Noise Combination Sampling (NCS)

• Noise Combination Sampling (NCS) is a technique that reformulates measurement and noise injection using simplex coding to achieve bias immunity and noise decorrelation.
• In single-pixel imaging, NCS constructs nonnegative measurement matrices via simplex-vertex codes, eliminating DC-bias and enabling artifact-free reconstruction.
• In diffusion-based generative models, NCS synthesizes optimally-aligned noise vectors for conditional sampling, improving reconstruction fidelity with fewer steps.

Noise Combination Sampling (NCS) refers to a family of frameworks for reformulating measurement and noise injection in both computational imaging and generative modeling. In single-pixel imaging, NCS constructs nonnegative measurement matrices using simplex-vertex codes to eliminate additive bias and decorrelate measurement noise. In generative models, particularly denoising diffusion probabilistic models (DDPMs), NCS synthesizes optimally-aligned noise vectors from a subspace to incorporate measurement constraints, enabling conditional sampling without the instability of additive guidance. NCS provides principled mechanisms for bias rejection, noise decorrelation, and robust conditional embedding, with applications spanning single-pixel imaging, inverse problems, and image compression (Czajkowski et al., 2018, Su et al., 24 Oct 2025).

1. Noise Combination Sampling in Single-Pixel Imaging

In the context of single-pixel imaging, NCS addresses robust measurement under ambient illumination and detector noise. Given an original real-valued sensing matrix $M \in \mathbb{R}^{k \times n}$ and observed measurements $y = Mx$, where $x \in \mathbb{R}^n$ is the scene, ambient noise and DC-bias degrade measurement fidelity. NCS replaces $M$ with a nonnegative-expanded matrix $M'$ such that measurements $y' = M'x$ are directly displayable on hardware intensity modulators (e.g., digital micromirror devices), and the original $y$ is exactly recovered via a fixed linear mapping.

The encoding proceeds by partitioning $M$ into blocks of $p$ rows and expanding each into $p+1$ rows, distributing the measurement pattern weights as nonnegative linear combinations of the $p+1$ vertices of a regular simplex. This construction ensures DC-bias from ambient illumination is mapped to the simplex's zero-sum property, enabling perfect algebraic cancellation in postprocessing. As a result, the final measurement is theoretically independent of constant illumination offsets (Czajkowski et al., 2018).

2. Simplex Coding, Measurement Model, and Bias Immunity

The NCS transformation is based on the properties of a regular $p$-simplex in $\mathbb{R}^p$, with vertices $v_1, ..., v_{p+1}$ satisfying $\sum_{i=1}^{p+1} v_i = 0$ and pairwise constant Euclidean distances. For each $p$-row block of $M$, the column vector $v$ is decomposed as a nonnegative linear combination of the simplex vertices (with exactly one coordinate set to zero), producing a corresponding nonnegative pattern in $M'$. This mapping is invertible through a block-diagonal matrix $Q = I_\ell \otimes V$ for $k = p\ell$.

A key property is that any constant offset in the measurement is algebraically eliminated: for observed $y'_{\text{obs}} = M'x + b\mathbf{1}$, one obtains $Qy'_{\text{obs}} = Q M'x + bQ\mathbf{1} = y + 0$. Thus, DC-bias immunity is ensured by construction (Czajkowski et al., 2018).

3. Extensions, Complementary Detection, and Measurement Noise

NCS accommodates complementary sampling using two detectors for measuring $M'$ and its complement, with differential signals $y' = y_A' - y_B'$. Nonnegativity of all entries in $M'$ further allows direct pattern modulation without offset, enabling binary display and efficient hardware implementation.

In the presence of i.i.d. Gaussian detector noise with mean $\mu$ and variance $\sigma^2$, classical shift-and-display methods introduce mean biases and correlated noise. Under NCS, the transformed noise remains zero-mean and uncorrelated, with covariance $$\operatorname{Cov}(y_{\text{obs}}) = \sigma^2 Q Q^T = \sigma^2 (1 + 1/p) I_k$$, scaling variance by a known but modest factor (Czajkowski et al., 2018).

