1. Introduction

It is commonly recognized that digital holography (DH) and holographic display constitute the best-known framework for truly 3D imaging and video as they aim to recreate the optical field emitted by a recorded scene [1]. With present state of technology, the digital reproduction of all 3D perception cues offered by classical holography is very difficult. Therefore, known implementations of holographic video involve some compromises made on the side of data capture or display, which influence visual properties of the reconstructed image. One common approach is to simulate holographic content with computer generated holograms (CGH) [2–5], which pose no restrictions to data capture system and focus on improving the display side [6–11]. Another proposition is to use integral photography for 3D recording [12], which feeds a standard holographic display based on Spatial Light Modulator (SLM) technology [13]. The technique facilitates 3D perception with a naked eye but is fundamentally limited in recorded image depth and resolution. A complete chain of end-to-end holographic imaging was attempted by a few groups. Most of the works involve an elementary hardware design consisting of a single holographic camera and a corresponding SLM [14–17], which projects a real image back to the object space. More advanced systems utilize a set of CDD cameras [18] or a synthetic aperture acquisition [19] coupled with a multi-SLM spherical display [20–22] to extend horizontal parallax. In all cases the obtained image suffers from inverted depth and a key-hole effect due to divergence of the reconstructed object field. In effect, only small images can be observed as whole from a single position. An alternative way is to project a virtual image of a Fresnel hologram [23]. In this setting the image is naturally orthoscopic, but the viewing zone coincides with the physical SLM, which means that the eye cannot be placed in the optimal region of image perception. Our recent work [24] provides a distortion-free monochromatic solution to pseudoscopic geometry of the real image. However, the method puts restriction on axial position of the image, requires additional lens and there are some losses of image quality at the processing stage.

In contrary to the state of the art this work proves that it is possible to obtain color reconstructions of large real orthoscopic objects, with arbitrary position and size, also without 3D deformations where the full image can be viewed with naked eyes. The geometrical manipulations of the image are especially simple, they require only additional hologram cropping. All these features are achieved by presented here Fourier holographic imaging, which is based on axial decoupling of both ends of the system. The system involves standard lensless digital Fourier holographic recording [25] with three wavelengths, an improved version of LED illumination display build using the viewing window (VW) concept [26] and introduced here Confocal Fresnel Propagation (CFP). The proposed imaging approach has small viewing angle, thus the position of viewer is spatially restricted, and in consequence viewer is unable to observe different perspectives of image. This feature can be improved using spatial or temporal multiplexing [9,18–22] or eye tracking system [26]. The Fourier holographic imaging is based on capturing, processing and displaying of the object wave without the corresponding spherical phase factor. This way of wave encoding is referred to as Compact Space Bandwidth Product (CSBP) representation [27]. Thanks to its full use the amount of Space Bandwidth Product (SBP) involved in the entire processing path is constant and determined by the display SLM. Thus, the characteristic advantage of the lensless Fourier holographic capture system in terms of SBP [28] is here extended for the entire imaging path. Experimentally, the numerical efficiency and flexibility of the solution is demonstrated by showing axial relocation and magnification of the orthoscopic holographic image.

The coupling between capture and display systems is carried out by means of the introduced CFP method for Fourier Holography where the carriers of propagated beams are confocal, i.e. have common focus point. The efficiency and accuracy of the solution is a result of fundamental feature of the algorithm where the size and resolution of the result is scaled according to the ratio of the radii of propagated beams. This important feature allows free manipulation of axial position as well as linear size of the image without any additional computational cost. Also, high quality image is obtained because the same readout curvature is introduced numerically and optically. The CFP method requires only two Fourier Transformations (FT) and no zeropadding. The technique takes a sampling idea from the non-paraxial CSBP Angular method [27], which on contrary requires four FTs, additional zeropadding and it is not based on confocal geometry.

A downside to all the aforementioned techniques is that they use laser light sources on the display side producing coherent speckles and being potentially hazardous to human vision. Color LED illumination is the preferred alternative [29–33], but it is not straightforward to employ due to its reduced spatial coherence and related image blur. The use of LED in VW display was already presented [30]. Here it is theoretically and experimentally demonstrated that high quality full color reconstruction can be obtained for objects with large depths, here 10 m. The theory indicates that the further improvement of reconstruction depth is possible when selecting the display magnification optimal for the visual perception. In our previous display based on Fresnel configuration and unitary magnification the low coherence of LEDs limited the reconstruction depth to 50 mm [31].

