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Is it possible to build a neural network that learns the connection between two images?

Let's say I have a number of X images that related to Y images. How can I build a neural network that takes an image as an input and outputs (generates) the output image?

The Y images are generated by applying some function to the X images.

Do I need a generate neural network for that? Are conventional neural networks capable of classification only?

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It is possible to have both input and output be images that differ in a predictable way. For example, architectures similar to autoencoders have been used to remove blur, change weather conditions, change between day and night photos etc. In these architectures, the training data is matching pairs of images. If your goal is to replicate some image enhancement, then often the input is artificially processed e.g. to reduce its quality in a hard to reverse way. A good example of this would be to remove distortion or noise from an image.

You can also use generative models. These are harder to get working, but can be more flexible in that you don't need image pairs in order to train, just a set of images labelled with the traits that you want to learn. Converting an image using a generative model involves using an encoder stage to get its embedding, altering the embedding based on label you require and then feeding the new embedding into the decoder stage. This is how you might alter a face portrait from male to female, or young to old, because it is not possible to find good natural image pairs for that task.

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Sure, for example for removing visual noise. Look up noise2noise by NVIDIA. The made a net that is capable of removing almost all noise from one picture. https://hothardware.com/news/nvidia-noise2noise-machine-learning-ai-magically-restores-your-grainy-photos

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