Spotlight Summary by David M. Paganin
Understanding the twin-image problem in phase retrieval
Time and again I find myself returning my mind’s eye to a small number of archetypal experiments in optics, examining them afresh from a (hopefully!) ever deeper perspective.
The Young’s double-slit experiment tops my list of optics scenarios which continually reward repeated pondering. Near the top of the list is the problem of coherent diffractive imaging (CDI), namely the question of determining the phase of a coherent complex scalar optical field given (i) the magnitude of its Fraunhofer diffraction pattern; and (ii) some “support” information, namely information regarding an area outside which the field to be reconstructed is zero.
There are evident parallels between the CDI problem and that of inline (Gabor) holography. Indeed, the problem of Gabor holography is in essence the same as that posed in the previous paragraph, with (i) “Fraunhofer diffraction pattern” replaced by “Fresnel diffraction pattern”; and (ii) no support information.
Mention of inline holography immediately brings to mind a famous limitation of the method, namely the twin-image problem. CDI, with its immense power as a means of lensless imaging in which (like inline holography) the process of image reconstruction is undertaken by using optical software to “decode” the measured hologram, also suffers from the twin-image problem. The question is, how can the twin-image problem of CDI – which has, at its root, the fact that the real-space complex wave functions f(x,y) and f*(-x,-y) have the same Fraunhofer diffraction pattern – be overcome?
The authors of this paper answer this question by first uncovering a remarkable new fact regarding twin-image induced stagnations in two-dimensional CDI reconstructions. They have shown that, when CDI reconstructions fail on account of the twin-image problem, the reconstructed Fraunhofer-space wavefunction is spatially separated into speckle-like “domains” where the phase is either close to that corresponding to f(x,y), or close to that corresponding to the twin-image f*(-x,-y).
This fact gives a powerful piece of new knowledge in the quest to overcome the twin-image problem of CDI. To this end, the authors show how a recognition of the previously-mentioned domain structure can be used to render CDI algorithms (including the famous “HIO” and “ER” algorithms of the senior author) more robust to twin-image induced stagnation. Please read the paper for details!
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The Young’s double-slit experiment tops my list of optics scenarios which continually reward repeated pondering. Near the top of the list is the problem of coherent diffractive imaging (CDI), namely the question of determining the phase of a coherent complex scalar optical field given (i) the magnitude of its Fraunhofer diffraction pattern; and (ii) some “support” information, namely information regarding an area outside which the field to be reconstructed is zero.
There are evident parallels between the CDI problem and that of inline (Gabor) holography. Indeed, the problem of Gabor holography is in essence the same as that posed in the previous paragraph, with (i) “Fraunhofer diffraction pattern” replaced by “Fresnel diffraction pattern”; and (ii) no support information.
Mention of inline holography immediately brings to mind a famous limitation of the method, namely the twin-image problem. CDI, with its immense power as a means of lensless imaging in which (like inline holography) the process of image reconstruction is undertaken by using optical software to “decode” the measured hologram, also suffers from the twin-image problem. The question is, how can the twin-image problem of CDI – which has, at its root, the fact that the real-space complex wave functions f(x,y) and f*(-x,-y) have the same Fraunhofer diffraction pattern – be overcome?
The authors of this paper answer this question by first uncovering a remarkable new fact regarding twin-image induced stagnations in two-dimensional CDI reconstructions. They have shown that, when CDI reconstructions fail on account of the twin-image problem, the reconstructed Fraunhofer-space wavefunction is spatially separated into speckle-like “domains” where the phase is either close to that corresponding to f(x,y), or close to that corresponding to the twin-image f*(-x,-y).
This fact gives a powerful piece of new knowledge in the quest to overcome the twin-image problem of CDI. To this end, the authors show how a recognition of the previously-mentioned domain structure can be used to render CDI algorithms (including the famous “HIO” and “ER” algorithms of the senior author) more robust to twin-image induced stagnation. Please read the paper for details!
Article Reference
Understanding the twin-image problem in phase retrieval
- J. Opt. Soc. Am. A 29(11) 2367-2375 (2012)
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