4. Simplex Dimension Selection and Adaptive Trade-offs

The choice of simplex dimension $p$ governs a trade-off between measurement overhead and noise robustness. Increasing $p$ reduces pattern expansion overhead (as $k/K = p/(p+1)$ approaches 1), but incurs a slightly larger noise variance scaling $(1 + 1/p)$. Empirical and numerical studies show the optimal $p$ (maximizing reconstruction PSNR for fixed detector noise) rises with decreasing noise: $p^* \approx 1$–$2$ for high-noise (single-detector) settings and $p^* \approx 10$ for low-noise (balanced, complementary detection). Reconstruction performance depends on the interplay between this factor and the compression ratio (Czajkowski et al., 2018).

5. Implementation and Experimental Results in Imaging

Experimental validation employs a 22 kHz DMD to project binarized DCT patterns via $M'$ or its complement. Detection is achieved with one or two bucket detectors. Reconstruction leverages Fourier-Domain Regularized Inversion (FDRI), using a precomputed generalized inverse $P' = P Q$. Real-time video at $\approx 11$ fps for 256×256 frames is attainable.

Measured results indicate that simplex coding improves PSNR under single-detector modes and achieves parity with balanced detection modes. In dynamic scenes, where direct (uncoded) sampling manifests brightness artifacts under fluctuating ambient conditions, NCS-processed reconstructions remain artifact-free (Czajkowski et al., 2018).

6. Noise Combination Sampling in Diffusion-Based Inverse Problems

NCS has been extended to generative modeling, particularly in inverse problems with pretrained Denoising Diffusion Probabilistic Models (DDPMs). Let $y = A x_0 + n$ with $x_0$ the unknown signal, $A$ the measurement operator, and $n \sim \mathcal{N}(0, \sigma^2 I)$. DDPMs apply a learned score model $s_\theta$ for denoising steps, but naïve conditioning on $y$ via additive measurement-score guidance often requires delicate step-size or guidance tuning, and risks manifold drift.

The NCS approach synthesizes an “optimal noise” vector $\epsilon_t^*$ at each reverse step from a Gaussian codebook $E_t = [\epsilon_t^1, ..., \epsilon_t^K]$. The optimal linear combination $\gamma^*$ is chosen to maximally align $E_t \gamma^*$ with the negative measurement score $-c$ (where $c = \nabla_{x_t} \log p(y|x_t)$), via the closed form $\gamma^* = (c^T E_t)/\|c^T E_t\|_2$, ensuring $\epsilon_t^*$ remains standard normal and maximally incorporates measurement constraints.

In the revised sampling update, the DDPM noise term is replaced as $$x_{t-1} = \mu_\theta(x_t, t) + \sigma_t \epsilon_t^*$$. As a result, conditional information is embedded into the noise, removing the need for step-wise hyperparameter adjustment and providing robustness even for small generation step counts $T$ (Su et al., 24 Oct 2025).

7. Empirical Performance, Limitations, and Open Questions

Empirical results on linear inverse vision tasks (e.g., FFHQ and ImageNet inpainting, super-resolution, deblurring) show substantial PSNR, FID, and LPIPS improvements of NCS-guided sampling over classical DDPM conditional solvers (DPS, MPGD), particularly at low $T$ (e.g., PSNR increase from 12.52 to 19.16 in FFHQ inpainting at $T=20$ for DPS vs. NCS-DPS). Similar compression gains with negligible degradation are observed when reducing the number of sampling steps.

Notable limitations include blurring with extremely simple codebooks and non-ideal noise-norm scaling as $K$ increases. Extending NCS to nonlinear measurement models or integrating adaptive, learned codebooks remains an open area. Future directions comprise merging NCS’s closed-form mechanisms with advanced quantization and semantic conditional models (Su et al., 24 Oct 2025).