How to detect the presence one- and two-dimensional patterns in distributions of spatial points?
Perhaps you could perform blob detection to find the individual dots and then map their location to a 2d space, that preserves their spatial structure, through the use of some measure for example the center of mass of the blob.
Once you have mapped each blob to its coordinates in the 2d space you could perform clustering or fit a regressor since the patterns somewhat resemble piecewise polynomials, you could model them as second or third degree polynomials, and build your algorithm upon RANSAC, it is a probabilistic algorithm that fits a model while discarding the outliers, you could also introduce some extra hyperparameters:

the percent threshold of points that minimize the cost function (you could use the sum of squared errors as a base cost function to build upon) and thus belong in the pattern.
the number of piecewise relations if you opt in the piecewise polynomial approach. Then you could exclude valid points that fit one branch of the piece wise polynomials from next iterations until all such polynomials are found.

Another classic algorithm that can cluster points based on their geodesic structure rather than just the distance would be DBSCAN, choosing a suitable $\epsilon$ for your dataset as the points that belong to the pattern are rather close to each other. A more complex classical machine learning algorithm is ISOMAP, it can also cluster points that based on geodesic properties