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📊 Statistical & Physical Formulations in LDCT Project

This document summarizes the key statistical and physical equations used in the project.
All formulations are presented in LaTeX and can be rendered directly in GitHub (via MathJax).

1. Noise Model in Low-Dose CT (Simulation)

In medical imaging, low-dose CT (LDCT) degradation can be simulated as:

  1. Photon statistics (Poisson noise):

$$ I_{\text{LD}} \sim \text{Poisson}(I_{\text{GT}}) $$

where ( I_{\text{GT}} ) is the ground-truth photon count image.

  1. Readout noise (Gaussian):

$$ I_{\text{LD}} ;=; I_{\text{Poisson}} + \mathcal{N}(0, \sigma^2) $$

This combination captures stochastic photon arrivals (Poisson) and detector/electronics noise (Gaussian).

2. Error Map (Difference Image)

To visualize reconstruction fidelity, we use pixel-wise error maps:

$$ E(x,y) ;=; |I_{\text{pred}}(x,y) - I_{\text{GT}}(x,y)| $$

where ( I_{\text{pred}} ) is the model prediction and ( I_{\text{GT}} ) is the ground truth.

3. Peak Signal-to-Noise Ratio (PSNR)

A common metric for image reconstruction quality:

$$ \text{MSE} = \frac{1}{N} \sum_{i=1}^N \left( I_{\text{pred}}^{(i)} - I_{\text{GT}}^{(i)} \right)^2 $$

$$ \text{PSNR} = 10 \cdot \log_{10} \left( \frac{I_{\max}^2}{\text{MSE}} \right) $$

  • ( I_{\max} ): maximum possible intensity value (here = 1.0 since images are normalized).
  • Higher PSNR → better reconstruction.

4. Structural Similarity Index (SSIM)

Captures perceptual similarity:

$$ \text{SSIM}(x,y) = \frac{(2\mu_x \mu_y + C_1)(2\sigma_{xy} + C_2)}{(\mu_x^2 + \mu_y^2 + C_1)(\sigma_x^2 + \sigma_y^2 + C_2)} $$

  • ( \mu_x, \mu_y ): mean intensities
  • ( \sigma_x^2, \sigma_y^2 ): variances
  • ( \sigma_{xy} ): covariance
  • ( C_1, C_2 ): stability constants

5. Learned Perceptual Image Patch Similarity (LPIPS)

A perceptual metric based on deep features:

$$ \text{LPIPS}(x,y) = \sum_{l} \frac{1}{H_l W_l} \sum_{h,w} | w_l \odot ( \phi_l(x)_{h,w} - \phi_l(y)_{h,w} ) |_2^2 $$

  • ( \phi_l(\cdot) ): deep feature maps at layer ( l )
  • ( w_l ): learned weights
  • Lower LPIPS → perceptually closer to GT

6. Line Profile Analysis (Edge Sharpness)

For a fixed row/column in the slice:

$$ L(j) = I(j,y_0), \quad j = 1,2,\dots,W $$

where ( W ) is the image width.
Sharpness is measured by gradient:

$$ \text{Edge Contrast} = \max_j \left| \frac{\partial L(j)}{\partial j} \right| $$

7. Noise Standard Deviation in ROI

To quantify noise level in a homogeneous region:

$$ \sigma_{\text{ROI}} = \sqrt{\frac{1}{N-1} \sum_{i=1}^N \left( I_i - \mu \right)^2} $$

  • ( I_i ): pixel intensities in the ROI
  • ( \mu ): mean intensity

In LDCT, ( \sigma_{\text{ROI}} ) is expected to be higher.

8. Diffusion Model Formulation (DDPM)

Forward diffusion process (adding Gaussian noise):

$$ q(x_t | x_{t-1}) = \mathcal{N}\left(x_t; \sqrt{1-\beta_t}, x_{t-1}, \beta_t I\right) $$

with variance schedule ( {\beta_t} ).

Closed-form relation:

$$ x_t = \sqrt{\bar{\alpha}_t} x_0 + \sqrt{1-\bar{\alpha}_t}, \epsilon, \quad \epsilon \sim \mathcal{N}(0,I) $$

where ( \bar{\alpha}t = \prod{s=1}^t (1-\beta_s) ).

9. Reverse Process (Sampling)

The neural network ( \epsilon_\theta ) predicts noise:

$$ p_\theta(x_{t-1} | x_t, I_{\text{LD}}) = \mathcal{N}\left(x_{t-1}; \mu_\theta(x_t, t, I_{\text{LD}}), \Sigma_t\right) $$

with mean:

$$ \mu_\theta(x_t, t, I_{\text{LD}}) = \frac{1}{\sqrt{1-\beta_t}} \left( x_t - \frac{\beta_t}{\sqrt{1-\bar{\alpha}_t}} \epsilon_\theta(x_t, t, I_{\text{LD}}) \right) $$

10. Bootstrap Confidence Interval

To provide statistical significance in evaluation metrics:

$$ \hat{\theta}^* = \frac{1}{B} \sum_{b=1}^B \theta^{*(b)} $$

where ( \theta^{*(b)} ) are bootstrap replicates (resampled metrics).

95% CI is given by the 2.5% and 97.5% quantiles of ( {\theta^{*(b)}}_{b=1}^B ).

✅ Summary

  • Noise models: Poisson + Gaussian (low-dose simulation).
  • Metrics: PSNR, SSIM, LPIPS (quality & perceptual fidelity).
  • Analysis tools: error maps, line profiles, ROI noise statistics.
  • Generative modeling: forward/reverse diffusion processes.
  • Statistical rigor: bootstrap confidence intervals for metrics.

These formulations provide the theoretical backbone of the LDCT reconstruction experiments in this repository.