Image Processing

Even with the best possible real microscope, the image of a precisely defined spot is no longer as sharp. During its passage through the microscope, a dot is turned into a disk as an inevitable consequence of the wave nature of light and the optics within the microscope. This is without even considering less fundamental issues such as lens imperfections.

Since each spot of a source is turned into a disk in the image, over the source as a whole the resulting image becomes fuzzy. To improve the image, we need to undo this inherent image smearing. The technique to do this is called regularised deconvolution. It is not a simple filtering technique that removes noise.

The technicals:

Below is a flow diagram showing the algorithm used to process images.

Reciprocal space

Real space

Fourier transform

Working from left to right:

• The optical transfer function (OTF) is defined for the microscope and mathematically describes the passage of "information" from the source object to the image. The OTF is frequency dependent, the excitation and emission stages are handled separately. For the confocal microscope, an explicit expression for the OTF can be derived in three dimensions.
• Each of these is Fourier transformed to generate the corresponding point spread function (PSF), one for the illumination stage, the other for the imaging stage.
• These are combined to give a total PSF, then Fourier transformed back to given a total OTF.
• The image data to be processed, once Fourier transformed, is deconvolved with the total OTF. This step removes the impact of the microscope optics both in the illumination process and the imaging process.
• In the above step, however, there is an additional challenge. Since the microscope has a finite spatial resolution, there are certain sized features it cannot resolve. This corresponds to a cut-off of information transfer at large spatial frequencies. Therefore the deconvolution problem is ill-posed (basically, there's a division by zero to deal with). To address this difficulty it is necessary to introduce a regularisation scheme to effectively fill in the gap left by the incomplete information. This also combats the problem of high frequency noise which otherwise wrecks the Fourier transformed data.
• Finally, the deconvolved image is Fourier transformed back to give the final real space image.

Here are some before and after images of an aggregated emulsion which show that this process works.

The image on the left is the raw image from the microscope. The version on the right has been deconvolved according to the prescription above. No other changes, for example filtering or contrast enhancement, have been made.

Although commercial packages are now available to do all this, we have developed our own version. One of the advantages of this is that we retain complete control over every stage of the image processing cycle. In particular our routine handles the full three dimensional problem and enhances confocality by incorporating information from neighbouring optical slices. We can also expand the technique into other domains.

Jargon busting: