Towards Deconvolution in Holography

Towards Deconvolution in Holography Nan Wang, Claas Falldorf, Christoph von Kopylow Bremer Institut f¨ ur Angewandte Strahltechnik, Klagenfurter Str. 2, 28359 Bremen, Germany ABSTRACT In this paper we present a way to formulate the holographic reconstruction of a wavefield throughout a volume by means of a sequence of convolutions. The discussion is based on the assumption that the field is generated by a set of real valued scattering sources within the volume. In analogy to two dimensional imaging this enables the application of deconvolution techniques to the holographic scheme. We show, that the proposed formalism can theoretically be used to perform a three dimensional deconvolution of the reconstructed amplitude in order to recover object information, e.g. the position of scattering sources. In the ideal case of an infinite aperture of the hologram the deconvolution may be employed by a simple inverse filter. However, for the more realistic case of finite apertures an iterative technique called Out of Hologram Extrapolation (OHE) is introduced, which is based on the projected Landweber method. Finally, the novel method is applied to a synthetic example in order to recover the positions of a set of distributed point sources. Keywords: 3D reconstruction, digital holography, holographic imaging, deconvolution, out of band extrapolation

1. INTRODUCTION Digital holography has been successfully used for shape and deformation measurement1–4 and in particle metrology,5–7 microscopy8–10 as well as in endoscopy11, 12 for years. Generally in digital holography the accomplished analysis is based on the phase information in the object plane using the reconstructed object wavefront. However, this type of analysis produces ambiguous results, if the object geometry contains increments, which are higher than half of the wavelength of the used light. In this case a measurement based on phase information requires the recording of multiple holograms with varying illumination directions, wavelengths or observation directions.13–15 This relative complex approach significantly reduces the measuring speed and thus strongly limits the applicability of digital holography especially in the shape measurement of dynamically changing objects. Another problem arises in the measurement of objects, which are greater than the depth of field of reconstruction. Also in this case, the analysis yields inaccurate values. The Caesar Institute in Bonn proposes a rather complex approach from the classical holography for measuring the shape of an object with a single hologram.16, 17 A hologram of the object is made at first and then reconstructed classically. Through this real three-dimensional reconstructed image, a screen is moving along the optical axis and a series of two-dimensional recordings is produced. An analysis of the contrast in these recordings delivers the contour of the object in the corresponding depth and thus the surface shape following the analysis of all individual recordings. This process is successfully applied in the fields of medicine and archaeology. However, it is very expensive through the use of classical holography and mechanical scanning in the analysis. Additionally, in this approach the individual recordings contain not only the information of the focused object plane but also blurred information in front of and behind the plane of the screen. Since the analysis is solely based on the intensity of the field distribution, the accuracy decreases. An analysis of the complex-valued field distribution on the basis of the image formation process of holography does not take place in this process. So far the determination of three-dimensional object information from digital holograms is only known from the particle measurement technology and is also based solely on the intensity distribution of the reconstructed field.7 In this process, the complex wavefront is reconstructed at various depths. The analysis is based on the assumption that the spatial position of a particle correlates with a local maximum of the intensity. The superposition of field amplitudes from adjacent particles leads to misinterpretation at this point and thus reduces the accuracy.

Modeling Aspects in Optical Metrology II, edited by Harald Bosse, Bernd Bodermann, Richard M. Silver, Proc. of SPIE Vol. 7390, 73900Y · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.827507

Proc. of SPIE Vol. 7390 73900Y-1

Source Point P (xP, yP, zP)

y(η)

