Bagged Deep Image Prior for Recovering Images in the Presence of Speckle Noise

TL;DR

The paper tackles recovering a real-valued image from undersampled, speckle-corrupted multilook measurements in a coherent imaging system by analyzing a likelihood-based estimator under the Deep Image Prior (DIP) hypothesis. It delivers a finite-sample, high-probability MSE bound that decouples into a term scaling with the DIP parameter count $k$, the measurement dimensions $m,n$, and the number of looks $L$, revealing how increased looks sharply improve accuracy when $m^2$ grows relative to $k\log n$. Algorithmically, it introduces Bagged-DIP to stabilise PGD by ensembling patch-based DIPs and employs the Newton-Schulz method to approximate large matrix inverses, yielding substantial computational savings with little loss in accuracy. Comprehensive simulations across varying undersampling ratios and number of looks validate the theoretical predictions, demonstrate state-of-the-art performance, and quantify the tradeoffs between model complexity, look count, and patch-based bagging. The work advances both the theory of DIP-based recovery under speckle noise and practical, scalable reconstruction methods for undersampled coherent imaging.

Abstract

We investigate both the theoretical and algorithmic aspects of likelihood-based methods for recovering a complex-valued signal from multiple sets of measurements, referred to as looks, affected by speckle (multiplicative) noise. Our theoretical contributions include establishing the first existing theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our theoretical results capture the dependence of MSE upon the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we introduce the concept of bagged Deep Image Priors (Bagged-DIP) and integrate them with projected gradient descent. Furthermore, we show how employing Newton-Schulz algorithm for calculating matrix inverses within the iterations of PGD reduces the computational complexity of the algorithm. We will show that this method achieves the state-of-the-art performance.

Figures (9)

• Figure 1: PSNR (averaged over 8 images) versus iteration count is depicted for four DIP models fitted to both clean (left panel) and noisy images with noise level $\sigma=25$ (right panel). The 4-layer networks are specified as follows: Blue - kernel size=1, channels [100, 50, 25, 10]; Orange - kernel size=3, channels same as Blue; Green - kernel size=1, channels [128, 128, 128, 128]; Red - kernel size=3, channels same as Green.
• Figure 2: Newton-Schulz approximation (red) compared with computing exact inverse (purple). Blue, orange and green curves correspond to stopping the update of the inverse after the first $5$, $10$, and $20$ iterations respectively.
• Figure 3: (Left) We compare a Bagged-DIP with three sophisticated DIP estimates. (Right) We compare PGD with simple and Bagged-DIPs across different looks on image "Cameraman".
• Figure 4: The structure of DIP and Output Blocks.
• Figure 5: (Left) The structure of Bagged-DIPs with $K$ estimates. (Right) Performance of fitting Bagged-DIP to clean images.

Theorems & Definitions (15)

• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Lemma 6.1
• proof
• Lemma 6.2
• Lemma 6.3: Concentration of $\chi^2$ jalali2014minimum
• Theorem 6.4: Hanson-Wright inequality
• Theorem 6.5: Decoupling of U-processes, Theorem 3.4.1. of de2012decoupling