A class of shift-variant reduction operations is introduced, that is useful for performing efficient and controllable shape and texture transformations between resolution levels. In their most general form, the operations proceed in three steps: (a) convolve a binary image with a kernel of arbitrary size; (b) threshold the result; (c) subsample to produce the reduced image. Taking a binary structuring element for the kernel, the threshold convolution on a binary image is equivalent to a rank order filter, and the full reduction operation is a threshold reduction. Threshold reductions that use convolution filters and subsample tiles of equal size are optimized by combining the three operations, using only logical raster operations and producing threshold convolution values only at the sampling points. For 2x reduction, the four possible threshold values (1, 2, 3, and 4) refer to the minimum number of ON pixels within each 2x2 tile for which a pixel in the reduced image will be ON. Algorithms for boolean raster operations are given for 2x, 3x, and 4x threshold reduction, and lookup tables that efficiently implement column raster operations are provided. Threshold reduction rates of 2.5x107 pixel/second can be achieved with a Sun SparcStation2TM . A maskforming image analysis cycle of threshold reduction, augmented by morphology and followed by replicative expansion to full resolution, is described, and some general properties of the cycle are derived. A simple application of threshold reduction to document image analysis, the extraction of halftone regions from scanned images that also contain text and line graphics, is illustrated. A sequence of 2x reductions with first low and then high thresholds is used to create a reduced image consisting of a mask over the halftone regions. In this way, the extraction occurs as a natural consequence of the reductions.