FIG. 4, coefficients in Region 2 form a linear rising edge with a positive gradient B. In particular, let P1_enddenote the index of the last (e.g., rightmost) coefficient in Region 1, and let P1_end+1 and P2_enddenote the indices of the first (e.g., the leftmost) coefficient and the last (e.g., the rightmost) coefficient in Region 2, respectively, and let C₁denote the first coefficient in Region 2, then for any index i within Region 2 (e.g., P1_end+1≤i≤P2_end), the coefficient DCF(i) is given as DCF(i)=B(i−P1_end−1)+C₁. In some embodiments, the rising edge in Region 2 is used to amplify the rising edge in the histogram.

FIG. 5 shows the coefficients in Region 3 (also referred to as the accumulation region). As illustrated in FIG. 5, coefficients in Region 3 form another linear rising edge with a positive gradient C. In particular, let P2_enddenote the index of the last coefficient in Region 2, and let P3_enddenote the index of the last coefficient in Region 3, and let C₂denote the last coefficient in Region 2, then for any index i within Region 3 (e.g., P2_end≤i≤P3_end), the coefficient DCF(i) is given as DCF(i)=C(i−P2_end)+C₂. Although the illustrated embodiment shows that the accumulation region has a positive gradient C, which is larger than the positive gradient B of the rising edge region, this is merely a non-limiting example. In some embodiments, the accumulation region has a zero gradient (e.g., a flat region). In some embodiments, the non-negative gradient C (e.g., C≥0) of the accumulation region is used to accumulate the peak region of the histogram.

FIG. 6 shows the coefficients in Region 4 (also referred to as the falling edge region). As illustrated in FIG. 6, coefficients in Region 4 form a linear falling edge with a negative gradient D. In particular, let P3_enddenote the index of the last coefficient in Region 3, and let P4_enddenote the index of the last coefficient in Region 4, and let C₃denote the last coefficient in Region 3, then for any index i within Region 4 (e.g., P3_end≤i≤P4_end), the coefficient DCF(i) is given as DCF(i)=D(i−P3_end)+C₃. In some embodiments, the falling edge region is used to amplify the falling edge of the histogram.

FIG. 7 shows that the coefficients in Region 5 (e.g., post-pulse region) have values that alternate (or oscillate) between a negative value E₂and a negative value E₁, such that the coefficients in Region 5 form a zig-zap pattern, a saw-tooth pattern, or a differential pattern. For example, coefficients in Region 5 with odd indices may be assigned the negative value E₂, and coefficients in Region 5 with even indices may be assigned the negative value E₁. The zig-zag pattern of coefficients in the post-pulse region is used to improve performance in low photon count scenarios, similar to the pre-pulse region. In addition, by choosing the mean of the E₁and E₂to be smaller than the mean of A₁and A₂(e.g., (E₁+E₂)/2<(A₁+A₂)/2), the differential correlator filter 200 amplifies changes in the histogram to trigger zero-crossing. The negative values E₁and E₂are set in accordance with the coefficient values in Regions 1, 2, 3, and 4 (e.g., to satisfy Equation (1)). In some embodiments, the difference ΔE=|E₁−E₂| is chosen to set the sensitivity to small changes in the histogram. For example, the smaller is the difference ΔE, the more sensitive is the filter output to smaller changes in the histogram.

FIG. 8 illustrates adjustments of the pre-pulse region and the post-pulse region to achieve certain performance targets. In the example of FIG. 8, let A denote the average of the negative values A₁and A₂(e.g., A=(A₁+A₂)/2, A<0), and let E denote the average of the negative values E₁and E₂(e.g., E=(E₁+E₂)/2, E<0), by choosing E to be smaller than A, the likelihood of a zero-crossing in the filter response, when a falling edge of a pulse in the histogram is present, may be increased. Note that by choosing E to be smaller than A, the overall filter response is likely to be negatively biased (e.g., due to the sum of all coefficients being negative), which may cause the idle response of the differential correlator filter 200 to be below zero.

The zig-zap pattern of the coefficients in the pre-pulse region and the post-pulse region may be well-suited for histograms with low photon counts, where statistics of the photon counts follow Poisson distribution. If the histogram is expected to have large photon counts and statistics of the photon count are approximately Gaussian, the shape of the differential correlator filter 200 may be simplified, e.g., by substituting the zig-zag patterns in the pre-pulse region and the post-pulse region with their respective mean value of A and E, respectively. In other words, the coefficients in the pre-pulse region may have a fixed value of A, and the coefficients in the post-pulse region may have a fixed value of E.

FIG. 9A illustrates a histogram 201 of a ToF imager, in an embodiment. Note that for simplicity, the histogram 201 shows the envelop of the histogram bins without illustrating individual histogram bins. FIG. 9A further illustrates two pulses 203A and 203B, which indicates the location of two targets. As a result of reflection from the targets, the photon count (e.g., signal count) of the histogram shows two corresponding pulses.

