Adaptive transformations for color hologram display

The advent of digital holography (whereby the interference pattern that constitutes the hologram is recorded using a digital detector array, rather than analog photographic media) has opened up new possibilities for developing spectacular 3D imaging and display systems.^1,2 Making these a reality requires capturing all the details of an object by recording three monochromatic digital holograms with different wavelengths (red, green, blue) that correspond to the primary colors of light.^3,4 These single-color holograms can then be numerically reconstructed and combined to display a fully detailed image of the object.

However, achieving good-quality results requires a strategy for correctly superimposing the reconstructed hologram images,^5–10 which may differ in scale or be spatially shifted relative to each other. To address this, we have recently developed a simple method based on adaptive affine transformations of holograms that allows full control over the object's position and size within a 3D volume.^11–13 This method ensures adaptive compensation of the three constituent color holograms, which are subsequently superimposed using a correlation-matching procedure.

The final step is to create a synthetic hologram, incorporating information from all the different-colored holograms recorded of the same object, using the NTSC (National Television Systems Committee) coefficients. The resulting hologram can be optically reconstructed by a spatial light modulator (SLM) at a single wavelength to display the image in 3D with all its original color features.

To illustrate the steps in our proposed method, we can consider the numerical reconstructions of two digital holograms of the same object, recorded at two different wavelengths, λ_red and λ_green. The red and green reconstructions will have different spatial (position and scale) characteristics, as can be seen in Figure 1(b) and (c). Thus, superimposing them does not yield a perfect spatial overlap, resulting in a loss of detail: see Figure 1(d).

Figure 1. Reconstruction of color holograms of a matryoshka doll (a), recorded with wavelengths λ_red=632 .8nm and λ_green=532nm. The reconstructions of the red (b) and green (c) holograms do not perfectly coincide spatially, producing loss of detail in their superimposition (d).

To appropriately match up the red and green reconstructions, we define a constant γ such that λ_green=γλ_red, and use it to apply an adaptive affine transformation (stretching) to the red hologram, as described in detail elsewhere.^11–13 In this way, the in-focus reconstruction of the red hologram can be obtained by reconstructing the transformed hologram at the wavelength λ_green, with suitable pixel sizes.¹³ However, we still cannot directly add these two digital reconstructions—shown in Figure 2(a) and (b)—due to a systematic shift error that is a consequence of the stretching. To correct this shift, we employ a correlation maximization (CM) approach,¹³ the end result of which is shown in Figure 2(c).

To obtain a single hologram that can be projected using a single-wavelength SLM, we compute a synthetic hologram based on the back-propagation of a suitable linear combination of the two-color numerical reconstructions (stretched red, and green) of Figure 2(a) and (b), according to the NTSC standard coefficients.¹³ Figure 3 shows the resultant 3D displays for both the stretched-red and green holograms, and for the synthetic hologram obtained by combining them. We can see that the display of the synthetic hologram in Figure 3(c) includes all the details of the two constituent color holograms.

Figure 2. (a) Numerical reconstruction of the red hologram after stretching. (b) Numerical reconstruction of the green hologram. Adding (a) and (b) together and applying the correlation maximization (CM) approach produces a perfect superimposition (c).

Figure 3. Optical reconstruction by a spatial light modulator. (a) Display of the stretched red hologram. (b) Display of the green hologram. (c) Display of the synthetic hologram created from (a) and (b) using the NTSC (National Television Systems Committee) coefficients.

In summary, our work has shown that applying a simple adaptive affine transformation to digital color holograms allows us to achieve a full-color numerical and optical reconstruction without losing any details of the color objects due to the monochromatic laser light. The method can be easily generalized to the case of three colors by also calculating the stretching of the blue hologram.¹³ We believe the presented technique represents a first step toward the future development of real 3D holographic displays or television.