GIPHOD (Group Invariant Persistent HOmology Demonstrator) is a program realized to show how persistent homology can be made invariant with respect to any given group G of homeomorphisms, still preserving non-invariance with respect to the homeomorphisms that do not belong to G. This property is important for applications in computer vision and pattern recognition.
GIPHOD uses a dataset of 10000 quite simple synthetic gray-scale images, represented as functions from the square [0,1]x[0,1] to the interval [0,1] (1=white, 0=black). 1000 images (call this set X) are randomly generated, whereas the rest 9000 images are obtained by applying transformations composed of translations, rotations and reflections to the images in the set X (let us call this set Y).
The user can choose one of the invariance groups and one of the query images that we propose. The query images that can be selected are taken from the set X, and it is possible to change their list by clicking on the “Random images” button. GIPHOD looks for the ten most similar images in our dataset (consisting of the sets X and Y), with respect to the invariance group that has been chosen. Among these ten images, the ones that are judged to be close to the query image are displayed in green.

GIPHOD is available online at