The reconstruction algorithm is divided in two stages. The first one is the usual back propagation of the recorded holograms, the second one is devoted to generate DIC images. Common back propagation is based on the diffraction integral in the Fresnel approximation to calculate the complex wavefield
in the image plane
is the hologram in the plane
. The intensity and the phase for the optical beam transmitted by the sample are calculated from the previous equation:
for a bovine spermatozoa, and preadipocyte 3T3-F442A mouse cells are displayed. In this first pictures phase maps are obtained using the standard procedure of the double exposure, that is, object and reference holograms curvature are subtracted each other to compensate the optical aberration in the setup [26
OPD computed starting from double exposure recording for (a) a bovine spermatozoa and (b), (c) and (d) preadipocyte 3T3-F442A mouse cells; (b) and (d) are, respectively, a pseudo 3D and 2D view of the same cell.
Double exposure method or, equally, LSI combined with DH are well suited and commonly used processes to retrieve quantitative information in DH [27
]. Nevertheless QPMs in present several difficulties for specimen visualization, for example, in the end of the spermatozoa tail is not much visible due to low contrast and high dynamic range of the phase map. Moreover, in , due to the high OPD dynamic range, visibility of some details in more complex specimen is hindered thus avoiding careful recognition of internal structure and/or external filaments. DIC imaging method is much better than QPM to discriminate details of the tail as well as the head of the spermatozoa. In fact DIC allows to distinguish better phase gradients corresponding to particles presence or different density areas into mouse cells.
DIC imaging, in common optical microscope, depends strongly from several parameters as the direction along which the interfering wavefronts are shifted, the amount of the lateral shift and the bias retardation eventually introduced between them. DH allows controlling all these factors in a posteriori
analysis of the complex wavefield. Here, DIC images are numerically obtained in the off-line analysis after the holograms recording has been performed. DIC images are recovered and optimized choosing the best values for the aforesaid parameters that are independent each others. Complex wavefield
is processed to obtain DIC images of the sample in several directions after the shift quantity and the bias retardation have been chosen. For each direction a replica of
is calculated digitally by numerical shifting it in the image plane (see
). The sheared wavefront is subtracted to the original one to compute the difference phase image.
Schematic representation of the digital shearing along a chosen direction
As described in Ref. [20
], if the defocus term is considered as the main contribution to the phase retardation and higher order aberrations are neglected, the calculated phase difference is given by
The shear quantities depend on the modulus of the vector
and on the angle,
, as in the following equations:
The equivalent DIC image is obtained by:
is an arbitrary and constant phase factor [13
A conceptual flow-chart of all steps for image reconstruction procedure is shown in
Flow chart for the linear DDHIC routine
In traditional DIC microscope the best values for the aforesaid parameters (i.e. shear, bias and direction) are chosen and remain fixed during the observation time. Each image of the specimen is recorded under specific settings of shift, bias and direction. The parameters are selected in situ and in real-time, through a subjective evaluation by the observer. Such parameters cannot be changed after the image has been recorded. The method proposed here, instead, allows to set them a posteriori avoiding to fix them in real-time. In this way, the best visualization condition can be found as post-processing step by manipulating the DH retrieved data. Furthermore a dynamic visualization can be displayable by fixing two parameters while one of the three is varying. In fact static DIC images with fixed parameters values could not be sufficient to discern all specimens details. Moreover, inside the same field of view, different regions of interest can have different phase variations that would require, for optimal visualization, a different parameters settings. Consequently, as it will be provided in the following paragraphs, dynamic visualization through DDHIC movies, can furnish a complete view of the sample allowing to detect all details and architecture of the phase-object under investigation. We tested the procedure on spermatozoa and mouse cells whose OPD was previously showed in in order to compare the resulting analysis for standard DH and DDHIC. The complete procedure of the method is illustrated in the flow chart of . Dynamic visualization provided from the first and second movies allow to optimize amount of shear and bias while the final dynamical PC-imaging is provided by varying the direction. Dynamic bias variation, shear quantities and shear direction will be showed.
3.1 Setting of shear pixels number
Linear DDHIC image processing is performed on bovine sperm and mouse cells to prove the routine feasibility in all its steps. One of the parameters to be set is the shear quantity. In
DDHIC images obtained by Eq. (3)
are displayed for different value of shear quantities
. Taking into account Eqs. (2)
, shear angle
is kept constant at 30° while the modulus of vector
is changed varying from 0.23μm to 1.38μm corresponding to a pixels variation from 1RP to 6RP. The phase factor
is kept constant.
Fig. 5 (a)-(f) DDHIC images of a mouse cell for different quantity of the shear pixels number changed from 1RP to 6RP (Media 1).