In the following we (i) present principle of our 3D Fourier imaging, (ii) develop the CFP method, (iii) analyze geometrical deformations remaining in the reconstructed 3D volume of real-world objects, (iv) show experimental implementation, (v) analyze coherence effects, and (vi) demonstrate display performance in reconstruction of holographic content with emphasis on image quality perceived by a naked eye for varying reconstruction depth.

2. 3D Imaging Principle

According to the sine condition in the Abbe’s theory of image formation [34], nonunitary magnification of a 3D image necessarily entails a nonlinear relation between transverse and longitudinal magnifications. Within the paraxial regime the longitudinal magnification is a square of the transverse one. Thus, a reconstructed volume is heavily distorted at larger magnifications. One example of 3D imaging is a direct reconstruction of coherent digital hologram of an object of size 100 mm in the LED display system [31], for which the maximum size of the image is only 10 mm. In this case the required scale manifests in an almost flat reconstruction as the transverse and longitudinal magnifications are 0.1 and 0.01, respectively, while our display has the capability of reconstructions of close 3D copies of recorded color objects.

Figure 1 shows a schematic of the image formation process in the proposed Fourier holographic technique. Figure 1(a) illustrates registration of the primary hologram realized in the lensless DH Fourier configuration, while Fig. 1(b) depicts reconstruction of a secondary hologram in the Fourier-based LED holographic display. The CFP routine provides transfer between the primary and secondary hologram plane. In the capture geometry the recorded object term OR* is a product of the object wave O and the conjugated reference wave R*:

 

Fig. 1 Imaging principle of Fourier a) DH capture and b) display geometry.

Download Full Size | PDF

Our holographic imaging is axially decoupled, which means that the axial position R₂ of reconstructed image can be set freely. Thus, the secondary hologram plane ξ₂, being the magnified SLM plane, does not coincide with the plane x₂ carrying the image content (FT of the hologram). Nevertheless, one can always set the image position to R₂ = R₁, which enables reconstruction of a holographic copy of an object with unitary magnification, even when its size exceeds the diameter of the magnified SLM. The arbitrary selection of R₂ is carried out numerically by means of the CFP for converging or diverging fields, described in detail in Sec. 3. The algorithm takes an image field O_c(x₂,R₂), and returns an object wave O_c(ξ₂,F_f) of secondary hologram plane with unchanged SBP. The sample size is modified according to F_f/R₂. The spherical carriers have a common focal point at VW, thus the propagation distance always equals to R₂ − F_f. On the display side, the secondary hologram plane ξ₂ contains the field lens L_f. The SLM is addressed with the CSBP field O_c(ξ₂,F_f) while the curvature of object field is supplemented by the lens L_f.

The full processing path of the capture-display system has four main steps: (i) numerical reconstruction of the object wave O_c(x₁,R₁) (Eq. (2)), (ii) numerical conversion of O_c(x₁,R₁) into O_c(x₂,R₂) (Sec. 4), (iii) propagation over the distance R₂ − F_f to plane ξ₂ resulting in O_c(ξ₂,F_f) (Sec. 3) and (iv) complex coding of O_c(ξ₂,F_f) [31,35]. The strength of our Fourier imaging relies mosty on steps (ii) and (iii) of the processing where axial position and dimension of reconstructed image are set. The axial locations of the object and its image are determined by R₁ and R₂ of the corresponding CSBP representations. Thus, the change of axial position of the image does not require additional processing. The position set as readout curvature R₂ is a parameter of CFP. There is another advantage coming from our imaging technique. Since the hologram and the CSBP representation of the image are related by FT, the magnification can be realized by clipping or zeropadding of the hologram realized in steps (i), (ii).

