Log(n)) per Plane Fast Discrete Focal Stack Transform

This paper presents a new fast discrete focal stack transform that can be used in the 3D depth recovery problem using a single image of a scene from a plenoptic camera. A plenoptic camera uses a microlens array between the lens of the camera and the image sensor to measure the radiance and direction of all the light rays in a scene. The microlens array creates many small images taken from slightly different viewpoints, which can be used to obtain conventional images focused on a frontal plane of desired depth. The collection of such images is the focal stack of the scene. The focal stack is of great importance to the depthfrom-focus approach since the elements in focus allow us to recover the depth of the objects in the scene. There are several approaches to the computation of the focal stack. Our approach is based on the Discrete Focal Stack Transform (DFST). We present a new algorithm that decreases the complexity of the DFST from O(n 3 ) per plane to O(n 2 log(n)) per plane. This allows the computation of a focal stack with O(n/log(n)) times the number of planes of the DFST with the same computational complexity. From a practical point of view, in typical scenes we are able to increase more than 200 times the number of planes of the DFST. Finally we provide some computational results that show the validity of the approach.