After you've trained a model on the MNIST set, how can I now classify an image as having two digits? More generally, how do I train a model to detect any number of digits on an image?
Training on MNIST, detecting sequences of numbers?
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There is a hot field called "object detection" that tries to do what you want. In general, you can detect anything (digits, people, cars, etc) from any images and even videos.
The state-of-the-art techniques roughly fall into two categories:
A well-known real-time object detection method is YOLO (You Only Look Once), which falls in the SSD category, and has a very impressive real-time demo here, to give you a sense of object detection. Search these methods' names and you will find a lot of example code that satisfies your needs.
If you are only looking for digit detection, also check out work surrounding Stanford's House Number Dataset. However, note that these works are generally from five and more years ago and do not necessarily beat general methods like Faster-RCNN and SSD.