How Do Image Filters Work? From Linear Color Adjustments to AI-Powered Repainting

1. Introduction

In everyday life, most images we see have been processed by filters, ranging from simple color adjustments on phones to complex AI‑generated style transfers. Mathematically, a filter can be viewed as a function that takes the original pixel RGB values and outputs transformed RGB values. This article analyzes three types of filters:

Single‑pixel input and output color filters;

Multi‑pixel input, single‑pixel output convolution filters;

AI‑based fully‑repainted filters.

2. Color Filters: Single‑Pixel Function Model

2.1 Basic Definition

Assume an image consists of a matrix of pixels, each with an RGB vector. A color filter maps each pixel’s RGB vector through a function f, producing adjusted pixel values.

2.2 Grayscale Filter

A grayscale filter forces the three channels to have the same value, typically using the formula Y = 0.299 R + 0.587 G + 0.114 B, where the coefficients reflect human eye sensitivity.

2.3 Brightness Adjustment

Brightness is adjusted by adding a constant c to each RGB channel: R' = R + c, G' = G + c, B' = B + c.

2.4 Hue Adjustment

Hue can be altered by modifying a single channel, for example reducing the green component.

2.5 RGB Curve Adjustment

RGB curve adjustment is a nonlinear transformation expressed by a piecewise function mapping input intensity to output intensity.

3. Convolution Filters: Multi‑Pixel Modeling

3.1 Definition of Convolution

Convolution filters consider a central pixel and its neighboring pixels. A convolution kernel K is defined and applied to compute the output value.

3.2 Blur Filter

A blur filter uses an averaging kernel.

3.3 Sharpen Filter

A sharpening filter emphasizes differences between pixels using a specific kernel.

3.4 Gaussian Blur

The Gaussian blur kernel is based on a normal distribution weight matrix.

4. Fully Repainted Filters: Deep‑Learning Modeling

4.1 Problem Definition

Fully repainted filters aim to generate images with a specific style. In style transfer, the objective is to minimize a weighted sum of content loss and style loss.

4.2 Content Loss

Given an input image p and a generated image x, their feature representations at a certain CNN layer are Fᵖ and Fˣ. Content loss is defined as the squared difference between these features.

4.3 Style Loss

Style loss is based on the Gram matrix of feature maps, measuring the difference between style representations of the target and generated images.

4.4 Overall Optimization

By back‑propagating the total loss, the generated image is iteratively updated to combine the desired content and style.

The article demonstrates how filters evolve from simple linear color transformations to sophisticated deep‑learning techniques, highlighting the mathematical models behind color filters, convolution filters, and fully repainted AI filters.