In exemplary implementations of this invention, two LCD screens display a multi-view 3D image that has both horizontal and vertical parallax, and that does not require a viewer to wear any special glasses. Each pixel in the LCDs can take on any value: the pixel can be opaque, transparent, or any shade between. For regions of the image that are adjacent to a step function (e.g., a depth discontinuity) and not adjacent to a sharp corner, the screens display local parallax barriers comprising many small slits. The barriers and the slits tend to be oriented perpendicular to the local angular gradient of the target light field. In some implementations, the display is optimized to seek to minimize the Euclidian distance between the desired light field and the actual light field that is produced. Weighted, non-negative matrix factorization (NMF) is used for this optimization.