John F. Restrepo

Magnified reconstruction of digitally recorded holograms by Fresnel–Bluestein transform

A method for numerical reconstruction of digitally recorded holograms with variable magnification is presented. The proposed strategy allows for smaller, equal, or larger magnification than that achieved with Fresnel transform by introducing the Bluestein substitution into the Fresnel kernel. The magnification is obtained independent of distance, wavelength, and number of pixels, which enables the method to be applied in color digital holography and metrological applications. The approach is supported by experimental and simulation results in digital holography of objects of comparable dimensions with the recording device and in the reconstruction of holograms from digital in-line holographic microscopy.

Automatic three-dimensional tracking of particles with high-numerical-aperture digital lensless holographic microscopy

We present an automatic procedure for 3D tracking of micrometer-sized particles with high-NA digital lensless holographic microscopy. The method uses a two-feature approach to search for the best focal planes and to distinguish particles from artifacts or other elements on the reconstructed stream of the holograms. A set of reconstructed images is axially projected onto a single image. From the projected image, the centers of mass of all the reconstructed elements are identified. Starting from the centers of mass, the morphology of the profile of the maximum intensity along the reconstruction direction allows for the distinguishing of particles from others elements. The method is tested with modeled holograms and applied to automatically track micrometer-sized bubbles in a sample of 4 mm3 of soda.

Interference in phase space

The phase-space representation of interference based on the marginal power spectrum gives new insight on interference, enlarging its potential applications by means of the principle of spatial coherence modulation. Carrier and (0,pi)-rays produced by three different types of supports are introduced for describing interference as the result of adding the radiant energy propagated by the carriers and the modulating energy (which can be positive or negative) propagated by the (0,pi)-rays. Numerical examples are presented.

Diffraction-based modeling of high-numerical-aperture in-line lensless holograms

Conventionally, for modeling in-line lensless holograms of systems with high numerical apertures and diverging spherical illumination, the samples are considered as an ensemble of secondary point sources. On following Huygenss principle, the in-line hologram is the result of the amplitude superposition of the secondary spherical wavefronts with the wavefront originated in the point source. Albeit simple, this approach limits the shapes of the objects that can be modeled and the computation time rises with the complexity of the sample. In this work, we present a diffraction-based approach to model in-line lensless holograms. Samples with any shape or size can be modeled for in-line holographic systems with numerical apertures up to 0.57. The method is successfully applied to model objects of intricate submicrometer structures and/or multiple samples lying within a unique sample volume.

Numerical evaluation of the limit of concentration of colloidal samples for their study with digital lensless holographic microscopy.

The number of colloidal particles per unit of volume that can be imaged correctly with digital lensless holographic microscopy (DLHM) is determined numerically. Typical in-line DLHM holograms with controlled concentration are modeled and reconstructed numerically. By quantifying the ratio of the retrieved particles from the reconstructed hologram to the number of the seeding particles in the modeled intensity, the limit of concentration of the colloidal suspensions up to which DLHM can operate successfully is found numerically. A new shadow density parameter for spherical illumination is defined. The limit of performance of DLHM is determined from a graph of the shadow density versus the efficiency of the microscope.

Real-time numerical reconstruction of digitally recorded holograms

The numerical reconstruction of digitally recorded holograms has constituted the bottle neck for real-time digital holography. The reconstruction process can be understood as the diffraction that undergoes a wavefront as it illuminates the digitally recorded hologram. As this process is done numerically, the reconstruction of a M × N pixels hologram into an image of similar dimensions is an operation with a Ο (M × N)2 complexity. The diffraction process can be represented by a Fresnel transform or a scalable convolution of the recorded hologram. In these representations the numerical reconstruction has a complexity of Ο (M × log N)2, still quite demanding computationally if the holograms are of 2048 × 2048 pixels. In this work, the power provided by a Graphics Processing Unit (GPU) is used to accelerate the numerical reconstruction of digitally recorded holograms. The methodology is supported on the parallelization of typical Fresnel transform and scalable reconstruction algorithms. On reconstructing holograms of 2048 × 2048 pixels, the reconstruction is speeded up 20 times for the former method and 11 times for the scalable convolution. For holograms of 1024 × 1024, the accelerated reconstruction methods allow for real-time digital holography.

Real time numerical reconstruction of digitally recorded holograms in digital in-line holographic microscopy by using a graphics processing unit

