Constant-Time Testing and Learning of Image Properties.

We initiate a systematic study of sublinear-time algorithms for image analysis that have access only to labeled random samples from the input. Most previous sublinear-time algorithms for image analysis were query-based, that is, they could query pixels of their choice. We consider algorithms with two types of input access: sample-based algorithms that draw independent uniformly random pixels, and block-sample-based algorithms that draw pixels from independently random square blocks of the image. We investigate three basic properties: being a half-plane, convexity, and connectedness. For the first two, our algorithms are sample-based; for connectedness, they are block-sample-based. All our algorithms have low sample complexity that depends polynomially on the inverse of the error parameter and is independent of the input size. We design algorithms that approximate the distance to the three properties within a small additive error or, equivalently, tolerant testers for being a half-plane, convexity and connectedness. Tolerant testers for these properties, even with query access to the image, were not investigated previously. Tolerance is important in image processing applications because it allows algorithms to be robust to noise in the image. We also give (non-tolerant) testers for convexity and connectedness with better complexity than our distance approximation algorithms and previously known query-based testers. To obtain our algorithms for convexity, we design two fast proper PAC learners of convex sets in two dimensions that work under the uniform distribution: non-agnostic and agnostic.