Consider the case when we have four layers to blend to produce the final image: F=A*(B*(C*D)) where A, B, C, D are partially transparent image layers and "*" denotes a compositing operator (with the left layer on top of the right layer). If only layer C changes, we should find a way to avoid re-blending all of the layers when computing F. Without any special considerations, four full-image blends would need to occur. For compositing operators that are commutative, such as additive blending, it is safe to re-order the blending operations. In this case, we might compute T=A*(B*D) only once and simply blend T*C to produce F, a single operation. Unfortunately, most operators are not commutative. However, many are associative, suggesting it is safe to re-group operations to F=(A*B)*(C*D), i.e. without changing their order. In this case we may compute S:=A*B once and save this result. To form F with an associative operator, we need only do two additional compositing operations to integrate the new layer S, by computing F:=S*(C*D). Note that this expression indicates compositing C with all of the layers below it in one step and then blending all of the layers on top of it with the previous result to produce the final image in the second step.