WO2013054088A1

WO2013054088A1 - Improvements in localisation estimation

Info

Publication number: WO2013054088A1
Application number: PCT/GB2012/052435
Authority: WO
Inventors: Joshua David Larkin; Peter Richard COOK
Original assignee: Isis Innovation Limited
Priority date: 2011-10-10
Filing date: 2012-10-02
Publication date: 2013-04-18

Abstract

A method of position localisation applicable to super resolution microscopy, but also to other localisation problems, which comprises assigning a probability density function based on the uncertainty in detected position to each detection event. The probability density functions are probabilistically aggregated to produce a joint probability density function representing the likely location of the signal emitter.

Description

IMPROVEMENTS IN LOCALISATION ESTIMATION

The present invention relates to a method of localisation, i.e. position determination, and, in particular, a method which improves the speed of computation of the relative position of a signal source and a signal detector based on the detection of plural signals by the detector without losing accuracy.

One application of localisation is in the field of imaging where the aim is to detect the position of a light source, and in particular the field of localisation microscopy where images are built up by detecting photons from individual light sources such as fluorophores attached to individual molecules. In recent years localisation microscopy has been developed as one of the so-called "super-resolution" microscopy techniques which allow images to be created with a spatial resolution much smaller than the diffraction limit. In normal optical microscopy diffraction effects establish a lower limit on the resolution which can be achieved. As is well-known, the effect of diffraction is that a point light source in a sample will be represented in the image plane by an "Airy disc" of finite width, the intensity profile in the image plane being termed the point spread function (PSF) of the microscope. Normally the diffraction limit is taken as being the width of the point spread function which is about 250 nm for visible light. Single-molecule localisation microscopy overcomes this problem by labelling molecules of interest in the sample with emitters such as fluorophores and then stimulating only a sparse subset of the emitters at the same time (e.g. using a low power laser) while collecting the image using a detector, such as a CCD camera. The resulting images of the isolated fluorophors (which are distorted by the point spread function) are fitted with a Gaussian distribution allowing the probable location of the emitter to be calculated (for example at the maximum of the fitted distribution). The process of stimulation of the sample and fitting of the distribution to the collected pixel intensities is repeated several thousand times and the full image is then reconstructed by a computer. There are different super-resolution techniques based on single molecule localisation such as photo-activated localisation microscopy (PALM), stochastic optical reconstruction microscopy (STORM), and others, and with these techniques two-dimensional spatial resolutions of the order of 20- 30 nm can be achieved.

However a problem with these techniques is that the process of fitting the Gaussian distribution to the pixel intensities is computationally intensive. Typically the fitting process involves standard techniques such as minimising least-square distances or maximum likelihood estimation (MLE), but because these are iterative techniques, they are not fast. The problem of fast calculation of position is, of course, also found in other fields. For example in the global positioning system (GPS) the position of a GPS receiver is calculated by computing the differences in delay of digital time data transmitted from multiple satellites orbiting the earth, and converting these differences into a position on the earth's surface. These time-delay measurements have associated uncertainties such as atmospheric and multi-path effects which cannot be corrected for and which determine the accuracy with which the position of the receiver can be calculated. Accuracy also depends on the fidelity of the mathematical model used to process the measurements and estimate the final uncertainty. Contemporary methods use a Kalman filter or a least-squares estimate, but again these techniques can be iterative and computationally intensive. Similar problems arise with localisation based on a (non-GPS) receiver measuring the amplitude of signals received from different antennas.

There is therefore a need for a method of position localisation based on the detection of multiple signals either from a signal source or from multiple sources detected by a single receiver, which allows for the uncertainty in the individual measurements, but which is computationally efficient and thus fast.

Accordingly the present invention provides a method of estimating the relative location of a signal source and a signal detector, the method comprising the steps of:

detecting each of a plurality of signals from the signal source;

measuring each of the plurality of received signals to a predefined resolution corresponding to detection intervals by recording in which detection interval each received signal is detected;

for each detection of a received signal assigning a respective probability density function (PDF) positioned within the detection interval in which each received signal is detected; and

aggregating the probability density functions to derive a joint probability density function representing the probable relative location of the signal source and detector.

