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J. Eur. Opt. Society-Rapid Publ. 22, 15( 2026)
We therefore refer to this process as hypercentric rendering. To the best of our knowledge, this represents the first demonstration of lensless hypercentric imaging. The approach combines the inherent advantages of digital holography with unconventional perspective imaging modalities. This combination opens new possibilities for applying hypercentric and related perspectives within interferometric measurement techniques, thereby significantly extending the methodological scope of coherent optical metrology.
2 Methods
2.1 Digital holographic hypercentricity
Figure 1 shows the basic idea of lensless hypercentric imaging, which displays a common configuration in digital holography. There a numerous ways to record digital holograms. The details of our recording process will be outlined in the next section. Here we will rather focus on the numerical reconstruction, i. e. the digital imaging of the object. For this, it is assumed that the wave field Uð ~ xÞ, scattered by the object, has been determined from the digital hologram across the sensor plane.
In digital holography, imaging involves a numerical representation of the wavefield propagation process to calculate the wavefield U O ð ~ xÞ directly in front of the object in a distance �z O from the camera sensor
U O ð ~ xÞ ¼P �zO Uð x~ 0 Þ; ð1Þ
where P z f::: g denotes the propagation operator, which propagates a wave field by a distance z. One common example of such a propagation operator is realized in the angular spectrum method, where
n Uð x~ 0 Þ ¼ F
�1
F Uð x~ 0 Þ H z ð ~ o kÞ; ð2Þ
P z
with the transfer function of free space propagation H z ð ~ kÞ¼exp ðik z zÞ; ð3Þ
and the z-component k z of the wave vector ~ k. The intensity I O ð ~ xÞ ¼jU O ð ~ xÞj 2 is the image of the object in focus. However, since we can freely select any distance z for the propagation in equation( 1), holography provides volumetric imaging.
To understand how this can be used for hypercentric imaging, let us assume that the surface of the object can be described as a large number of point-like scattering sources. One of these scattering sources is indicated by P A in Figure 1. The spherical wave emitted by P A is stored as part of the digital hologram and will be propagated through space upon reconstruction, as indicated by the spherical wave fronts. We can now choose to reconstruct the wave field in the aperture plane and apply a virtual stop aperture. This is done by multiplying a window function W d, which is 1 inside a disc of diameter d around the optical axis and 0 otherwise. If d is small enough, the aperture cuts out a segment of the spherical wave, which is strongly
Figure 1. Overview over the principle of hypercentric imaging using lensless digital-holography. The wave field is known across the camera plane x 0 and can be reconstructed in the object plane x to yield an image of the object. However, for hypercentric imaging, we numerically apply a virtual aperture with diameter d. The aperture cuts out a part of the spherical wave of each object point and turns it into a ray-like wave field. Upon propagation into the perspective plane, this lets object points further away from the camera shift outwards across the image, as seen from P A and P 0 A
. In contrast, object points closer to the camera, on the opposite side of the perspective plane, will appear shifted inwards. This behavior creates the hypercentric perspective.
directed and travels mainly in the direction � ~ k A. This effect of the aperture is crucial to hypercentric imaging. It turns an omnidirectional spherical wave into a directional, almost plane wave, similar to a ray.
After applying the aperture, the wave field is propagated back to a plane located in the center of the object, which we may refer to as the perspective plane. Due to its( now) almost ray-like nature, the wave field associated with P A will let appear a point at the position ~ P A in the perspective plane. Please note, how ~ P A is laterally shifted outwards when comparing it with its true lateral position. In Figure 1 we also find the point P B as a second example. If the same procedure is applied to it, it will appear at point ~ P B, shifted inwards across the perspective plane. This behavior is exactly what we expect from a hypercentric imaging system. Object parts far away from the observer, i. e. the camera sensor, appear larger than object parts close to it. The point P C in the center of the aperture is the convergence point, where all ray-like wave fields intersect. Its position defines the characteristics of the perspective and the amount of hypercentricity.
With the simple geometry given in Figure 1, wecan calculate the position ~ P A, depending on the true position P A and z P. For the sake of simplicity, we will describe P A =( x A, z A) in two dimensions, with its position along the optical axis z A and a single lateral coordinate x A. Since the problem is fully separable, extension two three