x(ξ) z

0 CCD Reference Wave R Figure 1. Coordinate system used to describe holographic reconstruction

In summary, it appears that all of the above mentioned processes extract only a fraction of the available information from the field distribution throughout the volume. In contrast to two dimensional imaging, an analysis based on the evaluation of the reconstructed complex amplitude in three dimensions is not considered currently. In order to provide a step towards this sort of analysis, we propose to reformulate the three dimensional holographic reconstruction in terms of a sequence of shift invariant multi dimensional convolutions. In case that the wave field in the hologram plane can be described as to be generated by a finite number of scattering sources, deconvolution techniques could be employed in order to recover object information within that scheme. Therefore, in the following section the holographic recording and subsequent reconstruction of a point source is reinterpreted as a three dimensional imaging process. It is shown that in the ideal case of an infinite aperture the imaging process can be expressed by a three dimensional shift invariant convolution. The disturbing effect of a spatially limited aperture of the hologram is investigated in section 2.2. It is shown that in this case the imaging process may be described by a set of coupled shift invariant convolutions. However, the corresponding transfer functions are band limited. In order to overcome this problem, an iterative technique based on the projected Landweber method is introduced and demonstrated by means of a synthetic example.

2. HOLOGRAPHIC IMAGING In this section the holographic recording and subsequent reconstruction of a point source will be interpreted as a three dimensional imaging process. Fig. 118 illustrates the corresponding holographic scheme being addressed throughout this work. The source point P has the coordinates (xP , yP , zP ). It is regarded as the source of a spherical wave g(x − xP , y − yp , z − zP ) with unit amplitude, which is superposed by a parallel reference wave R. The resulting interference pattern in the plane z = 0 is recorded by a CCD sensor and stored in a computer with x and y representing the pixel positions. In the following, it is assumed that this scheme is used to record a series of phase shifted holograms in order to determine the complex amplitude across the domain of the hologram.2

2.1 Reconstruction Based on a Hologram with Infinite Aperture The reconstruction of the complex amplitude u(x, y, z) throughout the volume is performed by numerically solving the Rayleigh-Sommerfeld diffraction equation,19 A2R u(x, y, z) = g ∗ (ξ − xP , η − yP , zP )g(x − ξ, y − η, z)dξdη (1) iλ Σ 2D A2R ∗ Σ→+∞ gzP (x − xP , y − yP ) ⊗ gz (x, y), = iλ where ξ and η are the coordinates of the hologram plane at z = 0, AR is the amplitude of the parallel reference 2D

wave, λ is the wavelength, ⊗ denotes the two dimensional convolution and the complex amplitude of the spherical


wave g is characterised by: g(x, y, z) =

2 + y2 + z 2 exp i 2π x λ . x2 + y 2 + z 2

(2)

In Eq.1 Σ represents the size of the hologram. In this case it is assumed that Σ → +∞, i.e. the hologram is infinitely large. Under the Fresnel approximation, g can be written as,19 g(x, y, z) ≈

π exp (i 2π λ z) exp i (x2 + y 2 ) . z λz

(3)

Using this approximation and the definition of the convolution, this non-limited reconstruction u(x, y, z) may be reformulated as a three dimensional convolution (see appendix A), u(x, y, z) = =

A2R g ∗ (x − xP , y − yP , z − zP )

(4) 3D

A2R δ 3 (x − xP , y − yP , z − zP ) ⊗ g ∗ (x, y, z).

This result shows, that the reconstruction from a hologram with infinite aperture can exactly rebuild the wavefield of the initial source point. As seen from Eq.4, the overall process is described by a shifted invariant three dimensional convolution and therefore can be regarded as a linear system with impulse response g. Hence, in case of multiple point sources we may write: u(x, y, z) = A2R g ∗ (x − xP,i , y − yP,i , z − zP,i ) (5) i

= A2R

3D

δ 3 (x − xP,i , y − yP,i , z − zP,i ) ⊗ g ∗ (x, y, z).

i

Evidently in this case, contributions of distributed point sources are superposed at any point of the reconstructed volume. However, the corresponding transfer function G = F {g ∗ } holds |G| = 1 in frequency domain. Therefore, in analogy to two dimensional imaging, a cloud of distributed point sources can be recovered by applying an inverse filter,20–23

1 −1 F {u} 3 δ (x − xP,i , y − yP,i , z − zP,i ) = F . (6) A2R F {g ∗ } i In practice, there is no possibility to have such an infinite hologram; this non-limited reconstruction has to be altered accordingly, when a limited aperture of the hologram is introduced. In the following section, this issue will be investigated in detail.

