Chapter 8 – Theoretical Biophysics  371

regions of these outputs and then reconstructing a new image using the inverse WT in the

same way as filtering of the DFT image output. Similarly various biological features can be

identified such as cellular boundaries (which can be used for image segmentation) and in

general image class recognition. PCA is ultimately more versatile in that the user has the

choice to vary the number of eigenvectors to represent a set of images depending on what

image features are to be detected; however, WT recognition methods are generally compu­

tationally more efficient and faster (e.g., “smart” missile technology for military applications

often utilize WT methods). A useful compromise can often be made by generating hybrid

WT/​PCA methods.

8.5.2  PARTICLE TRACKING AND MOLECULAR STOICHIOMETRY TOOLS

As discussed previously, localization microscopy methods can estimate the position of the

intensity centroid of a point spread function image (typically a fluorescent spot of a few

hundred nanometers of width) from a single dye molecule imaged using diffraction-​limited

optics to pinpoint the location of the dye to a precision, which is one to two orders of mag­

nitude smaller than the standard optical resolution limit (see Chapter 4). The way this is

achieved in practice computationally is first to identify the candidate spots automatically

using basic image processing as the previously to determine suitable hotspots in each image

and second to define a small region of interest around each candidate spot, typically a square

of 10–​20 pixels edge length. The pixel intensity values in this region can then be fitted by an

appropriate function, usually a 2D Gaussian as an approximation to the center of the Airy

ring diffraction pattern, generating not only an estimate of the intensity centroid but also the

total brightness I(t) of the spot at a time t and of the intensity of the background in image in

the immediate vicinity of each spot.

Spots in subsequent image frames may be linked provided that they satisfy certain criteria

of being within a certain range of size and brightness of a given spot in a previous image

frame and that their intensity centroids are separated by less than a preset threshold often set

close to the optical resolution limit. If spots in two or more consecutive image frames satisfy

these criteria, then they can be linked computationally to form a particle track.

The spatial variation of tracks with respect to time can be analyzed to generate diffusion

properties as described earlier. But also the spot brightness values can be used to determine

how many dye molecules are present in any given spot. This is important because many

molecular complexes in biological systems have a modular architecture, meaning that they

consist of multiple repeats of the same basic subunits. Thus, by measuring spot brightness,

we can quantify the molecular stoichiometry subunits, depending upon what component a

given dye molecule is labeling.

As a fluorescent-​labeled sample is illuminated, it will undergo stochastic photobleaching.

For simple photobleaching processes, the “on” time of a single dye molecule has an expo­

nential probability distribution with a mean on time of tb. This means that a molecular com­

plex consisting of several such dye tags will photobleach as a function of time as ~exp(−t/​tb)

assuming no cooperative effects between the dye tags, such as quenching. Thus, each track

I(t) detected from the image data can be fitted using the following function:

(8.108)

I t

I

t

tb

( ) =

−



^



^

0^exp

which can be compared with Equation 3.32 and used to determine the initial intensity I0 at

which all dye molecules are putatively photoactive. If ID is the intensity of a single dye mol­

ecule under the same imaging conditions, then the stoichiometry S is estimated as

(8.109)

S

I

ID

=

0