The paper presents the way that colour can serve solving the problem of calibration points indexing in a camera geometrical calibration process. We propose a technique in which indexes of calibration points in a black-and-white chessboard are represented as sets of colour regions in the neighbourhood of calibration points. We provide some general rules for designing a colour calibration chessboard and provide a method of calibration image analysis. We show that this approach leads to obtaining better results than in the case of widely used methods employing information about already indexed points to compute indexes. We also report constraints concerning the technique. Nowadays we are witnessing an increasing need for camera geometrical calibration systems. They are vital for such applications as 3D modelling, 3D reconstruction, assembly control systems, etc. Wherever possible, calibration objects placed in the scene are used in a camera geometrical calibration process. This approach significantly increases accuracy of calibration results and makes the calibration data extraction process easier and universal. There are many geometrical camera calibration techniques for a known calibration scene [1]. A great number of them use as an input calibration points which are localised and indexed in the scene. In this paper we propose the technique of calibration points indexing which uses a colour chessboard. The presented technique was developed by solving problems we encountered during experiments with our earlier methods of camera calibration scene analysis [2]-[3]. In particular, the proposed technique increases the number of indexed points points in case of local lack of calibration points detection. At the beginning of the paper we present a way of designing a chessboard pattern. Then we describe a calibration point indexing method, and finally we show experimental results. A black-and-white chessboard is widely used in order to obtain sub-pixel accuracy of calibration points localisation [1]. Calibration points are defined as corners of chessboard squares. Assuming the availability of rough localisation of these points, the points can be indexed. Noting that differences in distances between neighbouring points in calibration scene images differ slightly, one of the local searching methods can be employed (e.g. [2]). Methods of this type search for a calibration point to be indexed, using a window of a certain size. The position of the window is determined by a vector representing the distance between two previously indexed points in the same row or column. However, experiments show that this approach has its disadvantages, as described below. * E-mail: A.Nowakowski@ire.pw.edu.pl Firstly, there is a danger of omitting some points during indexing in case of local lack of calibration points detection in a neighbourhood (e.g. caused by the presence of non-homogeneous light in the calibration scene). A particularly unfavourable situation is when the local lack of detection effects in the appearance of separated regions of detected calibration points. It is worth saying that such situations are likely to happen for calibration points situated near image borders. Such points are very important for the analysis of optical nonlinearities, and a lack of them can significantly influence the accuracy of distortion modelling. Secondly, such methods may give wrong results in the case of optical distortion with strong nonlinearities when getting information about the neighbouring index is not an easy task. Beside this, the methods are very sensitive to a single false localisation of a calibration point. Such a single false localisation can even result in false indexing of a big set of calibration points. To avoid the above-mentioned problems, we propose using a black-and-white chessboard which contains the coded index of a calibration point in the form of colour squares situated in the nearest neighbourhood of each point. The index of a certain calibration point is determined by colours of four nearest neighbouring squares (Fig.1). An order of squares in such foursome is important. Because the size of a colour square is determined only by the possibility of correct colour detection, the size of a colour square can be smaller than the size of a black or white square. The larger size of a black or white square is determined by the requirements of the exact localisation step which follows the indexing of calibration points [3]. In this step, edge information is extracted from a blackand-white chessboard. This edge information needs larger Artur Nowakowski, Wladyslaw Skarbek Institute of Radioelectronics, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warszawa, A.Nowakowski@ire.pw.edu.pl Received February 10, 2009; accepted March 27, 2009; published March 31, 2009 http://www.photonics.pl/PLP

Variable magnification in numerical reconstruction of digitally recorded holograms

The numerical reconstruction of digitally recorded holograms can be done via different approaches. Convolution, angular spectrum and Fresnel transform are the most widely used. The size of the reconstruction pixel equals that of the recording device for the two former and for the latter that size is controlled by the experimental parameters; wavelength, number of pixels and reconstruction distance determine the achievable size of the of the reconstructed pitch hence the magnification of the holographic system. It has been a challenge to have a numerical reconstruction method in which the magnification can be chosen at will. In this work, a method for numerical reconstruction of digitally recorded holograms with variable magnification is presented. It is supported on the Fresnel-Bluestein transform that allows for changing the magnification, namely the size of the reconstruction pixel, independent of distances, wavelength and number of pixels. The method is applied to reconstruct holograms recorded in off-axis and in-line setups. The reached magnification is contrasted with that achieved as the holograms are reconstructed with Fresnel transform. Since the proposed method does not modify the number of pixels of the hologram, neither the wavelength nor the reconstruction distance, it suits for application like color digital holography, metrological application among others.

Concentration limit for mono-disperse colloids observable with numerical DIHM

Digital In-line Holographic Microscopy (DIHM) is a two-steps microscopy technique that allows for accessing to complex wave information of the optical field scattered by a sample. Initially, the sample is illuminated by a spherical wavefront such that the amplitude superposition of the portions of the spherical wavefront scattered and not by sample, is recorded on a digital camera; the recorded intensity is often referred as in-line hologram. On the second step, a numerical diffraction scheme is used to emulate the diffraction of a spherical wavefront by the in-line hologram therefore producing a reconstruction in amplitude and phase of the original object. Due to its evident experimental simplicity, DIHM is a widely used technique for in-situ applications and more recently on real time measurements. This widespread employment of the technique introduces the necessity of establishing the practical limits achievable with this imaging technique. Particularly, for the practical study of mono-disperse colloids, the critical concentration is a relevant factor to identify, in order to establish the optimal conditions up to which DIHM can successfully work. The reconstruction step produces a set of intensity images, at different axial distances, containing the information of all the recorded particles; in large study volumes and high concentrations the number of particles overcome the easiness of manual processing and therefore evidences the need of implementing more automatic tracking algorithms. In this way the limits of applicability of DIHM rely on both the experimental configuration and the digital processing. With the use of a modeling tool for DIHM and a semi-automatic tracking algorithm, a numerical estimation of the concentration limit for which DIHM can work is proposed, following the analysis for its dependence with the experimental configuration of the recording process.

Modelling High-NA In-Line Holograms

A diffraction-based approach is presented for modelling high-NA in-line holograms. This approach circumvents the limitations on size and shape of the modelled samples. The computation load is reduced by using Bluestein approach to DFTs.