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请你从头解释一下作者是怎么推导到公式9的

The derivation of Formula (9) from the document follows several key steps, each focusing on simplifying the probability density function (PDF) under certain assumptions related to the system's photon efficiency and background noise. Here's an outline of the derivation:

1. **Photon Detection Model**: The formula is derived in the context of modeling photon detection times. The rate function of the Poisson process, given by \(\lambda_{i,j}(t)\), is introduced to represent the time-varying intensity of the photon detections from the signal and the background.

2. **Conditional Probability**: The conditional cumulative distribution function (CDF) of the first arrival time \(U\) is expressed as:
   \[
   F(u) = \frac{1 - \exp \left( - \int_0^u \lambda_{i,j}(\tau) d\tau \right)}{1 - P_0(\alpha_{i,j})},
   \]
   where \(P_0(\alpha_{i,j})\) is the probability of zero detections in the period \(T_r\). This step uses properties of the Poisson process and conditions on at least one arrival during the interval.

3. **Probability Density Function (PDF)**: The PDF for the first arrival time \(U\), \(f_U(u)\), is derived by differentiating the CDF:
   \[
   f_U(u) = \frac{\lambda_{i,j}(u) \exp \left(-\int_0^u \lambda_{i,j}(\tau) d\tau \right)}{1 - P_0(\alpha_{i,j})}.
   \]
   This PDF represents the conditional probability density of the detection time given that there is at least one photon detected.

4. **Low-Flux Assumption**: The derivation proceeds under the low-flux assumption that the flux \(\eta \alpha_{i,j} S + B \ll 1\), where \(\eta\) is the system efficiency, \(S\) is the total signal count, \(\alpha_{i,j}\) is the reflectivity, and \(B\) represents background counts. This allows the exponential factor in the PDF to be approximated as negligible.

5. **Final Formula**: Using this approximation, the final PDF for the detection time is expressed as a mixture distribution, accounting for both signal and noise contributions:
   \[
   f_{T_{i,j}}(t_{i,j}; \alpha_{i,j}, z_{i,j}) = \frac{\eta \alpha_{i,j} s(t_{i,j} - 2 z_{i,j}/c) + B/T_r}{\eta \alpha_{i,j} S + B}.
   \]
   This is Formula (9), where the approximation follows from the simplification \(1 - e^{-x} \approx x\) for small \(x\).

This formula models the detection time as a combination of the signal pulse shape and uniform noise, reflecting the photon-efficient imaging setup under the given assumptions.

Would you like to explore further details about any specific part of this derivation?

Here are some related questions for deeper exploration:
1. What are the key assumptions behind the low-flux condition used in the derivation?
2. How does the exponential approximation \(1 - e^{-x} \approx x\) influence the final formula?
3. What is the physical meaning of the signal-to-background ratio in the context of photon detection?
4. How does the Poisson process model apply to time-of-flight measurements in imaging systems?
5. What alternative models could be used if the low-flux assumption does not hold?

**Tip**: When working with approximations like \(1 - e^{-x} \approx x\), ensure the range of \(x\) is sufficiently small to maintain accuracy.

Explore how the derivation of Formula 9 in photon detection systems involves Poisson processes, probability density functions, and low-flux assumptions. The solution outlines key steps such as deriving the PDF from the CDF under photon-efficient imaging setups.

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