Noniterative boundary-artifact-free wavefront reconstruction from its derivatives Pierre Bon,1 Serge Monneret,1,* and Benoit Wattellier2 1 2
Aix-Marseille Université, CNRS, Institut Fresnel, Campus de Saint-Jérôme, 13013 Marseille, France
PHASICS SA, XTEC Batiment 404, Campus de l’Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France *Corresponding author:
[email protected] Received 15 June 2012; revised 11 July 2012; accepted 11 July 2012; posted 12 July 2012 (Doc. ID 170633); published 8 August 2012
Wavefront sensors are usually based on measuring the wavefront derivatives. The most commonly used approach to quantitatively reconstruct the wavefront uses discrete Fourier transform, which leads to artifacts when phase objects are located at the image borders. We propose here a simple approach to avoid these artifacts based on the duplication and antisymmetrization of the derivatives data, in the derivative direction, before integration. This approach completely erases the border effects by creating continuity and differentiability at the edge of the image. We finally compare this corrected approach to the literature on model images and quantitative phase images of biological microscopic samples, and discuss the effects of the artifacts on the particular application of dry mass measurements. © 2012 Optical Society of America OCIS codes: 000.3860, 100.5070, 110.7348, 120.5050.
1. Introduction
Measuring the wavefront of a light beam is an important subject in many topics of experimental optics. It is widely used in laser beam characterization for beam shaping or quality improvement [1,2], optical metrology [3,4], atmospheric aberration measurements [5–7], ophthalmology [8,9], X-ray [10,11], and, more recently, quantitative phase imaging and thermal microscopy [12,13]. The wavefront sensors commonly used are based on Hartmann [14], Hartmann–Shack [15], or Shearing interferometry [16,17]. In the particular case of quantitative phase microscopy, methods based on differential interference contrast have also been developed [18,19]. All those techniques lead to the wavefront derivatives, assimilated to gradients for regular wavefronts, and then require a numerical integration to recover the actual wavefront in a quantitative manner. Numerous algorithms have been 1559-128X/12/235698-07$15.00/0 © 2012 Optical Society of America 5698
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developed in order to integrate the derivatives [11,18,20–23], and most of them are based on fast Fourier transform (FFT) algorithms [11,18,20–22]. FFT routines are very fast, but, as they are based on discrete Fourier transform (DFT), one of their intrinsic working hypotheses implies an image periodicity within the Fourier treatment to make the phase image continuous and differentiable at the edges. This may lead to some artifacts in the reconstructed phase, in particular when the actual phase distribution covers the boundary of the image. Some algorithms have been developed to overtake this limitation—completely [23] or partially [21,22]— but none of them to our knowledge are at the same time noniterative, low time-consuming, and easily implemented with native FFT toolboxes. We propose here a simple noniterative approach to integrate the gradients, avoiding boundary artifacts but still based on an FFT-based integration algorithm. Our approach looks similar to the one of Ghiglia and Pritt [22] (applied to biological images by Arnison et al. [18]), where the derivatives are mirrored before integration [mirrored derivative
integration (MDI)]. The role of this operation is to make the derivative continuous at the edges. This algorithm removes high-frequency boundary artifacts, but slowly varying artifacts are not removed. We show that this is due to the fact that MDI does not lead to a continuous differentiable integrated function. This explains why some artifacts are still observed. On the contrary, we demonstrate that it is necessary to mirror and antisymmetrize in the derivation direction each derivative [antisymmetric derivative integration (ASDI)] to ensure a continuous and differentiable integrated function. This operation creates new extended data that are compatible with the DFT process, fulfilling the requested periodicity of the final integrated phase map. As a consequence, we completely remove the boundary artifact even when phase objects cross the actual image boundaries. This process is particularly well adapted in quantitative phase imaging, where some of the target objects very often stick out of the nominal field of view. In our case, we consider phase microscopy applied to biology. In the following sections, we first justify the boundary artifact existence in conventional approaches based on DFT, thanks to mathematical circulation considerations. We then propose our ASDI approach before comparing it with a basic noniterative one and MDI, for model objects and for experimental microscopic quantitative phase measurements made with a quadriwave lateral shearing wavefront sensor used in combination with a microscope [12]. We finally discuss the effects of the boundary artifacts on a given application of quantitative phase measurement: dry mass monitoring within a biological sample [24]. 2. Origin of the Gradient Integration Boundary Errors with FFT Approaches
A good trade-off between calculation speed, error propagations and implementation simplicity for wavefront reconstruction from its derivatives is to use FFT-based approaches. We can cite algorithms based on iterative least-square minimization [20,21,25] [Eq. (1)] or noniterative complex plane integration that has been developed for imaging purposes [10,11,18] [Eq. (2)]. In this paper, we will always use the complex plane method (CPM) to recover phase distributions, but the problem discussed here also exists with the least-square approach.