The detection intervals correspond to the quantised steps in the detection process. Thus where the detector is an image detector the detection intervals are the pixels. Where the signals are time signals as in a GPS system the detector intervals correspond to the resolution of the time signals. Where the amplitude of the signals is being measured the detector intervals correspond to the smallest amplitude that can be resolved. In each case the true value might not be in the centre of the relevant detector interval, or even in that interval at all, and this possibility is represented in the invention by assigning a PDF at some position within the detector interval, e.g. with the centre of the PDF at the centre of the detection interval, with the PDF having a width based on the likely error. The PDF may well extend across several adjacent detector intervals. In some cases the PDF will be centered on the centre of the detection interval but it may be that in some circumstances it is appropriate to offset it. For instance, when neighbouring pixels indicate that the PDF should be nearer a pixel edge in the case of imaging, or if knowledge about signal time lag dictates that the PDF is more likely to be centred at the leading/trailing edge of a time signal.

Thus rather than fitting a single probabilistic distribution to all the received signals in detector space (for instance across the pixels in an imaging detector, or over the different times in a time detector, or over the different amplitudes in an amplitude detector), an individual probability density function is assigned to each signal detection and then the individual probability density functions are probabilistically aggregated to construct a joint probability density function which directly represents the spatial or temporal emitter location. Because this does not involve a fitting process, it is not iterative and is thus computationally fast.

As well as being fast, the method also offers increased accuracy for low signal strengths, e.g. low photon counts in microscopy, where conventional techniques such a minimising least-square distances and maximum likelihood estimation struggle.

The probability density function is obtained or calculated in advance based on the characteristics of the detector and the medium through which the signal passes and can, for example, be a normal or Gaussian distribution with an arbitrary amplitude and a width based on the measured or estimated characteristics of the detector and medium.

In an imaging application the detection intervals are pixels of the detector and the signals are light. As applied to localisation microscopy the signals can be individual photons and the probability density function is the point spread function of the microscope.

In the case where the point spread function of the microscope is the same for each received signal, the probability density function has the same amplitude and width for each received photon. Consequently the step of aggregating the probability density functions amounts simply to finding the mean of photon positions, or put another way the mean of the centre positions of the pixels which have received a photon, weighted by the number of photons that pixel has received.

Rather than centering the PDFs on the centre of the detector interval, e.g. pixel, they can be shifted from the centre. In one embodiment the PDFs assigned to detector intervals surrounding a particular detector interval with the strongest signal are shifted by an amount, e.g. a fraction of a detector interval width, towards the detection interval with the strongest signal. Thus in the case of imaging, PDFs for pixels surrounding the brightest pixel will be shifted towards the brightest pixel by, for example, half or one quarter of a pixel width.

Similarly the PDFs assigned to the detection interval with the strongest signal can be shifted towards neighbouring detection intervals in proportion to the signal strength in those neighbouring detection intervals. Thus, again in the case of imaging, the PDFs in the brightest pixel would be shifted away from the centre towards neighbouring pixels by an amount depending on (e.g. proportional to) the intensity in the neighbouring pixels.

The width of the PDFs can also be varied, for example depending on distance from the detection interval with the strongest signal (e.g. the brightest pixel). Thus PDFs assigned to detection intervals far from the brightest pixel can be given a much greater width (e.g. proportional to that distance).

Both the width of the PDF and its position within the detection interval can be varied by combining these techniques.

In the case of imaging the process can be executed in one, two or three dimensions

(localisation microscopy can return the depth (z axis) position as well as lateral (x and y axis) position).

The method can include a step of removing background. When imaging a sample photons may be emitted from positions other than the emitter of interest and detectors also create noise. One way of removing background is to identify some part of the image which has no source of interest (for example at the periphery), and to set a background threshold based on the signal intensity there. This threshold can then be removed from the whole image.