Even though the capture and display geometries are analogous, the distortion-free 3D imaging of the entire object may not always be possible. Both systems may use different wavelengths or have different angular field of views (aFoVs). The effect of the wavelength mismatch is straightforward and its minimization is discussed in Sec. 4. However, the aFoV₁ is determined by the pixel size of CCD while aFoV₂ by the focal length F_f and the dimension of the image of the SLM B_ξ2:

So far, the discussion was focused on the optical conjugation of the captured object with its image. We can notice that the planes u₁ and u₂, being the Fourier counterparts of the object/image planes, are optically conjugated as well. Plane u₁ is the hologram capture plane while u₂ is known as a VW [26]. Its size is related to the parameters of capture as:

3. Confocal Fresnel Propagation Method

The coupling between capture and display systems is carried out by means of the paraxial propagation algorithm for CSBP fields of common focus. The algorithm takes the display geometry of the Fig. 1(b), where the field O_c(ξ₂,F_f) at the SLM-conjugated plane ξ₂ is obtained out of O_c(x₂,R₂) of the image plane x₂. The propagation is derived from the well-known Fresnel diffraction formula

Let us compare memory requirements of the CFP method of this work to the classical propagation algorithms, which involve direct sampling of the diffraction field. The complete object wave is a product of CSBP representation of the wave and the corresponding spherical carrier. The CSBP representation samples only the wave changes in reference to this spherical carrier. For our experimental setup, where the registration parameters are: Δ_u1 = 3.45 µm, λ = 0.5 µm, R₁ = 1.06 m and N₁ = 2048, SBP of the CSBP representation of the object wave is SBP[O_c(x₁,R₁)] = 2048 × 2048. For the complete object wave the spherical carrier must be also sampled thus its SBP is larger: SBP[O(x₁)] = FoV₁ × (Δ_x1⁻¹ + Δ_u1⁻¹). For the considered experimental parameters SBP the complete of object wave is SBP[O(x₁)] = 463 × 2048 × 2048. The example indicates that the majority of memory resources are here used for sampling spherical carrier and not the wave information itself.

4. Image magnification, axial shift and the capture-display mismatch

This work empathizes color imaging of 3D real objects with unitary 3D magnification. Nevertheless, the imaging path also enables geometrical manipulation of the image: axial shift and transverse magnification m_D. The subscript _D indicates that this magnification is introduced digitally. In the case of 1:1 imaging (m_D = 1) small distortions in reconstruction of the 3D volume are unavoidable due to slightly different wavelengths used in recording and reconstruction. Introducing m_D ≠ 1 leads to increased distortions due to different radii or transverse magnifications.

Geometrical manipulation of the holographic image, discussed in Sec. 5, is especially simple within the pipeline of the proposed Fourier holography. In this case the processing path needs to be supplemented with rescaling of CSBP representation of the holographic image. This takes care of the transverse magnification. The axial position of the reconstructed image R₂ is a parameter of CFP propagation, which is a part of the processing path.

Figure 1 presents coordinate convention used here for the sake of analysis of the reconstructed volumetric distortions, where the axial coordinates z₁ = 0 and z₂ = 0 indicate central planes of the corresponding 3D volumes. The numerical part of the imaging process is targeted to reconstruct the center region of the 3D object (plane x₂) without distortions and chromatic aberrations. This is realized by introducing parameters of the mismatch between the capture and display systems (curvatures, wavelengths, transverse magnification) to the CSBP field at x₂, which contains the object field captured with R₁ and λ₁. The magnification m_D is technically introduced by resampling a holographic image O_c(x₁,R₁) simply by cutting and zeropadding of the primary hologram and its FT, respectively. Since the SLM displays a diffracted wave that is calculated with parameters of the reconstruction system, there will be no distortions or chromatic aberrations in the central plane of the 3D object. The object at planes z₂ ≠ 0, on the other hand, will be affected to some degree.

Let us consider reconstruction of an image from the secondary hologram plane in Fig. 1 with arbitrary wavelength λ₂, transverse magnification m_D, and at reconstruction distance R₂ ≠ R₁. In this general case the reconstructed image at z₂ is related to the object field at z₁ via:

The first case considers only the wavelengths mismatch. The obtained distortions are shown as plots of volumetric axial deformations Fig. 2(a) and transverse magnification Fig. 2(b) for R₂ = R₁ = 1 m, m_D = 1, with object depths −200 mm > z₁ > 200 mm. The largest distortions are obtained for channel G, which is characterized by the largest wavelength mismatch. For this channel the object captured at −1.2 m will be reconstructed at −1.208 m and with small magnification M = 1.007. The distortions are certainly small but deserve a comment. One can notice that the complex amplitude distributions at corresponding planes x₁, x₂ are unequal, i.e. O₂(x₂) ≠ O₁(x₁). This is a consequence of reconstruction of CSBP representation O_c(x₁,R₁) at x₂, which, when reconstructed with λ₂, equals $O_c\left(x_2,{\lambda }_1{\lambda }_2{}^{-1}{R}_1\right)$. Since the change in λ affects the original field, while the reconstruction distance is defined by R₂, the reconstructed field O₂(x₂) is effectively deformed by a small phase distortion with curvature $\kappa =\left({\lambda }_2-{\lambda }_1\right){\lambda }_2^{-1}{R}_1^{-1}$.

 

Fig. 2 Axial deformation D_z and transverse magnification M for three cases of imaging: a), b) R₂ = R₁, m_D = 1, c), d) axial shift to R₂ = 2 × R₁, m_D = 1, and e), f) axial shift to R₂ = 2R₁ and magnification m_D = 2.

Download Full Size | PDF

The second example considers an axial shift of the reconstructed object to R₂ = 2R₁, while keeping m_D = 1. The volumetric distortions illustrated in Figs. 2(c) and 2(d) are small. The distortions are larger than in the previous case; they are mainly determined by the mismatch in the curvature radii. The third example illustrates deformations obtained when imaging with large magnification. The values R₂ = 2R₁ and m_D = 2 are selected to preserve the angular size of the reconstructed object. The obtained deformations are plotted in Figs. 2(e) and 2(f) and in the analyzed volume all axial, chromatic and transverse distortions are noticeable. The majority of distortions are due to m_D ≠ 1.

5. Realization of the Fourier holographic imaging system

This Section gives details on experimental realization and numerical implementation of our Fourier holographic imaging. The system consists of capture and display setups, which are shown in Fig. 3. The full-color imaging is obtained through superposition of three monochrome holograms corresponding to a chosen RGB colors through time multiplexing.

 

Fig. 3 a) Synthetic aperture Fourier capture setup, and b) Fourier holographic display based on VW and LED source.

Download Full Size | PDF

Let us first analyze the lensless Fourier DH capture setup, where we use RGB lasers with wavelengths λ_R1 = 637 nm, λ_G1 = 532 nm, λ_B1 = 457 nm. In the primary hologram plane the CCD camera records interference of the object wave with the spherical reference wave. The two separate beams are formed after the beam splitter. The half-wave plate controls the contrast of the fringe pattern. The reference point source is formed with an aspheric lens illuminated with a plane wave. The beam collimation is realized with a pinhole situated in the focal plane of the lens C (focal length F_c = 300 mm). The second part of the working beam hits a glass diffuser providing uniform illumination of the object scene. The 3D test object is the “Lowiczanka” color figurine placed at a distance 1060 mm from the hologram plane. Our object has 120 mm in height, 60 mm in width and approximately 50 mm in depth. The primary holograms are captured on a Basler CCD camera (2448 × 2050, 3.45 µm pixel pitch) mounted on a motorized stage with 200 mm of the horizontal scanning range. For the blue wavelength we have aFoV₁ = 7.6° in X, which is a vertical axis. Due to the off axis geometry in Y direction aFoV₁ is twice smaller. Consequently, for chosen distance 1060 mm, the lateral size of the 3D object can reach 70 × 140 mm. Here we use a 1D synthetic aperture hologram of size 2540 × 60000 pixels, where a set of holograms was registered for different positions of CCD in Y direction. This gives access to extended and continuous range of horizontal perspectives encoded in the synthetic aperture hologram.