2.2 Reconstruction Based on a Hologram with Finite Aperture In this work a rectangular window aperture with width A in x-axis and B in y-axis for the hologram is assumed. Replacing the infinity with these two limits in Eq. 1 the limited reconstruction u (x, y, z) is given as, +∞ η A2R ξ u (x, y, z) = g ∗ (ξ − xP , η − yP , zP )rect( )rect( )g(x − ξ, y − η, z)dξdη. iλ A B −∞ y 2D x A2R ∗ g (x − xP , y − yP )rect( )rect( ) ⊗ gz (x, y). (7) = iλ zP A B By using the same technique as in the last section this u gives (see appendix B), u (x, y, z) =

A2R AB g(x, y, z) 2 λ

2D 3D B A 3 ∗ ∗ sinc( x)sinc( y) ⊗ [δ (x − xP , y − yP , z − zP ) ⊗ g (x, y, z)]g (x, y, z) . λz λz


(8)

This modified representation of u shows the influence of the spatially limited aperture of the hologram. The wavefield is slice by slice blurred due to the convolution with the sinc product. Furthermore, from the representation in Eq. 8, the following can be concluded: 1. The limited reconstruction u can be reformulated as a sequence of two coupled shift invariant convolutions, where one is three dimensional while the other is two dimensional. 2. For A, B → ∞, the sinc product becomes δ(x)δ(y) and consequently the 2D deconvolution disappears. This accords with the non-limited case. 3. The resolution in z direction is not affected by the limited aperture of the hologram. 4. The two dimensional convolution with the sinc product is band-limited, since the corresponding transfer function is a rect in any case. This property makes simple inverse filtering for deconvolution not feasible.

3. OUT OF HOLOGRAM EXTRAPOLATION As seen from the preceding paragraph, deconvolution of the reconstructed complex amplitude by means of an inverse filter is impracticable. The reason is the specific structure of the corresponding transfer function, which in any case is band limited according to the dimensions of the hologram. Therefore, in the following a method will be presented which makes use of a priori knowledge in order to extrapolate the wavefield outside of the area defined by the hologram. It is based on the assumption that the complex amplitude uz (x, y) to be recovered is generated by a set of real valued point sources distributed across a plane z perpendicular to the optical axis:

uz (x, y) =

N −1

u0,i · δ(x − xi , y − yi ).

(9)

i=0

The amplitude u0,i denotes the strength of an individual source i. The suggested approach starts with the spatially limited complex amplitude H0 in the hologram plane {ξ, η}, which by introducing the operator of the Fresnel propagation P and the characteristic function of the hologram DH (ξ, η) can be expressed as follows: H0 (ξ, η) = P{uz (x, y)} · DH (ξ, η).

(10)

Here, DH equals 1 across the hologram and 0 elsewhere. Regarding Eq.9 and Eq.10, the following constraints can be defined in the hologram plane as well as in the reconstruction plane: • The amplitude to be recovered in the reconstruction plane is expected to be real valued and non negative. • The known complex wavefield H0 (ξ, η) in the hologram plane is considered true. These assumptions may be implemented into an iterative scheme based on the projected Landweber method.24 The goal is to extrapolate the complex amplitude outside the area defined by DH in the hologram plane. Adapting the Landweber method to the specific case of the Fresnel propagation and considering the above mentioned constraints, the amplitude Hk+1 of the k + 1-th iteration is estimated from Hk as follows:

Hk+1 = P |P −1 {Hk }| · τ (1 − DH ) + H0 .

(11)

Here, the norm | · · · | projects the reconstructed amplitude P −1 {Hk } = uk onto the subset of real valued, non negative distributions while preserving the energy. After re-propagation of |uk | into the hologram plane, the iterative step is then finalized by re-inserting the known amplitude H0 across the area defined by DH . Proper choice of the regularization parameter τ may reduce the number of required iterations. However, convergence is observed even in the case of τ = 1.