An alternative background removal technique is to adjust the width of the probability density function based on the difference between the local pixel intensity and a determined background level. This effectively negates the effect of those pixels whose intensity is close to the background level.

Thus one aspect of the invention, applied to microscopy, provides a method of super- resolution localisation microscopy in which a probability density function is assigned to each photon received in a detector pixel, positioned within the pixel and having a width equal to the point spread function of the microscope, and the probability density functions are aggregated to calculate a joint probability density function representing an estimate of the location of the source of the photons. In the application of the invention to a global positioning system (GPS) receiver, the GPS satellites constitute signal sources and the receiver forms the detector with the detection interval being the time resolution in the GPS time signals.

The localisation method of the invention may also be applied to the problem of location of a receiver such as a mobile telephone (without GPS) whose position is determined by the relative amplitudes of electro-magnetic signals it receives from different antennas. The detection interval is then the resolution with which the amplitude is measured, and the net uncertainty for each amplitude measurement is due to multiple errors (a, b, etc.) such as digitization, signal integrity/quality, atmospheric conditions, electro -magnetic interference etc.

The invention will be further described by way of example with reference to the accompanying drawings in which:-

Figure 1 schematically illustrates a localisation microscopy system to which the invention can be applied;

Figure 2 show example localisation microscopy images;

Figure 3 schematically compares the approach of the invention and the prior art localisation microscopy methods;

Figure 4 illustrates the prior art fitting of a distribution to pixel intensities in localisation microscopy and the assignment of probability density functions in accordance with an embodiment of the present invention;

Figure 5 schematically compares the image processing of an embodiment of the present invention to the image processing of the prior art;

Figures 6 A and B are flow diagrams of alternative embodiments of the present invention applied to localisation microscopy;

Figure 7 schematically illustrates a GPS system to which the present invention can be applied; and

Figure 8 schematically illustrates the effects of tuning the position and width of the probability distributions with distance from the brightest pixel.

Figure 1 illustrates schematically a super resolution microscopy system. The sample 1, for example a biological sample labelled with fluorescent emitters, is illuminated by stimulation light 2, and is viewed through a microscope 3 equipped with a CCD camera 5. The signals from the CCD camera 5 are processed by a data processor 7 and can be stored in the data store 9 and displayed on the display 11. To optimise the balance between spatial resolution and signal-to-noise ratio, the pixel size in the CCD camera and microscope optics are chosen so that 2.8 times the linear dimension of a pixel is equal to the resolution limit of the microscope (which depends on the numerical aperture of the objective and the

wavelength of light emitted by the fluorophor). In a typical single molecule or

PALM/STORM experiment an image is acquired by collecting photons from temporally and spatially isolated photon emitters in the sample by integrating the number of photons incident on each pixel in the CCD camera. This is analogous to binning data into a histogram and thus loses any sub-pixel spatial information.

Figure 2 illustrates typical images from such a CCD. In the conventional techniques, given a pixel that has registered one photon, the assumption is made that the uncertainty (or probability density) of the location of the source of that photon is uniformly flat over the area of the pixel which registered the photon, and uniformly zero elsewhere. This is illustrated in Figure 3 as the "top hat" bar with a probability of 1 that the emitter location is within the area of the pixel and a probability of zero that it is elsewhere. After several photons have been registered by the CCD camera, the sum of these "top hat" distributions effectively provide a bar chart as illustrated in Figure 4 representing pixel intensities and so in the conventional techniques a Gaussian distribution indicated by the heavy line in Figure 4 is fitted to the histogram of pixel intensities and then the probable location of the emitter is calculated from this fitted distribution (the most probable location being at the peak of the distribution illustrated as μο in Figure 4). However as mentioned above this fitting process is

computationally intensive.

