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Axial Transmitted Light Transformation Imaging: A version of RTI for oblique illumination?

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I've been looking into oblique transmitted light microscopy as an inexpensive way of generating both optical contrast and resolution closer to the Abbe limit. A recent review and evaluation of the technique can be found here:

https://www.sciencedirect.com/science/article/pii/S0968432817303888

Oblique illumination is a technique where the specimen is illuminated via axial light from the condenser via a patch stop with an open slit or crescent, placed on the bottom of the condenser.

Oblique illumination is unusual in that it shifts the placement of the zero order and higher orders of light on the rear focal plane off-center, allowing higher orders of light to enter into image formation on one side of the zero order, but losing orders of light on the other side of the zero order. This actually enhances resolution, but only in part of the image, and oblique illumination has the unusual effect of having better resolution on the illuminated side of the image than on the shaded one.

It is also possible to set up a movable slit on the condenser such that one can illuminate the specimen from several directions. My thought is that this in combination with some version of RTI might be able to generate a composite image which can recreate illumination from multiple directions, and perhaps even a composite image that could give a single image with optimal resolution throughout the image. Obviously, this is a novel technique using a directional form of transmitted light rather than reflected light and may require some modification of the RTI algorithm.

I'll note that my background is in biological microscopy and that I am far from an specialist in optics or imaging algorithms, however, this is an idea I'd like to bounce off of others and see if it could be developed.

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Eleni Koutoula and colleagues at the Archaeological Computing Research Group at U. of Southampton did some work on transmitted RTI using visible light and infrared radiation some years ago, for example, this demonstration at the CAA conference in 2013: 
https://uk.caa-international.org/2013/02/19/reflectance-transformation-imaging/

As you mentioned, among the challenges of this method would be that the setup doesn't follow a basic assumption on which RTI is based, which is that the reflected light can be modeled using a biquadratic function in the case of PTMs or hemispherical harmonics in the case of RTIs. That doesn't mean you couldn't find a way to process transmitted "RTI" microscopy and get an interesting result (once you supply a set of light positions) or that the information gained using transmitted "RTI" microscopy wouldn't be useful.  With more transmitted light positions, you should get more information than is possible using a single light position. 

There are examples of reflected RTI microscopy using a light array dome, for example, see Paul Messier's "monkey brain": 
http://www.paulmessier.com/single-post/2014/10/01/The-first-batch-of-studios-Monkey-Brain-light-arrays-for-RTI-microscopy-have-shipped-to-museum-conservation-departments

Also, there are some interesting results using an LED array microscope with phase contrast, bright-field, and dark-field illumination to achieve extremely high resolution and 3D imaging:
http://www.laurawaller.com/research/

 

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