Wx; y
FT−1
Wx; y FT
∂∂x W ∂∂y W ! ∂x W rot ; ≠ ≠ 0⇔ ∂y W ∂x ∂y
W h z φ0 ·
Arg
Q
j z
π
− zj rj
;
(4)
with z the considered point affix, zj the affix of the singularity j, φ0 a constant, rj a relative integer called residue, and Arg the complex argument function. Then the real wavefront is the sum of this W h function plus the regular wavefront component calculated from its derivative integration [Eq. (1) or Eq. (2)]. In the case of a unique singularity, W h is a pure helicoid function. Both reconstruction models given by Eqs. (1) and (2) are based on Fourier space calculations and, as an image is a discrete representation of the space, they are usually implemented using FFT routines. The main hypothesis when using such routines is the required periodicity of the considered function. When this function is limited to a finite extension (in our case, we consider two-dimensional images), the DFT algorithm
FT∂W ∕ ∂xνx ; νy i · FT∂W ∕ ∂yνx ; νy ; νx i · νy
where Wx; y is the wavefront at the point x; y, FT and FT−1 represent the Fourier transform and the inverse Fourier transform operations, and νx ; νy are
(3)
where ∂x W represent the measured derivative of W along the x direction and ∂y W represent the measured derivative along the y direction. In the general case, the wavefront, viewed as a vectorial field, is composed of the integration of a gradient field [using Eq. (1) or Eq. (2)] plus a vector potential responsible for the branch cuts. This vector potential integration leads to a so-called hidden phase [27], which can be represented with the function W h :
νx · FT∂W ∕ ∂xνx ; νy νy · FT∂W ∕ ∂yνx ; νy ; ν2x ν2y
−1
the frequency coordinates conjugated respectively with the x and y coordinates. The main hypothesis to write Eqs. (1) and (2) is that the wavefront function W is differentiable everywhere. It can be wrong in some cases, especially when dealing with strongly fluctuating atmospheric wavefront measurements [26] where some singularities, also known as branch cuts, may arise. Branch cuts are discontinuities in the wavefront phase that occur at places of zero amplitude in the optical field. The singularities are located where the circulation of the measured derivatives is not equal to zero:
(1)
(2)
considers the function as replicated in each dimension and so periodizes it. In the general case, such a replication induces artificial singularities in the image 10 August 2012 / Vol. 51, No. 23 / APPLIED OPTICS
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(a)
(b)
(c)
(d)
(e)
(f)
+
=
Fig. 1. (Color online) (a) Simulated object: a sharp and uniform object is placed at one image boundary. (b) Reconstructed image [CPM, derivative data calculated from (a)]. (c) Error map (b)–(a) where artifacts are clearly visible. (d) Same as (c) but flipped with respect to vertical dashed line visible on (c); a dipole radiationlike shape is visible. (e) Monopole spiral phase distribution centered on the bottom intersection between the object disc and the left border of image (a). (f) Opposite spiral phase compared to (e), centered on the top intersection between the object disc and the left border of image (a). In this particular object, the sum of (e) and (f) gives (d).