H0

y

η

x

a)

y

ξ

b)

x

c)

Figure 2. a) The structure of the synthetic object: 30 point sources are equally distributed across the circumference of a circle with diameter d = 0.5mm. The visible area is 1 × 1mm. b) The real amplitude of the propagated wavefield in the hologram plane in a distance of z = 100mm to the object plane. The box indicates the limits of the characteristic function DH (ξ, η) of the hologram. The wavefield H0 (ξ, η) inside the box is assumed to be known. The visible area is 13.62 × 13.62mm. c) The real amplitude of the wavefield u0 (x, y) in the object plane, which has been reconstructed from the known complex amplitude H0 (ξ, η). All distributions are normalized and sampled by 256 × 256 pixels.

η

η

ξ

a)

y

ξ

b)

x

c)

Figure 3. a) The real amplitude |H200 (ξ, η)| of the complex wavefield in the hologram plane after k = 200 iterations and b) the amplitude |H500 (ξ, η)| after k = 500 iterations. c) The amplitude |u500 (x, y)| in the object plane, which has been reconstructed from H500 (ξ, η). In the later, the grey scale has been inverted for illustrational purposes. All distributions are normalized and sampled by 256 × 256 pixels.

The method will be demonstrated by means of a synthetic object whose structure is seen from Fig.2a. The black marks indicate the positions of 30 point sources which are equally distributed along the circumference of a circle with diameter d = 0.5mm. The distance between the object plane and the hologram plane is z = 100mm and the wavelength is set to λ = 532nm. In Fig.2b the normalized real amplitude of the propagated wavefield in the hologram plane is shown. The aperture of the hologram, which is marked by the box, is given by 1.7 × 1.7mm. The wavefield H0 (ξ, η) inside the box is considered to be known. Fig.2c shows the real amplitude of the reconstructed wavefield u0 = P −1 {H0 } in the object plane. It is seriously distorted due to the spatial limits of the hologram, causing contributions of adjacent point sources to overlap. However, in Fig.3a and Fig.3b the real amplitude of the wavefield in the hologram plane is presented after 200 and 500 iterations of the suggested procedure, where the regularization parameter was set to τ = 1.015. It is seen, that the wavefield Hk is gradually extrapolated with progressing number of iterations k and finally shows very good agreement with the original spectrum as given by Fig.2b. Eventually, in Fig.3c the real amplitude |u500 (ξ, η)| in the object plane is shown. Please note, that in order to provide a better comparison to Fig.2a, the grey scale is inverted. The high values of the amplitude are localized at positions which almost coincide with those of the point sources thus allowing for an automatic detection of particles for example.


4. SUMMARY In this publication, we have shown that in specific cases the holographic reconstruction of a complex amplitude may be regarded as a three dimensional imaging process. This is especially true if the complex amplitude observed in the hologram plane can be described as to be generated by a set of real valued scatters. A careful analysis of this scheme yields, that the corresponding imaging process can be mathematically expressed by a sequence of two shift invariant convolutions. Therefore, in analogy to the normal 2D imaging deconvolution techniques become applicable to the holographic scheme. Recovering the position of point sources by a 3D deconvolution was theoretically shown within the proposed formalism. In the case of the reconstruction using an hologram of infinite size, a simple inverse filter could be employed. However, for the more realistic case of a hologram with limited apertur an iterative technique called Out of Hologram Extrapolation was introduced and demonstrated by means of a synthetic example. The positions of a set of real valued point sources were successfully recovered from the complex amplitude as provided by a spatially limited phase shifted hologram.

ACKNOWLEDGMENTS The authors are grateful to the Deutsche Forschungsgemeinschaft (DFG), Germany, for supporting the project ”HoloReForm” and the sub-project B5 ”Sichere Prozesse” in the Collaborative Research Centre SFB 747 ”Micro Cold Forming” (”Mikrokaltumformen”). Prof. Dr. Ralf B. Bergmann has made valuable suggestions during reviewing.