With the present invention instead of using a top hat probability distribution for each pixel, a probability density function which emulates the point spread function of the microscope is assigned to the centre of each pixel which detects a photon. This is illustrated for one photon in Figure 3. As can be seen in Figure 3 this probability density function spreads well beyond the limits of the pixel itself. Because the photon originally fell somewhere in the pixel, but we do not know where, the distribution is centred on the middle of the pixel. This introduces a small error, but this can be minimised by using smaller pixels. In this embodiment a normal distribution is used for the probability density function, this being the usual representation of a point spread function of a microscope.

This same assignment of a probability density function is repeated for each detected photon. In consequence rather than a histogram type representation of pixel intensities, a set of probability density functions is assembled, each PDF being centred on the pixel which detected the corresponding photon. Such a set of PDFs is illustrated in Figure 4. These PDFs can be probabilistically aggregated as explained below to produce a joint probability density function from which the most probable location of the emitter can be calculated. This is indicated by the light line in Figure 4 and it can be seen that in this example it is centred at the same position as the Gaussian distribution fitted to the histogram of pixel intensities. Thus both techniques give the same result for the most likely emitter position. However the probabilistic aggregation of the probability density functions is much quicker to compute as explained below.

Figure 5 illustrates the two different processing routes, of the prior art on the left and the present invention on the right. As illustrated both produce the same estimate of emitter location, but in the prior art route this is achieved by fitting a Gaussian distribution to a histogram of intensities, whereas in the present invention it is by aggregating probability density functions centred on each pixel which detected a photon.

In this embodiment the uncertainty in photon position is represented as a normal distribution and each photon distribution is independent from all other photon positions. This means that the joint probability density function can be represented as a product of individual PDFs as follows:

in one dimension, x, where μ; is the centre position of a photon, i, and σ; is the uncertainty of the photon position. Ignoring constant terms for simplicity (because they only affect the amplitude of the function, which is not of immediate interest here), we get:

Next we define:

so that

Factoring and, once again, ign ring constant terms

After replacing ki and k₂

The equation resembles a simple Gaussian distribution:

^{1 1} I \

G(x^₀ , cy₀ ) = e (?)

whose centre is equal to the centre of the joint distribution:

In a very simplified embodiment an assumption can be made that σ; = σ (all photon position uncertainties are equal) and in this case the location of the emitter adopts a more intuitive form as the mean of the individual photon positions:

In this simplified case the emitter position, represented by Equation 9, is obtained by taking the mean of the individual photon positions. Put another way, it is the mean of the pixel centres which detected a photon, weighted by the number of photons they detected. This is a simple calculation and does not require iterative processing, unlike the fitting of the Gaussian distribution of the prior art. However in general the values of σ; are not all set to the same value. Nevertheless the calculation of the centre of the joint distribution according to equation 8 is still quickly and efficiently computable because it is not an iterative process. Returning to the Gaussian distri tion, the width of the joint distribution is:

therefore,

This, too, adopts a familiar form when σ; = σ:

With the present invention, therefore, the point spread function of the microscope serves as an estimate of the uncertainty imparted on the position of every photon by the microscope, a probability based on the point spread function is assigned to each individual photon, and then these probabilities are aggregated in such a way that the product is also a probability. The result is a distribution that describes the probability of the location of the emitter.

In contrast, prior art fitting methods, for example using minimising least-square distances or maximum likelihood estimation, describe the probability of emitter location by first fitting a curve to the pixel intensities, then deducing location information from that curve.

Figure 6A illustrates the steps in one version of the method as carried out under the control of the processor 7. In step SI the sample 1 containing fluorophores is stimulated by stimulation light 2 and in step S2 the image is acquired by the CCD camera 5. As is typical in single molecule localisation microscopy the stimulation is at a low level in order that only a sparse set of emitters in the sample are stimulated. Therefore the steps of low level stimulation and image acquisition are repeated multiple times until a full image has been acquired. If in step S3 the required number of acquisitions have been completed, then in step S4 the number of photons detected by each pixel is estimated. Any of the existing techniques for estimating from the pixel intensity the number of photons which hit the pixel can be used, such as disclosed in [Thompson, R.E., Larson, D.R. & Webb, W.W. Precise Nanometer Localization Analysis for Individual Fluorescent Probes. Biophys. J. 82, 2775-2783 (2002)] and [Pawley, J.B. Points, Pixels, and Gray Levels: Digitizing Image Data. Handbook of

Biological Confocal Microscopy (2006)]. In the latter example, photon count is simply estimated by dividing pixel intensity by the gain applied by amplifiers in the CCD.