boundaries, whereas the real wavefront does not present any singular aspect [10]. Figure 1 shows an example of an artifact arising from simulated derivatives, integrated using CPM, when a model object crosses the image boundaries. The reconstructed image of Fig. 1(b) presents artifacts that can be described, for this simple but sharp object, as the sum of two pure helicoid functions of opposite amplitudes [Figs. 1(e) and 1(f)]. This leads to a dipolelike artificial shape in the phase distribution, which implies a strong error in the phase reconstruction at the object place and also at the opposite image boundary, as these two borders are artificially linked during the working DFT process in the Fourier space. Artificial degradation of the actual wavefront is also observed with real data. As an example, we present the optical path difference (OPD) map introduced by a living COS-7 cell, as it has been measured with a quadriwave lateral shearing interferometer (QWLSI) mounted on a conventional wide-field
(a)
(c)
(d)
transmission microscope with white-light illumination [12]. Figure 2 shows that artifacts on the measured OPD image clearly depend on the COS-7 cell position with respect to the image boundaries. The left part of the figure shows that when no OPD information is crossing image borders, the observed phase sample is correctly reconstructed from its derivatives using CPM. On the right part, the same cell is observed but is now going over the image limits, leading to artifacts both at the cell place and at the opposite image boundary. This kind of boundary error can obviously be a real problem when the exact local quantization of a phase distribution is required. One solution to avoid those problems is to detect the singularity points, using Eq. (3), and to add the hidden phase emerging from the border branch cuts. This approach becomes heavy and relatively imprecise when dealing with a complex object in a border, such as a biological sample like the one in Fig. 2(f). That is why we developed a simple correction scheme that has been used in quantitative phase imaging with biological application, as we will see in the last part of this paper. 3. Boundary Error Correction by Antisymmetrization of the Derivatives
As explained before, boundary integration errors mainly occur because of discontinuities in the periodized phase map W obtained with DFT routines. To limit this effect, Ghiglia and Pritt [22] proposed to extend the original derivatives ∂x W and ∂y W to a four-times bigger data set ∂x W MDI and ∂y W MDI by joining to the original derivative its mirrored data in the two directions (MDI): 8 > ∂x W−x; −y ∂x W−x; y > > < ∂x W MDI ∂x Wx; y ∂x Wx; −y : ∂ W−x; −y ∂ > y y W−x; y > > ∂ W y MDI : ∂y Wx; −y ∂y Wx; y
5
The extended even derivatives lead, after integration, to an integrated phase map that is odd. The DFT routines impose the continuity to the integrated phase and, as it is an odd function, this extended
(f)
(g)
(e)
(b) 10µm
Fig. 2. (Color online) Reconstruction of the OPD distribution of a living COS-7 cell from its gradients using CPM. (a) Measured OPD derivative along the vertical direction. (b) Measured OPD derivative along the horizontal direction. (c) Reconstructed OPD when the cell is fully included in the microscope field of view. (d), (e), (f) Same as (a), (b), and (c) with the cell clearly going over the image limits. (g) Error map of (f) due to boundary effects. 5700
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(b)
(a)
(c)
(e)
(h)
(i)
(f) (d)
(g)
(j)
Fig. 3. (Color online) Comparison between reconstructed phase distributions using a basic integration (left), an MDI [22] (middle), and ASDI (right). (a), (b), (c) Derivative data along x (left) and y (right) axes. (d) Phase reconstruction using Eq. (2), where boundary artifacts are clearly visible due to the hidden phase. (e) Extended derivative data along x (left) and y (right) axes using a symmetrical approach. (f) Phase reconstruction from (e) using CPM. (g) Final phase reconstructed using MDI; boundary artifacts are limited but still here. (h) Same as (e) using a antisymmetrical approach, in the derivative direction, to fulfil Eq. (7) requirements. (i) Same as (f) with (h) data. (j) Final phase reconstruction using ASDI; no boundary-induced artifact is visible.
