In this work we propose a scanline optimization procedure for computational stereo using a linear smoothness cost model performed by programmable graphics hardware. The main idea for an efficient implementation of this dynamic programming approach is a recursive scheme to calculate the min-convolution in a manner suitable for the parallel stream computation model of graphics processing units. Since many image similarity functions can be efficiently calculated by modern graphics hardware, it is reasonable to address the final disparity extraction by graphics processors as well. Our timing results indicate that the proposed approach is beneficial for larger image resolutions and disparity ranges in particular.