In our work the laser digital holograms of real 3D objects are optically reconstructed in the display with RGB LED sources (Doric LEDs, center wavelengths λ_R2 = 635 nm, λ_G2 = 515 nm, λ_B2 = 465 nm, fiber core 960 µm). The design of the holographic display system presented in Fig. 2(b) is also based on the Fourier geometry. The name Fourier is used because the reading wave is a spherical converging wave that matches the geometry of the capture system, and the display SLM is addressed with a diffracted object wave without this spherical phase factor. The sphericity of the reading wave is added by the lens L_f while the SLM is illuminated with a collimated beam generated by the LED sources located in the backfocal plane of the collimator lens (F_c = 400 mm). Thus, the display setup can be divided into two parts realizing two separate operations: (i) complex coding of CSBP representation, and (ii) conversion to the Fourier geometry. The first part is realized with lenses L₁ (F₁ = 100 mm) and L₂ (F₂ = 600 mm) creating a 4F afocal imaging system, which conjugates the SLM (Full-HD SLM, Holoeye 1080P, 1920 x 1080 pixels, pixel pitch 8 µm, refresh frame-rate 60 Hz) with the 3D hologram reconstruction volume with magnification m = 6, and the vertical absorbing cut-off filter in the Fourier plane of the SLM. The filter enables the complex field encoding from a phase only hologram [31]. It is also responsible for the reduced resolution in y direction. The second part relies on the additional lens L_f (F_f = 600 mm) in the output of the setup, which focuses the reconstructed object beam into the observer’s eye. In order to reconstruct the primary digital hologram of a real 3D object at arbitrary R₂ the SLM is addressed with the secondary hologram carrying the corresponding diffraction field in the CSBP representation O_c(ξ₂,F_f).

The size of the reconstructed object is limited by angular field of view aFoV₂. In our setup aFoV₂ = 8.8° > aFoV₁ (Eq. (3)), hence the color imaging of full captured object with unitary magnification is possible. The off-axis geometry of the capture gives smaller image in y direction. Thus, the full-HD SLM is aligned vertically in the display, i.e. with 1920 pixels dimension in the x direction. Consequently, for this direction we have aFoV₂ = 8.8°. In this way the off-axis geometry does not reduce capabilities of the display and we are able to reconstruct the object in size very close to its natural 3D dimensions. For this we first reconstruct the hologram from its 2160 × 1920 pixels with Eq. (1) and take upright image (left part of the reconstruction), then we propagate it over the distance R₂ - F_f using the CFP technique developed in Sec. 3, where R₂ = R₁. As a result the CSBP distribution of the secondary hologram O_c(ξ₂,F_f) is obtained with resolution of the plane ξ₂. This discussion is correct for the B channel. For the remaining λ_n the hologram is numerically reconstructed from λ_n/λ_B (1080 × 1920) pixels, and then the upright image is cropped to 1080 × 1920. In this way the pixel size of numerical reconstructions at x₁ is the same for all wavelengths. To reconstruct the hologram at arbitrary axial position R₂ and magnification m_D, the required change of size of pixel pitch have to consider the physical change of resolution in the display geometry that is accounted for by the CFP method. At a given distance R₂ = αR₁ the pixel pitch is magnified by the factor α, hence m_D = αβ. Here, β is the part of the digital magnification that is realized through cropping of the hologram. For instance, consider image reconstruction for R₂ = 2R₁ and m_D = 4. In this case β = 2. Thus, the hologram has to be first reconstructed from (2 × 1080) × (2 × 1920) samples and then cropped to 1080 × 1920.

In our implementation of the system the color imaging is achieved through the time multiplexing technique as in our experimental verification we put emphasis on the highest possible quality of reconstructed holograms. In the display RGB LED sources are synchronized with the SLM using SLM synchronization signal. The SLM works at 60 Hz thus full-color frame is obtained at frequency 20 Hz. It is also possible to obtain full-color imaging from a single frame when implementing Frequency Division Method (FDM) [31], but with a decreased resolution. The capture system provides R, G, B synthetic aperture data. Thus, moving across this large hologram it is possible to reconstruct 3D objects with continuously changing perspective.

6. Coherence analysis of a LED driven display

Here, the study of influence of spatially partially coherent illumination on an image resolution is provided. It is assumed that the display SLM is capable of complex coding by itself and T(χ) is a distribution of complex wavefield that is given by this SLM at the plane χ (Fig. 3). Let the SLM be illuminated a the statically stationary wave that is fully characterized by a spatial coherence degree g, which is related by FT with the source intensity S as g(χ’) = ∫S(ν)exp{2πiχ’ν/λF_c}dν. Then, using difference variables χ’ = χ₂ - χ₁ and $\overline{\chi }=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.({\chi }_1+{\chi }_2)$the mutual intensity at plane χ is $J(\overline{\chi },\chi \text{'})=g(\chi \text{'})T(\overline{\chi }+\raisebox{1ex}{$\chi \text{'}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)T^{*}(\overline{\chi }-\raisebox{1ex}{$\chi \text{'}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$.

