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edfas.org ELECTRONIC DEV ICE FA I LURE ANALYSIS | VOLUME 23 NO . 3 28 was used for both time and space dimensions which led to difficulties in choosing it and possibly not optimum results. Some variants of DBSCAN such as ST-DBSCAN, [15] allow users to choose different clustering parameters for space and time, providing more consistent outputs. Hence, we will use this algorithm in this study. As a brief summary, ST-DBSCAN requires three input parameters to compute clusters: two maximum distance parameters to search for neighbors that can forma cluster ( ε XY and ε T ) and a minimum number of points, MinPts, to start forming a cluster. Points that have fewer neighbors than the MinPts parameter and do not belong to any cluster are labelled as noise and are discarded. Readers can refer toM. Ester et al. [14] andD. Briant [15] formoredetails about the algorithms. After clusters have been computed, a probability density function (pdf) in the ( x,y ) plane can be estimated for each cluster. Assuming the photon emission source has dimensions below the optical resolution limits of the microscope, we approximate the emission profile by a 2D Gaussian function. This ismotivatedby the fact that actual image of a point source is supposed to be an Airy disk; however, a 2D Gaussian function should give a relatively good approximation for this image due to all the various errors introduced along the measurement chain. In other words, it can be assumed that photons distribution in the ( x,y ) plane belong to a normal/Gaussian distribution. The pdf of multivariate normal distribution is defined by: (Eq 1) where µ designates the vector of the coordinates of the center of the distribution and Σ is the variance-covariance matrix. The amplitude parameter A is defined as: (Eq 2) where the operator det () designates the determinant of the matrix. The process is summarized in Fig. 2. From a cluster of photons (a), two approaches can be considered. The first one involves generating the emission spots image by counting the number of photons per pixel, (b) and (c). This is what is done in standard dynamic photon emission image generation process. The alternative, suggested in this paper, is to compute a probability density function (pdf) for each cluster and then sum all the pdfs to gener- ate a higher resolution image of photon emission profile (d) and (e). In this process, precision used to compute the pdf defines the image resolution. PROOF OF CONCEPT DETECTION SCHEME All acquisitions reported in this sectionwereperformed using a customized setup of the Semicaps SOM1100. Photons are detected by a free running single photon sensor (ID230 from ID Quantique [16] ) connected to one of the fiber ports normally used for laser scanning. The scanner position defines the ( x,y ) locations in pixelswhere the photons are recorded. Use of the scanner allows multiple choices of image resolution or zoom factor by changing the appropriate set of parameters in the scanner control UI. The typical measurement scheme for time-correlated single photon counting applications includes a counting module which records the time of arrival of photons with regard to a reference signal (e.g., test loop trigger in IC analysis). For this study, such a module was not available and thus an oscilloscope (LeCroy WaveMaster Zi-808B) was used to register the time of arrival of each photon. A synchronization signal was connected to both the scanner of the scanning optical microscopes (SOM) and the oscilloscope, to record the photon ( x,y ) location. The oscilloscope was set to trigger on the detection pulse generated by the sensor instead of the time reference signal. This approach minimizes the number of missed detection pulses. The oscilloscope then measured the duration between the rising edge of the test loop signal and the rising edge of the photon detection pulse and stored this value. A schematic of the measurement setup is shown in Fig. 3. The optical path includes a polarizing beamsplitter Fig. 1 Schematic of the display of photons in ( x,y,t ) space. Photons generated by switching of transistors (i.e., signal) are colored in redwhile photons not related to any electrical activity (i.e., noise) are colored in blue. MACHINE LEARNING FOR TIME-RESOLVED EMISSION (continued from page 25)