IDEAS home Printed from https://ideas.repec.org/
MyIDEAS: Login to save this paper or follow this series

Measure-valued images, associated fractal transforms and the affine self-similarity of images

  • Davide LA TORRE

    ()

  • Edward R. VRSCAY

    ()

  • Mehran EBRAHIMI

    ()

  • Michael F. BARNSLEY

    ()

We construct a complete metric space (Y,dY) of measure-valued images, μ:X→M(Rg), where X is the base or pixel space and M(Rg) is the set of probability measures supported on the greyscale range Rg. Such a formalism is well-suited to nonlocal image processing, i. e. , the manipulation of the value of an image function u(x) based upon values u(yk) elsewhere in the image. In fact there are situations in which it is useful to consider the greyscale value of an image u at a point x as a random variable that can assume a range of values Rg of R. One example is the characterization of the statistical properties of a class of images, e. g. , MRI brain scans, for a particular application, say image compression. Another example is statistical image processing as applied to the problem of image restoration (denoising or deblurring). Of course, it is not enough to know the greyscale values that may be assumed by an image u at a point x: one must also have an idea of the probabilities (or frequencies) of these values. As such, it may be more useful to represent images by measure-valued functions. We then show how the space (Y,dY) can be employed with a general model of affine self-similarity of images that includes both same-scale as well as cross-scale similarity. We focus on two particular applications: nonlocal-means denoising (same-scale) and multiparent block fractal image coding (cross-scale). In order to accomodate the latter, a new method of fractal transforms is formulated over the metric space (Y,dY). Nonlocal image processing has recently received a great deal of attention, fuelled in part by the exceptional success of the nonlocal means image denoising method. Fractal image coding is another example of a nonlocal image processing method. Both of these methods, which will be described briefly below, may be viewed under the umbrella of a more general model of affine image selfsimilarity, in which subblocks of an image are approximated by other sublocks of the image. Indeed, a number of other image processing methods that exploit self-similarity and the various example-based methods, also fit naturally under this nonlocal, self-similar framework.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: http://wp.demm.unimi.it/tl_files/wp/2008/DEMM-2008_045wp.pdf
Download Restriction: no

Paper provided by Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano in its series Departmental Working Papers with number 2008-45.

as
in new window

Length:
Date of creation: 22 Dec 2008
Date of revision:
Handle: RePEc:mil:wpdepa:2008-45
Contact details of provider: Postal: Via Conservatorio 7, I-20122 Milan - Italy
Phone: +39 02 50321522
Fax: +39 02 50321505
Web page: http://www.demm.unimi.it

More information through EDIRC

No references listed on IDEAS
You can help add them by filling out this form.

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:mil:wpdepa:2008-45. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (DEMM Working Papers)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.