  One of the main challenges in property testing is to characterize those
properties that are testable with a constant number of queries. For unordered
structures such as graphs and hypergraphs this task has been mostly settled.
However, for ordered structures such as strings, images, and ordered graphs,
the characterization problem seems very difficult in general.
  In this paper, we identify a wide class of properties of ordered structures -
the earthmover resilient (ER) properties - and show that the "good behavior" of
such properties allows us to obtain general testability results that are
similar to (and more general than) those of unordered graphs. A property P is
ER if, roughly speaking, slight changes in the order of the elements in an
object satisfying P cannot make this object far from P. The class of ER
properties includes, e.g., all unordered graph properties, many natural visual
properties of images, such as convexity, and all hereditary properties of
ordered graphs and images.
  A special case of our results implies, building on a recent result of Alon
and the authors, that the distance of a given image or ordered graph from any
hereditary property can be estimated (with good probability) up to a constant
additive error, using a constant number of queries.