In step S5 background noise is removed. Again, there are a variety of known techniques for removing background noise, for example the perimeter of the image can be examined and a background threshold decided from that area, with this threshold then being removed from all other pixels.

Alternatively a probabilistic technique can be used in which background photos are included directly in the calculation of emitter location in steps S6 and S7, but with an uncertainty based on their intensity. In more detail photons which are definitely derived from an emitter (because they are found together in a cluster of pixels) are assigned a probability density function equal to the point spread function of the microscope just as explained above. Photons which can definitely be said to be background (because they are geometrically- isolated e.g. from a position >3 σ from the brightest pixel) or whose intensity is less than median intensity, are assigned a probability density function with infinite width (σ =∞). This effectively means that the influence of that particular photon on the joint distribution becomes zero. In between these two extremes the width σ of the distribution is weighted depending on the intensity of the pixel the photon came from compared with the median intensity of pixels proximal to the emitter. Thus:

[^a psF + <5 _PSF (I I ^median{I)) otherwise where / is the intensity of the pixel the photon came from, aps_F is the width of the PSF, R is proportional to the signal to noise ratio of the image (which dictates the slope between the PSF and infinity) and was heuristically set as lOOOx the signal-to-noise ratio in this embodiment, and median is the background level below which photons are deemed to be noise.

In step S6 a PDF is assigned to each photon (either the photons remaining after the removal of background or after setting the width of the PDFs to take account of whether the photon is regarded as background or not), with these probability density functions being centred on the centre of each pixel. In step S7 the probability density functions are aggregated to produce a probability density function for the emitter location. Having calculated the emitter locations, the processor can construct and display an image in steps S8 and S9.

Figure 6B illustrates an alternative flow of steps, which nevertheless reach the same end. In step S2' an individual photon position is acquired by the CCD detector and a PDF is immediately assigned to that pixel centre in step S6'. This is repeated until the full image has been acquired. In step S5' the background can be removed, unless it has already been accounted for by setting the widths of the PDFs as mentioned above (which would occur in the assignment of each PDF in step S6'). The PDFs are then aggregated and an image is formed and displayed as before. In the embodiment described above the pixels had, as conventional, optional dimensions for a microscope with a 250 nm full width half-maximum (FWHM) point spread function to meet the Nyquist criterion applied to the full- width at half-maximum of the PSF divided by 2.8. However with the present invention as each photon is treated as if recorded at the centre of a pixel, it becomes advantageous to use smaller pixels, with shorter centre-to- centre distance.

The mathematical treatment above represented probable emitter locations in one dimension given photon positions in one dimension. However the method is simply applicable to two and three dimensions by projecting the photon counts onto the different axes and repeating the method separately for each Cartesian component. This method retains the computational simplicity of the invention, and works for radially asymmetric geometries (as geometries with rotational symmetry have equal centres).

In the embodiment above the probability density functions were positioned in the centre of the pixels and the photon position uncertainties σ; were not varied. However in more sophisticated versions of the invention either or both of these can be tuned

independently to optimise the precision and/or speed achieved at a given signal-to-noise ratio. In this case, to maximise localisation precision in noisy images, the values of μ; and σ; can be varied depending on distance from (and position relative to) the brightest pixel.

