Boundary completion is one of the desired properties of a robust object boundary detection model, since in real-word images the object boundaries are commonly not fully and clearly seen. An extreme example of boundary completion occurs in images with illusory contours, where the visual system completes boundaries in locations without intensity gradient. Most illusory contour models extract special image features, such as L and T junctions, while the task is known to be a difficult issue in real-world images. The proposed model uses a functional optimization approach, in which a cost value is assigned to any boundary arrangement to find the arrangement with minimal cost. The functional accounts for basic object properties, such as alignment with the image, object boundary continuity, and boundary simplicity. The encoding of these properties in the functional does not require special features extraction, since the alignment with the image only requires extraction of the image edges. The boundary arrangement is represented by a border ownership map, holding object boundary segments in discrete locations and directions. The model finds multiple possible image interpretations, which are ranked according to the probability that they are supposed to be perceived. This is achieved by using a novel approach to represent the different image interpretations by multiple functional local minima. The model is successfully applied to objects with real and illusory contours. In the case of Kanizsa illusion the model predicts both illusory and real (pacman) image interpretations. The model is a proof of concept and is currently restricted to synthetic gray-scale images with solid regions.