phase map tends to zero in the boundary. This property creates low-frequency errors compared to the real phase map [as is visible in Figs. 3(g) and 4(e)]. MDI still creates artifacts in the boundaries when using DFT routines, but the error is reduced with respect to a basic integration. Actually, the highfrequency errors are erased using MDI [as visible by comparing Figs. 3(d) and 3(g)]. Nevertheless, the low-frequency errors strongly modify the reconstructed phase map and need to be corrected in order to allow rigorous quantitative studies. As we just discussed, the low-frequency problem rises from the fact that DFT artificially periodizes the integrated phase function. We consider here a modified approach to completely suppress the boundary artifacts. The phase has to be continuous at the boundary, and so differentiable as we are working on a discretized space, to avoid boundary artifacts. We propose to work with an extended phase map W e with the property to be perfectly even by construction. This extended even phase is thus continuous at the border of the image and differentiable. Let us consider that the initial support of W is 0; N × 0; M with N; M ∈ N2 . We also suppose that W is fully differentiable on its support. We build from W an even function W e with a support of −N; N × −M; M: W−x; −y W−x; y We : (6) Wx; −y Wx; y W e is a fully periodic function. That implies that W e can be written as a Fourier series and, so, the
DFT coefficients are perfectly equal to the Fourier series coefficients. This limits the risks of the boundary artifact using DFT. As W is differentiable on its support, so is W e. As W e is perfectly even, its partial derivatives along the x and y directions are each odd in the derivative direction and even in the opposite direction. This implies a necessary condition on the derivatives to make the integrated function even, and consequently continuous at the edges: 8 > −∂x W−x; −y −∂x W−x; y > > < ∂x W e ∂x Wx; y ∂x Wx; −y : −∂ W−x; −y ∂ > y y W−x; y > > W ∂ : y e ∂y Wx; y −∂y Wx; −y
7
We apply this idea to measured derivatives ∂x W and ∂y W by extending them to a four-times bigger area following the conditions of Eq. (7). In this way, each final extended derivative image is fourtimes bigger in area and is odd in its derivative direction and even in the opposite. That leads to a couple of derivatives ∂x W e and ∂y W e : each one of them is first extended along its derivative direction by joining its antisymmetrical image, and then the resulting image is duplicated one more time by adding its symmetrical along the other direction. The interest is that the extended derivatives fulfil the necessary conditions to lead to an even perfectly periodic phase distribution after integration using Eq. (2). This implies that artificial border-induced artifacts are 10 August 2012 / Vol. 51, No. 23 / APPLIED OPTICS
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Table 1. Comparison of Relative Standard Deviation error of OPD Reconstruction of the Fig. 3 Simulated Object for Three Integration Methods
Algorithm
Relative standard deviation
Basic CPM integration Mirrored derivative int. (MDI) Antisymmetric derivative int. (ASDI)
16.9% 14.3% 0.15%
discarded. Figure 3 gives the flow chart of this algorithm. Because the data that are really introduced into the CPM integration are doubled along the two space directions, and also because the complexity of the best FFT routines is in On · ln n, our ASDI is 41 ln 4 ∕ ln n → 4 times slower (for big images) than one iteration of the basic one, but has the same easy-to-use characteristics and mathematically no border artifact. We characterize the reconstruction error of the model object of Fig. 3 for the three methods (basic CPM, MDI, and ASDI). Table 1 sums up the results, clearly showing that our ASDI approach is more than 100× better than the state-of-the-art algorithms. We then checked the interest of our algorithm on real data. Figure 4 presents a comparison between the basic integration using CPM, the MDI algorithm proposed by Ghiglia and Pritt [22] and used for phase imaging by Arnison et al. [18], and our new ASDI. The imaged-sample is a drosophila embryo that is larger than the microscope field of view. As for COS-7 cells, derivative data are obtained with a QWLSI mounted on a conventional wide-field microscope. Figure 4 demonstrates that ASDI leads to a much more realistic OPD map: the embryo is optically thicker in the upper-left image corner [Fig. 4(f)] as expected [Fig. 4(a)]. On the other hand, the basic integration using CPM or MDI gives a maximal OPD approximately in the center of the image, which is not realistic [Figs. 4(d) and 4(e)]. The difference between the basic integration and our ASDI is presented in Fig. 4(g). For this sample, a huge number of singularity points makes it difficult to describe (a)