Let us denote I(x₂) as intensity of the partially coherent image reconstructed at a distance d = F_f - R₂ from the secondary hologram plane (Fig. 1(b)). Negative value d is the case of virtual imaging, while positive value corresponds to real imaging. Determination of I(x₂) requires introduction of two physical elements of the display system from Fig. 3(b), that is (i) the afocal conjugation between χ and ξ₂ by ξ₂ = mχ, where m = F₂/F₁, and (ii) the field lens with focal F_f. This gives the observed intensity image I(x₂) as:

Figure 4 presents a geometrical interpretation of the partial coherence effect. In this illustration the limited coherence is represented by coherence area A_c at the plane ξ₂, where A_c = mA_c0 with A_c0 representing the coherence area of the field actually illuminating the SLM plane, e.g. taking the distance between first zeros of the mutual intensity. For quantification of the reduction of image quality let us introduce two angles: α_A, the angular size of A_c, and α_VW the angular size of the VW, both measured from the axial image point P at d. Quantity α_A/λ expresses the maximum bandwidth of a spherical wave emitted from point P in conditions imposed by the given coherence area A_C. Hence α_A can be regarded as a factor, which limits the imaging resolution due to the source coherence. Equivalently, α_VW/λ can be regarded as the maximum bandwidth of the spherical wave in the coherent case. Thus, the α_VW represents maximum resolution that can be generated by the resolution of SLM. In Fig. 4(a) three cases of imaging are illustrated by points P_C, P_PC and P_clim for virtual imaging. The point P_clim, which is obtained for α_A = α_VW represents a limiting case of the coherent imaging, because image of finer resolution cannot be reconstructed. Thus, the point P_C (α_A > α_VW) will be reconstructed at full resolution while for the P_PC (α_A < α_VW) there will be a drop of resolution due to partial coherence. The axial coordinate d_clim of P_clim is given by:

 

Fig. 4 a) Geometrical interpretation of the effect of partially coherent imaging as an effect of source coherence area A_c; b) limits of reconstruction depth for partially coherent imaging at maximum resolution; c) resolution drop as a function of d for C = 0.15, 0.4 (blue curve indicates experimental setup); the parameters for calculations: λ = 0.515 μm, F_f = 600 mm.

Download Full Size | PDF

Suppose now that a reconstructed image is perceived by an observer. Let’s take an example of day vision with pupil size B_eye = 4 mm and consider two cases, one for B_VW > B_eye (our experimental case), and second with B_VW = B_eye. The second case could be obtained in our display by increasing afocal magnification by 1.61. In the first case not all details of the reconstructed image will be perceived, even with laser light. The second case optimizes visual performance of the display and improves coherence parameter C by 1.61².

In Figs. 4(b) and 4(c) the coherence effect on the resolution of 3D image is investigated. Figure 4(b) visualizes the reconstruction depths${\Delta }_d=d_{\text{clim}}^{+}-d_{\text{clim}}^{-}$as a function of coherence parameter C for which viewer will see holographic images with all details. In Fig. 4(b) two cases of C = 0.15 mm, 0.4 mm are additionally marked. The case of C = 0.15 mm describes coherence effect in the display for the selected wavelength λ = 0.515 μm, where A_C = 0.97.

The value of C = 0.4 mm defines an optimized system with afocal magnification increased by 1.61. In Fig. 4(b) we have Δ_d = 184 mm and 571 mm for C = 0.15 and 0.4, respectively. Both cases are further investigated in Fig. 4(c), which illustrates how the image resolution defined here as ${\alpha }_A/{\alpha }_{VW}$ falls with increasing distance d of the image from the field lens. Notably, imaging with intentional decrease of resolution by limited coherence (e.g. ${\alpha }_A/{\alpha }_{VW}$ = 0.5) may be profitable due to further speckle reduction.

7. Experimental results

Here, the proposed two step Fourier imaging is experimentally investigated in two parts. The first one gives evidence of high quality imaging of real 3D color object. The second shows that imaging path enables geometrical manipulation of the reconstructed image and proves that high quality imaging is preserved for large object depths. Here, up 1.5 m for the DH of a real object and up to 10 m for CGH calculated from 3D cloud of points object, using LEDs in the display step. All images presented in this Section were captured with a zoom lens camera (Canon EOS 5D Mark II) placed in the VW plane.

