# Stable Diffusion for Image Synthesis

## Denoising Score Matching

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Denoising score matching, a fundamental technique employed by Stable Diffusion, plays a crucial role in the generation of high-quality images. It operates on the principle of refining noise-corrupted images iteratively, gradually improving their quality through a series of denoising steps.

The process begins with a noise-corrupted image, denoted as X, which serves as the starting point for the denoising procedure. The objective is to remove the noise present in the image while preserving its essential features and details. This is achieved through an iterative optimization process guided by denoising score matching principles.

At each iteration, denoising score matching aims to minimize the discrepancy between the distribution of the denoised image and the underlying true distribution of clean images. This is accomplished by iteratively updating the denoised image in such a way that it better matches the characteristics of clean images while simultaneously minimizing the impact of noise.

<mark style="background-color:blue;">**Stable Diffusion**</mark> utilizes denoising score matching to generate high-quality images. Given a noise-corrupted image X, Stable Diffusion refines it through <mark style="background-color:blue;">**iterative denoising steps.**</mark>&#x20;



<table data-view="cards"><thead><tr><th></th><th></th><th></th><th data-hidden data-card-cover data-type="files"></th></tr></thead><tbody><tr><td></td><td><mark style="background-color:purple;">Beyond text-to-image, our features can be used for inpainting and so much more.</mark></td><td></td><td><a href="https://592342210-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FCHwFZt06MRSjU0aF2Fiy%2Fuploads%2FOMR9VCe80eoJ3kBykvBI%2Fphoto_2024-02-09_19-08-58-removebg-preview.png?alt=media&#x26;token=54604689-8861-4c40-aed6-4b28ebf8a20c">photo_2024-02-09_19-08-58-removebg-preview.png</a></td></tr><tr><td></td><td><mark style="background-color:purple;">Useful for creative tasks like concept art generation, illustration, and photo editing.</mark></td><td></td><td><a href="https://592342210-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FCHwFZt06MRSjU0aF2Fiy%2Fuploads%2FHSkQwDwQ9inmDdCBI4F1%2Fphoto_2024-02-09_19-03-42-removebg-preview.png?alt=media&#x26;token=08136d86-0a96-41aa-9a1c-fb125eb6014e">photo_2024-02-09_19-03-42-removebg-preview.png</a></td></tr><tr><td></td><td><mark style="background-color:purple;">creating realistic images based on textual descriptions and so much more.</mark></td><td></td><td><a href="https://592342210-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FCHwFZt06MRSjU0aF2Fiy%2Fuploads%2Few8mZzPJrwNB8L5rgUjJ%2Fphoto_2024-02-09_19-03-41-removebg-preview.png?alt=media&#x26;token=825a7121-fccb-4166-ad03-fc9217421e92">photo_2024-02-09_19-03-41-removebg-preview.png</a></td></tr></tbody></table>

The process can be expressed mathematically as

$$X_{k~-1} = X_k + \sqrt{2\eta}\nabla_x log p(X) + \sqrt{\eta\in_k},$$

Here,  represents the _<mark style="background-color:blue;">**step size**</mark>_, $$\nabla_xlogp(X_k)$$ is the score function, and $$\in_k$$, \~ $$N(0, I)$$ is <mark style="background-color:blue;">**Gaussian noise.**</mark>



The denoising process typically involves two key components:

1. **Score Function**: A score function is employed to capture the gradient information of the underlying data distribution. It provides valuable insights into the structure and features of the data, enabling the denoising algorithm to make informed decisions during the optimization process.
2. **Optimization Algorithm**: An optimization algorithm is utilized to iteratively update the denoised image based on the information provided by the score function. This algorithm seeks to minimize the discrepancy between the denoised image and the true distribution of clean images, effectively reducing the effects of noise and enhancing image quality.

By iteratively refining the denoised image through successive denoising steps, Stable Diffusion leverages denoising score matching to generate high-quality images with improved clarity, sharpness, and fidelity. This iterative approach allows the algorithm to gradually eliminate noise while preserving important structural and semantic features, resulting in visually pleasing and realistic image outputs.

<figure><img src="https://592342210-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FCHwFZt06MRSjU0aF2Fiy%2Fuploads%2FlwtR2PsVkxezbYSWUpxu%2FPSX_20240325_200539.jpg?alt=media&#x26;token=2245a397-d3b3-495b-a6cd-f24a38d56e0b" alt="" width="563"><figcaption><p><strong>DALL - E Difusion in play through Text-to-image feature of Textopia.ai</strong></p></figcaption></figure>