FIG. 9B illustrates an output signal 205 of the differential correlator filter 200, in an embodiment. The histogram 201 of FIG. 9A is processed by the differential correlator filter 200 (e.g., filtered by the differential correlator filter 200, or convolved with the differential correlator filter 200 through a convolution operation) to produce the output signal 205 of FIG. 9B. As illustrated in FIG. 9B, the output signal 205 shows two pulse regions corresponding to the two target locations (see 203A and 203B in FIG. 9A). In particular, the first pulse region of the output signal 205 is defined between zero-crossing points 207A and 207B, and the second pulse region of the output signal 205 is defined between zero-crossing points 209A and 209B. Note that the zero-crossing point 207A (or 209A) is a zero-crossing from negative to positive, which indicates the start of a pulse region; and the zero-crossing point 207B (or 209B) is a zero-crossing from positive to negative, which indicates the end of a pulse region. In other words, each pair of the zero-crossing points (e.g., 207A and 208B, or 209A and 209B) indicate the start point and stop point of a respective pulse region in the output signal 205.

A zero-crossing algorithm for finding the zero-crossing points (e.g., start points and stop points of pulse regions) in the output of the differential correlator filter 200 is described herein. The algorithm monitors the output of the differential correlator filter 200 for a change from negative to positive, and marks the position of the change as a potential start point. The algorithm then monitors the output of the differential correlator filter 200 for a change from positive to negative, and marks the position of the change as a potential stop point. The corresponding histogram data (or the corresponding output of the differential correlator filter 200) between the potential start point and the potential stop point are summed together and used as a confidence measurement. If the confidence measurement is above a threshold value (e.g., a user defined threshold), the potential start point and the potential stop point are considered valid and stored as a pair of start point and stop point. Denote the output of the differential correlator filter 200, or filter response, as FR, and denote the histogram as h, the above processing may be described by the following pseudo code:

for i=2: length(FR)

if FR(i−1)<0 and FR(i)>0

start(n)=i;

end

if FR(i−1)>0 and FR(i)<0

confidence(n)=Σh(start(n):i);

end

if confidence(n) > threshold

stop(n)=i;

n=n+1;

end

end

Once the start points and stop points are found, a “center-of-mass” calculation can be performed to calculate a median phase for each pulse region, which indicates the location of the target in that pulse region. In particular, for each pair of start point and stop point (which defines one pulse region in the filter response that corresponds to a respective target), the center-of-mass calculation calculates a median phase, which corresponds to a weighted histogram bin location for the target. The operation may be described by the following pseudo code:

- for n=1: length (stop)

$median_phase (n) = \frac{Σ_{i = start (n)}^{stop (n)} h (i) \times i}{Σ_{i = start (n)}^{stop (n)} i}$

- end
- The zero-crossing algorithm and the center-of-mass algorithm discussed above are simply non-limiting examples. Other suitable algorithms are also possible and are fully intended to be included within the scope of the present disclosure.

FIG. 10 illustrates a flow chart for generating a differential correlator filter, in an embodiment. In block 1010, a differential correlator filter, same as or similar to the example illustrated in FIG. 2, is generated in accordance with system specification. For example, depending on system level considerations such as the expected signal count, noise level, type of targets, number of targets, the width of the pulse shape of the light source, and so on, certain design aspects of the differential correlator filter, such as the width of the pulse region of the filter, the number of coefficients in the pre-pulse region and the post-pulse region, and the shape (e.g., zig-zap vs. constant value) of the pre-pulse region and the post-pulse region, may be determined. Next, in block 1020, various parameter for the differential correlator filter are adjusted to, e.g., satisfy Equations (1) and (2) above, and to tune the sensitivity of the filter to changes in the histogram. Next, in block 1030, the coefficients of the differential correlator filter are stored for use in the hardware implementation of the filter, or for use in computation in a software implementation of the filter.

FIG. 11 illustrates a flow chart for processing the histogram of a ToF imager using a differential correlator filter, in an embodiment. The differential correlator filter may be the same as or similar to the example of FIG. 2, and may be designed according to the flow chart of FIG. 10. As illustrated in FIG. 11, in block 1110, each pixel (e.g., a SPAD) in the ToF imager accumulate photon count to generate a histogram h. In block 1120, the histogram h from each pixel is convolved with the differential correlator filter to generate the filter response FR. In block 1130, the zero-crossing algorithm described above is used to compute the confidence measurement and find the start/stop points in the histogram. In block 1140, a median-phase for each pulse region defined by a pair of start/stop points is calculated, using the calculation described above.

To appreciate the performance advantage of the disclosed differential correlator filter, comparison with some reference filters are discussed hereinafter with reference to FIGS. 12A and 12B.

FIG. 12A illustrates three different filters for processing the histogram of a ToF imager, in an embodiment. Curve 301 shows a weighted filter, which is a square pulse with the sum of the filter coefficients being approximately zero. Curve 303