A.1 Derivation of Eq. (3) By expanding Eq. (2) with the definition of εli,t = xli,t µli,t, we have: Et = We note that each xli,t influences Et in two ways: (i) it occurs in Eq. (6) explicitly, but (ii) it also determines the values of µl 1k,t via Eq. Considering also the special cases of l = Land l = 0, we obtain Eq. (3). We note that θl+1i,j affects the value of the function Et of Eq. (6) by influencing µli,t via Eq. Here, we provide further details about training PCNs, useful to reproduce them. Furthermore, we have applied a decay factor of 0.9 to γ, applied each time the energy failed to decrease.

Image similarity measures are also crucial in the field of robust machine learning.

We describe a novel method for training high-quality image denoising models based on unorganized collections of corrupted images. The training does not need access to clean reference images, or explicit pairs of corrupted images, and can thus be applied in situations where such data is unacceptably expensive or impossible to acquire. We build on a recent technique that removes the need for reference data by employing networks with a blind spot in the receptive field, and significantly improve two key aspects: image quality and training efficiency. Our result quality is on par with state-of-the-art neural network denoisers in the case of i.i.d.

We present the first diffusion-based framework that can learn an unknown distribution using only highly-corrupted samples. This problem arises in scientific applications where access to uncorrupted samples is impossible or expensive to acquire. Another benefit of our approach is the ability to train generative models that are less likely to memorize any individual training sample, since they never observe clean training data. Our main idea is to introduce additional measurement distortion during the diffusion process and require the model to predict the original corrupted image from the further corrupted image. We prove that our method leads to models that learn the conditional expectation of the full uncorrupted image given this additional measurement corruption. This holds for any corruption process that satisfies some technical conditions (and in particular includes inpainting and compressed sensing). We train models on standard benchmarks (CelebA, CIFAR-10 and AFHQ) and show that we can learn the distribution even when all the training samples have 90\% of their pixels missing. We also show that we can finetune foundation models on small corrupted datasets (e.g. MRI scans with block corruptions) and learn the clean distribution without memorizing the training set.

This paper studies the removal of salt-and-pepper noise from images using median filter (MF) and simple three-layer autoencoder (AE) within recursive threshold algorithm. The performance of denoising is assessed with two metrics: the standard Structural Similarity Index SSIMImg of restored and clean images and a newly applied metric SSIMMap - the SSIM of entropy maps of these images computed via 2D Sample Entropy in sliding windows. We shown that SSIMMap is more sensitive to blur and local intensity transitions and complements SSIMImg. Experiments on low- and high-resolution grayscales images demonstrate that recursive threshold MF robustly restores images even under strong noise (50-60 %), whereas simple AE is only capable of restoring images with low levels of noise (<30 %). We propose two scalable schemes: (i) 2MF, which uses two MFs with different window sizes and a final thresholding step, effective for highlighting sharp local details at low resolution; and (ii) MFs-AE, which aggregates features from multiple MFs via an AE and is beneficial for restoring the overall scene structure at higher resolution. Owing to its simplicity and computational efficiency, MF remains preferable for deployment on resource-constrained platforms (edge/IoT), whereas AE underperforms without prior denoising. The results also validate the practical value of SSIMMap for objective blur assessment and denoising parameter tuning.

Image similarity measures are also crucial in the field of robust machine learning.

While the performance of DDGMs is impressive, not all of their aspects are fully understood.