Firstly, as illustrated in Figure 8A the left hand side illustrates two photon distributions each positioned at the centre of a pixel (arrowed) to either side of the centre bright pixel. However in accordance with this modification to the technique, shown on the r hand side, these two distributions are shifted towards the position of the brightest pixel (towards the centre in Figure 8 A) by a distance of between 0.2 and 1 pixels width. The illustration is of a half pixel width shift (solid lines compared to the original position dotted). Furthermore, distributions from the brightest pixel are also shifted from the central default location by a distance proportional to the intensities of adjacent pixels.

In more detail, and remembering that the number of distributions attributed to each pixel is based on the CCD intensity for that pixel, the value of μ; for each distribution is (in one dimension) as follows:

where μ_£ is the location of the pixel centre, dj is the distance between the pixel and the brightest pixel, S₀ is the pixel width, and C is an arbitrary scaling constant (in Figure 3 A C = 2) . The shift of the distributions from the brightest pixel by a distance proportional to the intensities of the adjacent pixels can be expressed (in one dimension) as:

where Io is the intensity of the brightest pixel and IR and are the intensities of the adjacent pixels to the right and left respectively. A similar shift can be applied in the Y- dimension relative to the pixel intensities above and below.

This modification of shifting the position of the distributions can achieve

approximately a 5% reduction in error for low signal-to-noise ratios.

It is also possible to tune the width of the distribution σ,. As an emitter is most likely to be positioned in the brightest pixel, the distributions from other pixels can be expanded (i.e. Oj increased), with the distributions being progressively more expanded as they lie further away from the brightest pixel. Figure 8B illustrates this for two example

distributions. On the left hand side the two distributions are given the same width, whereas on the right hand side the distribution which is furthest away from the bright pixel is expanded (which in 2D makes it look more diffuse). In one dimension this can be expressed as:

where OPSF is the width of the point spread function in terms of σ, /UMAX is the centre location of the brightest pixel, 2.5 is the scaling factor chosen such that σ, = OPSF for the maximum pixel, and G(x) is a piecewise Gaussian distribution with a flat top:

These expressions ensure that the distributions of photons from, and adjacent to, the brightest pixel are equal to the point spread function whereas those from surrounding pixels become progressively wider the further away the pixels are, and those from distant pixels (i.e. > 3σ PSF) become infinitely wide. The tuning of σ in this way can yield up to 36% less error than other techniques at lower signal to noise ratios.

Both strategies of tuning μ and σ can be combined. This can further improve the performance by reducing error, especially at low signal-to-noise ratios, compared to other techniques.

Additional tuning of μ and σ, for example as functions of pixel intensity, and optimisation of the rate at which the values change with distance are also possible.

The embodiments above applies the invention to super resolution microscopy.

However the principle of assigning a PDF to each detection event, centred within the smallest division in detector space is applicable in other fields of localisation as well. For example in the global positioning system (GPS), as illustrated in Figure 7, the position of a GPS receiver 70 on the earth's surface is determined by computing differences in delay of digital time data t_a, tb, t_c transmitted from multiple satellites 72, 73 and 74, and converting those differences into a position on the earth's surface. These time delay measurements have associated uncertainties - residual uncertainties that cannot be corrected for, such as atmospheric and multi path effects, but can be accounted for as was done for the point spread function in the imaging application described above - that determine the accuracy of calculating the position of the receiver.

Receiver accuracy further depends on the fidelity of the mathematical model used to process the measurements and estimate the final uncertainty. Contemporary methods used a Kalman filter, or the more conventional least-squares estimate, to produce a mathematical model. These techniques can be iterative and, therefore, computationally intensive.

Applying the method of the invention yields a direct computation resulting in a reduction in computational time required to obtain receiver position and accuracy.