(b)
the hidden phase as a sum of helicoid functions [Eq. (4)]. It is nevertheless true that such an artificial additive phase component has to be banished when the target application of the phase gradients measurement clearly requires quantitative phase data. As an example, the last part of the paper will focus on one of this kind of application. 4. Application to Dry Mass Measurement of Living Cells
One important application of quantitative phase imaging is the dry mass determination within a sample [24,28]. Integrated OPD on a whole living cell thus gives a quantity that is proportional to the cell dry mass, i.e., the mass of everything but water inside the cell. Equation (8) formalizes this approach: m
1 α
Z OPD · dS;
(8)
cell
where m is the dry mass, α is the so-called specific refractive index increment (equal to approx. 0.18 μm3 · pg−1 [24]), and dS is an elementary surface. Let us consider a pseudo-artificial patterned sample, composed of a mosaic of the single cell presented in Fig. 2(c). Theoretically, as each cell is identical to the others in the pattern, the dry mass should be the same everywhere in the image, as long as the cell is completely in the field of view. Figure 5 shows the dry mass distribution over the cell population with respect to the used algorithm (basic integration, MDI or ASDI). It is clearly visible in Fig. 5(f) that the histogram obtained on complete cells with the basic algorithm or with mirrored derivatives is much larger than the one with our optimized algorithm. Corresponding statistics are presented in Table 2. All algorithms give the same average dry mass for the cell population, but the standard deviation is much better (more than 2 orders of magnitude) with our ASDI algorithm. As a consequence, this improved algorithm becomes essential for dry mass
(d)
(f)
10µm
(c)
(e)
(g)
Imaged Zone
Fig. 4. (Color online) (a) Drosophila embryo (scheme): the darkness is proportional to the sample thickness, and the imaged zone is represented by a black dash rectangle. (b) Drosophila OPD gradient along x. (c) Drosophila OPD gradient along y. (d) OPD obtained by basic FFT integration algorithm of gradients (a) and (b); boundary artifacts are present. (e) Same as (d) using MDI; boundary artifacts are reduced but still visible. (f) Same as (d) using ASDI; no artifact is visible. (g) OPD error of (d) [obtained by subtraction with (f)]. 5702
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(a)
(c)
(e)
90
Basic Integration Mirrored Derivative Int. Antisym. Derivative Int.
80 70
(b)
Count
60
(d)
50 40 30 20 10 0 0
100
200
300
400
500
600
Dry mass (pg)
Fig. 5. (Color online) (a) OPD image obtained with the basic CPM integration; the image presents artifacts. (c) OPD image obtained with the antisymmetrized derivative algorithm; the image is boundary artifact-free. (c) Error map (a)–(c). (d) Segmentation image; the background is in black, and each other color represents a distinct cell. (e) Histogram of the segmented cell dry mass of the image (d). Brown, using (a); red, using MDI (image not presented here); orange, using (c).
Table 2.
Comparison of Dry Mass Study on the OPD Image Obtained with Three Different Algorithms (Data of Fig. 5)
Classical integration Mirrored derivative int. (MDI) Antisymmetric derivative int. (ASDI)
Average dry mass (pg)
Standard deviation (pg)
297.1 296.5 296.3
55.1 51.3 0.3
study in order to avoid artifact effects in the useful final phase distribution. 5. Conclusion
We propose a simple solution to avoid a boundary artifact on a reconstructed wavefront from its derivatives, using a simple FFT-based complex plane integration algorithm. The derivatives have to be duplicated and antisymmetrized in their specific shear direction in order to artificially periodize the integrated wavefront map. Our noniterative ASDI is approximately four times slower than a single iteration of a basic integration approach but allows a much higher precision in the phase recovery. ASDI is also much more exact than the state-of-the-art MDI with the same calculation time. Those precise results are clearly essential when dealing with quantitative phase imaging and may also be useful in truncated beam analysis and correction. We checked our algorithm in the particular study of dry mass monitoring within a biological sample, and the integration error was reduced by a factor of more than 100, compared to the classical integration algorithms. This new approach for wavefront reconstruction from its derivatives become powerful when the
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