify the dot candidates.

• use A simple (stupid) classification of each dot to say if it is relevant or not. This is based on an Otsu threshold on the log(value)

• lat_circularity the largest over the smallest eigenvalue of the covariance matrix of the local pixels.

Properties returned from the fitting routine:

• f_bg the background level of the spot.

• f_signal_count the total number of photons in the spot (assuming a quantum efficiency of 1).

• f_peak_signal the largest value of the fitted spot (note: can be higher than the maximum pixel value).

• f_x, f_y, f_z the fitted sup pixel location of the spot.

• f_sigma_lateral, f_sigma_axial the size, i.e., the estimated sigma value for the gaussian in the lateral plane and along the axial direction.

• f_status the status value returned from the fitting function.

• f_error $-log(p)$ of the spot.

• f_corr the correlation between the pixel values and the fitted model. This information is really useful to reject spot candidates that are found on line- and edge- like structures in images.