Thus to apply the principle of the present invention the magnitude of the error from each error source is estimated, either theoretically or numerically. If all error sources behave normally (as a Gaussian distribution) then the equations derived above for microscopy are directly applicable. Each error source is represented by a probability distribution describing the true time delay distributed over temporal space. The width of each distribution corresponds to the magnitude of the individual error source. All error sources from each time point and/or multiple time points are aggregated using Equations 8 and 12 above to compute a net probability describing the actual time delay and thus allowing the calculation of the position of the receiver 70. Thus, in an example embodiment equation 8 is modified such that μ₍ is the time measurement at time t detected by the receiver, where t = 1, 2, Nare individual time measurements received sequentially from the same satellite and/or in parallel from different satellites, and o_t is the net uncertainty of the time measurement t, and the position of the receiver is

The net uncertainty for each time point can be comprised of multiple sources of error (a, b, n) - including atmospheric conditions digitization, etc. - such that

Similarly, it follows from equation 11 that the uncertainty of the position of the receiver is given by

A third embodiment of the invention applies the localisation method of the invention to the problem of location of a receiver such as a mobile telephone (without GPS) whose position is determined by the relative amplitudes of electro-magnetic signals it receives from different antennas. The same equations that apply for GPS apply here, but t is replaced by another variable (s) that describes discrete samples of electro-magnetic signal amplitude, and the net uncertainty for each amplitude measurement is due to multiple errors (a, b, etc.) such as digitization, signal integrity/quality, atmospheric conditions, electro -magnetic interference etc. In more detail therefore equation 8 is modified such that μ₅ are the amplitude measurements made by the receiver, where s = 1, 2, N, and a_s is the net uncertainty of the amplitude measurement, and the p ition of the receiver is

The net uncertainty for each amplitude can comprise multiple sources of error (a, b, n) including digitization, signal integrity/quality, atmospheric conditions, electro-magnetic interference, etc - such that (18)

Claims

1. A method of estimating the relative location of a signal source or a signal detector, the method comprising the steps of:

detecting each of a plurality of signals from the signal source;

for each detection of a received signal assigning a respective probability density function whose shape describes the location of the source based on that received signal alone, with its centre within the detection interval in which each received signal is detected; and aggregating the probability density functions to derive a joint probability density function representing the probable relative location of the signal source and detector.

2. A method according to claim 1 wherein the probability density function is calculated or measured based on the characteristics of the detector.

3. A method according to claim 1 or 2 wherein the step of aggregating the probability density functions comprises calculating the mean of centre of the detection intervals in which signals are received, weighted by the number of signals received in that interval.

4. A method according to any one of the preceding claims wherein the detection intervals are pixels of an image detector.

5. A method according to any one of the preceding claims wherein the localisation is executed in one, two or three spatial dimensions.

6. A method according to any one of the preceding claims wherein the signals are photons.

7. A method according to any one of the preceding claims wherein the detector is an image detector mounted on a microscope for super resolution microscopy.

8. A method according to claim 7 wherein the probability density function is the point spread function of the microscope.

9. A method according to claim 7 wherein the probability density function is a Gaussian distribution.

10. A method according to any one of the preceding claims further comprising the step of eliminating background by varying the width of the probability density function according to the difference between the signal received at each detection interval and the median signal.

11. A method according to any one of the preceding claims wherein the width of the probability density functions are varied in dependence upon at least one of: (a) distance from the detection interval with the strongest signal, or (b) signal strength in the detection interval.

12. A method according to any one of the preceding claims wherein the respective probability density function assigned to a received signal is centered at the centre the corresponding detection interval.

13. A method according to any one of claims 1 to 11 wherein the assigned probability density functions are shifted from the centre of the detection interval towards the detection interval with the strongest signal.

14. A method according to any one of claims 1 to 11 or 13 wherein the assigned probability density functions in the detector interval with the strongest signal are shifted from the centre of the detection interval by a distance dependent upon the signal strength in the neighbouring detection intervals.

15. A method according to claim 1, 2 or 3 applied to the global positioning system wherein the signal source is a plurality of GPS satellites and the signals are time signals and detector intervals correspond to the temporal resolution of the time signals.

16. A method according to claim 1, 2 or 3 applied to the localisation of a receiver of radio waves wherein the signal source comprises a plurality of radio transmitters emitting signals whose amplitude is measured at the receiver and detection intervals correspond to the amplitude resolution of the measurements at the detector.