Speckle images characterized by a non-Rayleigh amplitude probability density function (pdf) are formed when the assumptions underlying fully developed speckle are violated, such as when the number of scatterers in a resolution cell is small, or scatterers are organized with some periodicity, leading to nonuniform scatterer phasing. Various random-walk models have been developed to explain how non-Rayleigh cases such as the K, Rician, and homodyned-K pdf's arise. Marked regularity models provide an alternative to specialized random-walk models, because they describe the physical spatial placement of point scatterers in one dimension in a parametric manner. Spatial distributions ranging from clustered to random to periodic can be obtained by adjusting the model parameters governing the scatterer's density and ``regularity.'' It is shown that the amplitude statistics of the coherently detected speckle pattern of a collection of scatterers described by the model may be either Rayleigh or non-Rayleigh, and that it is possible to generate the special cases of Rician, K, and homodyned-K pdf's through various settings of the regularity parameters. The model's second-order properties are known and can be varied, eliminating the need to incorporate nonstochastic variations into the random-walk models.