Identifying sparse salient structures from dense pixels is a long-standing problem in visual computing. Solutions to this problem can benefit both image manipulation and understanding. In this paper, we introduce an image transform based on the $L_1$ norm for piecewise image flattening. This transform can effectively preserve and sharpen salient edges and contours while eliminating insignificant details, producing a nearly piecewise constant image with sparse structures. A variant of this image transform can perform edge-preserving smoothing more effectively than existing state-of-the-art algorithms. We further present a new method for complex scene-level intrinsic image decomposition. Our method relies on the above image transform to suppress surface shading variations, and perform probabilistic reflectance clustering on the flattened image instead of the original input image to achieve higher accuracy. Extensive testing on the Intrinsic-Images-in-the-Wild database indicates our method can perform significantly better than existing techniques both visually and numerically. The obtained intrinsic images have been successfully used in two applications, surface retexturing and 3D object compositing in photographs.