On the recovery of images from partial information using [delta]²G filtering
Creators: name: Reimer, James Allen
Publisher: name: University of British Columbia
Year: 2010

description: This thesis considers the recovery of a sampled image from partial information, based on the 'edges' or zero crossings found in ∇²G filtered versions of the image. A scheme is presented for separating an image into a family of multiresolution images, using low pass filtering, subsampling, and ∇²G filtering. A scheme is also presented for merging this family of ∇²G filtered images to rebuild the original. The recovery of each of the ∇²G filtered images from their 'edges' or zero crossings is then considered. It has been suggested that ∇²G filtered images might be characterized by their zero crossing locations. It is shown that ∇²G filtered images, filtered in 1-D or 2-D are not, in general, uniquely given within a scalar by their zero crossing locations. Two theorems in support of such a suggestion are considered. The differences between the constraints of Logan's theorem and ∇²G filtering are considered, and it is shown that the zero crossings which result from these two situations differ significantly in number and location. Logan's theorem is therefore not applicable to ∇²G filtered images. A recent theorem by Curtis on the adequacy of zero crossings of 2-D functions is also considered. It is shown that the requirements of Curtis' theorem are not satisfied by all ∇²G filtered images. Further, it is shown that it is very difficult to establish if an image meets the requirements of Curtis' theorem. Examples of different ∇²G filtered images with the same zero crossings are also presented. While not all ∇²G filtered images are uniquely characterized by their zero crossing locations, the practical recovery of real camera images from this partial information is considered. An iterative scheme is developed for the reconstruction of a ∇²G filtered image from its sampled zero crossings. The zero crossing samples are localized to the original image sample grid. Experimental results are presented which show that the recovered images, while retaining many of the features of the original, suffer significant loss. It is shown that, in general, the full recovery of these images in a practical situation is not possible from this partial information. From this experimental experience, it is proposed that ∇²G filtered images might be practically recovered from their zero crossings, with some additional characterization of the image in the vicinity of each zero crossing point. A simple, non-iterative scheme is developed for extracting a characterization of the ∇²G filtered image, through the use of an image edge model and a local estimation of a contrast figure in the vicinity of each zero crossing sample. A redrawing algorithm is then used to recover an approximation of the ∇²G filtered image from its zero crossing locations and the extracted characterizations. This system is evaluated using natural scene and synthetic images. Resulting image quality is good, but is shown to vary depending on the nature of the image. The advantages and disadvantages of this technique are discussed. The primary shortcoming of the implemented local estimation technique is an assumption of edge independence. A second approach is developed for characterizing the ∇²G filtered image zero crossings, which eliminates this assumption. This method is based on 2-D filtering, and provides a new technique for the recovery of a ∇²G filtered image from its sampled zero crossings. The method does not involve iteration or the solution of simultaneous equations. Good image reconstruction is shown for natural scene images, with the ∇²G filtered image zero crossings localized only to the original image sample grid. The advantages and disadvantages of this technique are discussed. The application of this recovery from partial information technique is then considered for image compression. A simple coding scheme is developed for representing the zero crossing segments with linear vector segments. A comparative study is then considered, examining the tradeoffs between compression tuning parameters and the resulting recovered image quality.