Gradients in the optical path along the selected direction are better visible for higher values of the shear pixels number. Inside the cell perimeter a rising shadow-cast effect bestows a pseudo three-dimensional realism. Some structures in the cell are much more visible by increasing the shear. Nevertheless, for rising shift values the noise around the cell grows up too. Moreover, even if the contrast inside is improved, augmenting the shear there is, of course, a reduction of the spatial resolution. A compromise between contrast visibility and noise level is desirable. On the other hand a single shear value is not the optimal parameter value for all phase variations in the sample. For example some structures are much more visible for high shear values. A movie for varying
values is supplied.
3.2 Bias setting
Another parameter to be chosen is the bias retardation introduced to enhance the contrast between the specimen and the background. The cell investigated is a mouse cell whose QPM was displayed in . DDHIC images for different values of the bias are displayed in
–, where the shear direction is kept fixed at 45° and the shear quantity, selected before, at 4RP. It is clear, by observing that appropriate bias values allow to enhance, in correct way, the phase-contrast. In particular, phase values for the bias of 0.0 rad and 3.0 rad are the best choice. Also for this parameter a movie is provided showing the contrast variation as function of the bias. Eventually the choice can be evaluated automatically.
Fig. 6 DDHIC images of a mouse cell for different bias retardation values. (a)-(i) the same mouse cell is displayed for bias values ranging from to with step of 1rad. (Media 2).
3,3 Shear direction setting
As final step, for a complete visualization of the sample, a DDHIC routine is implemented with aim to build-up a movie with dynamic DIC along all directions. In fact differences in the light optical path are dependent on the direction of the shear and consequently diverse phase gradients are detectable for each different shear direction. DDHIC is accomplished in fast and effective way by applying the routine just modifying one parameter, the shear angle
(). Images of the mouse cell for different shear directions are reported in
, Shear quantity is fixed at
and bias at
. Red arrows indicate the shear directions while green arrows point-out various cell structures that are visible or not depending from the direction of shear. It is clear that all the structures are visible only if a dynamic phase contrast along all direction is provided to the observer. DDHIC furnish optimal dynamic visualization that allows to detect all structures (see Media 3
As further example, DDHIC is applied to another biological structure, a cow spermatozoa cell. The optimized shear value is kept constant and equal to 2RP corresponding to 364nm in the image plane while the bias is fixed at
. Shear angle (or direction) is changed and resulting images are showed in
. The shear angles range is 360° with step of 30°C. From the picture is clear that, depending on
, different specimen details and regions are enhanced. For example, at angles
the last portion of the tail is better contrasted and visible in respect to that of the QPM obtained by DH shown in .
Fig. 8 DDHIC images in different shear direction of a sperm cell; (a)-(n) the same spermatozoa is displayed for shear angle values ranging from to with step of 30°. (Media 4).
Furthermore for angles
it is possible to recognize the separation between the acrosome and postacrosom regions of spermatozoa cell while the high dynamic range of the QPM in does not allow to distinguish it.
The advantage offered by DDHIC stands in the possibility to obtain DIC along any direction “a posteriori” even if the object is fast moving or experience changes during the observation time. In fact in such cases it is not possible to operate rapidly mechanical movement to optimize the three parameters for a good and effective high contrast observation: amount of shear bias retardation and shear direction. DH allows to record dynamically the sequence of digital hologram during the observation time and if needed the focus can also be adjusted a posteriori too. Moreover shear and bias can be adjusted during the numerical reconstruction to obtain the highest obtainable contrast.
Furthermore, the possibility to visualize dynamically the DIC of object by changing continuously the shearing direction offers one more advantage in visualizing better the various details for the observer.
3,4 Time dependent DDHIC and QPM
An evaluation of the difference between traditional DH phase image and DDHIC visualization is reported for a sample whose position is time dependent. Preadipocyte mouse cell during differentiation is the sample investigated. Hundred of holograms are recorded to detect cell displacement and modification. Several of them are selected and processed to investigate specimen temporal behavior. Movies are realized to study cell morphology alteration through DDHIC as well as QPM to furnish a complete a-posteriori specimen analysis. A Movie related to
is made of four sub-movies. Two displays the QPM in 2D and pseudo 3D representation while the others two show DDHIC images for two perpendicular shear directions, respectively. In two frames of that movie are shown at two different instants of time. Red circles indicate some elements inside cell perimeter observable in DDHIC configuration but not visible in the QPM. High OPD range hinder the visualization of such details that are completely unnoticeable from the OPD image but turn out discernible in DDHIC images.
Fig. 9 Sample modification tracking realized by QPM and DDHIC methods; (a) is made of four sub-figures correspondent to the istant of time : two images display DDHIC for different shear angles while the other two are the quantitative phase distributions (more ...)
is a clear example of the usefulness of the post processing procedure because different details are visible for different shear directions and, in case of floating object, such choice can be realized only after image recording. Thanks to the parameters control obtainable by DH features the observer has a wide-ranging statement of cell structure even when it is moving.