In the first experiment the optical reconstruction of 3D color object with its original dimensions is realized. A lensless Fourier hologram of “Lowiczanka” figurine recorded at distance R₁ = 1.06 m was used as an input. The captured data was then reconstructed at distance R₂ = R₁ to ensure imaging with unitary magnification, leading to close 1:1 copy of the object. Since our Fourier imaging system is characterized by aFoV₂ > aFoV₁, the full object could be displayed. Figure 5 presents images of the obtained 3D reconstruction. In Figs. 5(b) and 5(c) the camera is focused on different planes (jacket and skirt), which are separated in depth by approximately 18 mm. We enlarge those regions to illustrate detailed reconstruction of the object. In Fig. 5(b) a hand, colorful bracelet and flowers on the jacket are sharp while in Fig. 5(c), when defocused, even the stitch pattern of the skirt is visible. Only one of the two regions is clear in each picture, therefore our imaging technique is precise enough to distinguish between the two planes. Also, the skirt is visibly closer to observer than the jacket, therefore the 3D image is orthoscopic, as if it was a real object viewed directly. This feature is not provided in other works presenting a TV-like display of DHs, where only pseudoscopic images are reconstructed [14,16,18–21].

 

Fig. 5 a) Photo of 3D “Lowiczanka” figurine and its optical reconstruction with time multiplexing method in two focus positions b) jacket, and c) skirt. See Visualization 1 illustrating rotating figurine.

Download Full Size | PDF

In an additional experiment we show different perspectives obtained from the same “Lowiczanka” hologram. The result is presented in Visualization 1. The SLM was addressed in time sequence by different parts of the synthetic aperture hologram corresponding to different transverse positions of camera in capture setup, which span object observation angles in range ± 15°. Since the experimental results emphasize the imaging quality of the display each color frame of visualization was captured separately and merged together into movie.

The second experiment, documented in Fig. 6, illustrates that our Fourier holographic imaging enables reconstruction of 3D scenes, in which axial location and transverse magnification of the reconstructed image can be set freely. Additionally it shows unnoticeable loss of quality when reconstructing 3D scenes with large depth, here up to 2 meters. Since registration of a scene of this size on a single optical table would be a technically challenging task, the large-scene hologram (for one SLM) has been synthesized numerically from holograms of two objects captured separately. The first object was a “Rooster” figurine placed at R₁ = 0.5 m from the CCD, and the second was the “Lowiczanka” figurine from before, placed at R₁ = 1.06 m. Next, two holograms were synthesized with different R₂ position of the “Lowiczanka” object. In the first synthesized hologram complex amplitudes displaying each object were calculated with R₂ = R₁ in each case and added together before addressing theobject were calculated with R₂ = R₁ in each case and added together before addressing the SLM. The LED-driven reconstruction of the synthesized hologram was photographed in two focal positions corresponding to in-focus observation of each object. Figures 6(a) and 6(b) present the obtained images. Visualization 2 shows a continuous sweep of the focus position from one object to the other one. To demonstrate that 3D object can be reconstructed at arbitrary axial location, the second synthesized hologram was generated from the same DH data. In the 3D scene reconstruction distance of “Lowiczanka” was almost doubled (R₂ = 2 m) while “Rooster” occupies the same axial location as previously. To keep the angular size of the figurine the image magnification was set to m_D = 2. The result is shown in Fig. 6(c). It can be noticed that there is no visible change in reconstruction quality when comparing to Fig. 6(b). In terms of the imaging resolution, this corresponds to a drop of ${\alpha }_A/{\alpha }_{VW}$ by factor of 0.81 (see marking points on the blue curve in Fig. 4(c)). Finally, to investigate possibility of transverse magnification of the reconstructed image the hologram form Fig. 6(c) was magnified by factor m_D = 4. Results presented in Fig. 6(d) shows that fine details can be reconstructed with high quality.