APPENDIX A. DERIVATION OF THE NONLIMITED RECONSTRUCTION Substituting the Fresnel approximated functions g (see Eq. 3), the Eq. 1 gives, u(x, y, z) = =

2D A2R ∗ gzP (x − xP , y − yP ) ⊗ gz (x, y) iλ A2R exp i 2π λ (z − zP ) iλ zzP

π 2D π · exp −i x2 + y 2 (x − xP )2 + (y − yP )2 ⊗ exp i . λzP λz

(12)

According to the definition of convolution it can be altered into, A2R exp i 2π λ (z − zP ) (13) u(x, y, z) = iλ zzP +∞ π π (x − ξ)2 + (y − η)2 dξdη. (ξ − xP )2 + (η − yP )2 exp i · exp −i λzP λz −∞ After removing the exponential parts without variable ξ and η out of the integral signs it gives, π A2R exp i 2π π 2 2 λ (z − zP ) x2 + y 2 u(x, y, z) = xP + yP exp i (14) exp −i iλ zzP λzP λz +∞ π(zP − z) 2 2π 2π · (xξ + yη) dξdη. exp i (xP ξ + yP η) exp −i ξ + η 2 exp i λzzP λzP λz −∞ The integral of this equation is the Fourier transform of a product with two exponential parts. After replacing the first three exponents with spherical waves g at the same time, Eq. 14 gives, u(x, y, z) =

A2R g(x, y, z)g ∗ (xP , yP , zP ) iλ

λz(zP − z) 2 z 2 2 2 ·λ z F exp iπ (xP ξ + yP η) ξ +η exp i2π zP zP


=

=

A2R g(x, y, z)g ∗ (xP , yP , zP ) iλ zzP zP z z 2 2 ·iλ exp −iπ xP ) + (y − yP ) (x − zP − z λz(zP − z) zP zP 2π zP zP − z A2R exp i λ (z − zP ) 2 2 2 2 exp −iπ (x + y ) + iπ (x + y ) iλ zP − z λz(zP − z) λz(zP − z) zP − z 2xP x + 2yP y z 2 2 2 2 (x + yP ) − iπ (x + yP ) exp iπ exp −iπ λzP (zP − z) P λzP (zP − z) P λ(zP − z)

=

A2R g ∗ (x − xP , y − yP , z − zP )

=

A2R δ 3 (x − xP , y − yP , z − zP ) ⊗ g ∗ (x, y, z).

(15)

3D

With it the derivation from the 2D to the 3D convolution for the non-limited reconstruction is finished.

APPENDIX B. DERIVATION OF THE LIMITED RECONSTRUCTION This derivation uses the same technique as in appendix A. At first by substituting the Fresnel approximated functions g (see Eq. 3), the Eq. 7 becomes, y 2D A2R ∗ x gzP (x − xP , y − yP )rect( )rect( ) ⊗ gz (x, y) iλ A B (z − z ) A2R exp i 2π P λ = (16) iλ zzP

π π y 2D x · exp −i x2 + y 2 (x − xP )2 + (y − yP )2 rect( )rect( ) ⊗ exp i . λzP A B λz

u (x, y, z) =

After implementing the same process from Eq. 13 to Eq. 15, the u gives, u (x, y, z) =

=

A2R g(x, y, z)g ∗ (xP , yP , zP ) iλ

z λz λz λz(zP − z) 2 2 2 2 exp i2π ·λ z F rect( ξ)rect( η) exp iπ (xP ξ + yP η) ξ +η A B zP zP

AB B A2R A g(x, y, z)g ∗ (xP , yP , zP ) (17) sinc( x)sinc( y) iλ λ2 z 2 λz λz

2D zP zzP z z exp −iπ ⊗ iλ xP )2 + (y − yP )2 . (x − zP − z λz(zP − z) zP zP

Since g(x, y, z) contains variable x and y, they are not allowed to be replaced into the second braces within the 2D convolution. But the 1/z 2 in the first braces can be moved into the second and substituted under Fresnel approximation by, 1 z2

1 + y2 + z 2 = g(x, y, z)g ∗ (xP , yP , zP ).

≈

x2

(18)

With it the limited reconstruction u in Eq. 8 is obtained.

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