Holography, tomography and 3D microscopy as linear filtering operations

Abstract

In this paper we characterise 3D optical imaging techniques as 3D linear shift invariant filtering operations. From the Helmholtz equation that is the basis of scalar diffraction theory we show that the scattered field, or indeed a holographic reconstruction of this field, can be considered to be the result of a linear filtering operation applied to a source distribution. We note that if the scattering is weak, the source distribution is independent of the scattered field and a holographic reconstruction (or in fact any far-field optical imaging system) behaves as a 3D linear shift invariant filter applied to the refractive index contrast (which effectively defines the object). We go on to consider tomographic techniques that synthesise images from recordings of the scattered field using different illumination conditions. In our analysis we compare the 3D response of monochromatic optical tomography with the 3D imagery offered by confocal microscopy and scanning white light interferometry (using with quassi-monochromatic illumination) and explain the circumstances in which these approaches are equivalent. Finally, we consider the 3D response of polychromatic optical tomography and in particular the response of spectral optical coherence tomography and scanning white light interferometry.

Depth-resolved Imaging and Displacement Measurement Techniques Viewed as Linear Filtering Operations

“…There are other realizations of TSI: in biology, Lauer microscopy is used to produce tomographic views of cells [13,18]. In a backscattering configuration this is essentially the same as TSI, the only difference being that in TSI the central angle is non zero, which greatly increases depth resolution and also allows the separation of inplane and out-of-plane sensitivity by combining phase measurements obtained for different illumination directions, symmetric with respect to the observation direction.…”

“…In common with all far-field optical instruments, however, they only collect a fraction of the scattered field by the entrance pupil, defined by the objectspace numerical aperture of the system N A = n sin() where  is the half-angle subtended by the cone of rays accepted by the aperture from a point in the object. It has been shown in [13] that when an axially symmetric system with finite N A is used to measure the complex scattered field, the transfer function can be written as 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 12…”

“…Equations (4) and (5) express the reconstruction problem as a linear, shift invariant, filtering operation [13]. H(r) is a key function for all the depth-resolving techniques since it determines their spatial resolution and sensitivity.…”

“…holography) was first considered by Wolf [8] and expressed in a frequency space representation [9,10], while synthesis of 3-D images has been advanced by others [11,12]. More recently a mathematical framework was proposed that considered holography, tomography and methods of 3-D microscopy as linear filtering operations [13] and this approach will be adopted to compare the depth-resolved, displacement measuring systems that are the subject of this paper.…”

High performance computing for a 3-D optical diffraction tomographic application in fluid velocimetry

“…Let us remark that, E m (r) is a far-field measurement and therefore, the forward solution E sth (r) needs to be filtered to remove the non-propagating waves [17], thus obtaining the expected measured field, E c (r). The optimization problem involved in the tomographic reconstruction consists in a minimization of the cost function defined by the square root difference between the measured and the computed field.…”