 

Fig. 6 Optical reconstruction of synthesized DH including a pair of 3D scenes with “Rooster” and “Lowiczanka” figurine captured in different focus positions a) “Lowiczanka” 1.06 m b) “Rooster” 0.5 m, c) “Lowiczanka” 2 m (“Rooster” 0.5 m), d) “Lowiczanka” 2 m (“Rooster” 0.5 m) digitally magnified by factor m_D = 4. See Visualization 2 illustrating change of focus between “Rooster” and “Lowiczanka”.

Download Full Size | PDF

To fully present the capabilities of the display CGH calculated from 3D cloud of points representing computer graphic of a dog was used as a holographic content. The dotted structure makes it a good test for visualization of imaging quality. The hologram displayed on the SLM was generated using the phase-added stereogram method [37]. Figures 7(a)-7(d) illustrate holographic reconstructions of the CGH designed and captured for different reconstructions distances: 600 mm, 1 m, 2 m, and 10 m. The used 3D model consists of 7000 points and for R₂ = 600 mm has dimensions: width 50 mm, height 70 mm, and depth 50 mm. For other values of R₂ the 3D object was rescaled to keep the same angular size. The camera is focused on the right ear of the 3D model. This part of the image is enlarged. For each hologram the 3D object was rescaled to keep the same angular size. It can be noticed that the size of the single point is growing with the reconstruction distance. The highest difference is obtained between 600 mm and 1 m. However, the blur of the points is small in all the images, even for 10 m, and the quality of the reconstructed CGH is high. The theory outlined in Sec. 6 supports this observation. For example, shifting the reconstruction distance from 2 m to 10 m involves a drop of imaging resolution expressed as ${\alpha }_A/{\alpha }_{VW}$by factor of 0.76.

 

Fig. 7 Optical reconstruction of CGH calculated from 3D cloud of points with enlarged in-focus position captured at different reconstruction distance a) 600 mm b) 1 m, c) 2 m d) 10 m.

Download Full Size | PDF

8. Conclusions

This paper presents an end-to-end Fourier holography solution of real world objects with high quality full-color 3D orthoscopic imaging, where axial position and magnification of the image can be set freely without additional computational load. The imaging technique is referred to as Fourier holography since the FT relation between distributions at the object and hologram planes is preserved in capture, numerical processing and optical reconstruction.

The proposed 3D imaging concept is based on holographic recording and reconstruction of a converging object beam. The reconstructed object wave is backpropagated directly to the observer’s eye that is placed at VW, which is an image of the CCD of capture. The capture step is realized in a laser-based lensless Fourier DH system while the LED-based display system relies on a flat SLM illuminated by collimated waves, a magnifying telescopic system and a field lens, which brings back natural convergence of the object field. The SLM is addressed by complex amplitude taken from a arbitrarily defocused plane with its spherical phase factor removed, thus correct 3D depth is maintained in the reconstruction.

The coupling between capture and display systems is carried out in a flexible way by means of an original numerical method, named CFP, within which the carriers of propagated beams are confocal. The CFP routine propagates the object field without the need for sampling of the spherical carriers, which assures high accuracy and efficiency of the solution. The size and resolution of the result is scaled according to the ratio of radii of the spherical carriers. It means that the Fourier digital holographic content can be reconstructed in the display at arbitrary location with the same pixel count and without loss of quality.

The complete processing framework provides an easy way for image magnification, which also plays an important role in axial relocation of the 3D image as well as synchronization of dimensions of RGB channels. In this case the input for CFP needs to be linearly rescaled, here realized through hologram cropping or zeropadding. When it comes to the full-color imaging the processing allows for chromatic aberration free reconstruction of a selected plane of the 3D object, which is also free of distortions. As a special case, this work explores possibility of 3D imaging of large undistorted objects with 1:1 magnification, theoretically and experimentally. The theory shows that the parasitic 3D magnifications caused by the wavelength mismatch between the capture and display systems are negligible.

Additionally, the proposed display configuration is capable of high quality reconstruction of large 3D color objects using LED illumination. We show experimentally that despite the limited spatial coherence of the light source high image quality is preserved across large object depths, here up to 10 meters. The results are supported by theoretical analysis of the influence of partially coherent illumination on image resolution. The study shows that in our present implementation of the system for the reconstruction distance of 10 m the observed resolution drops down to 0.164 of the maximal value. This value can be raised to 0.425 only by increasing the afocal magnification in the system by factor 1.61, which brings the VW size down to a typical diameter of an eye pupil.