Coherent imaging has a wide range of applications in, for example, microscopy, astronomy, and radar imaging. Particularly interesting is the field of microscopy, where the optical quality of the lens is the main limiting factor. In this article, novel algorithms for the restoration of blurred images in a system with known optical aberrations are presented. Physically motivated by the scalar diffraction theory, the new algorithms are based on Haugazeau POCS and FISTA, and are faster and more robust than methods presented earlier. With the new approach the level of restoration quality on real images is very high, thereby blurring and ringing caused by defocus can be effectively removed. In classical microscopy, lenses with very low aberration must be used, which puts a practical limit on their size and numerical aperture. A coherent microscope using the novel restoration method overcomes this limitation. In contrast to incoherent microscopy, severe optical aberrations including defocus can be removed, hence the requirements on the quality of the optics are lower. This can be exploited for an essential price reduction of the optical system. It can be also used to achieve higher resolution than in classical microscopy, using lenses with high numerical aperture and high aberration. All this makes the coherent microscopy superior to the traditional incoherent in suited applications.
© 2017 Optical Society of America
1. Introduction and context
Image restoration is an important task in image processing with numerous applications. A typical application consists of the correction of optical errors in imaging systems. These optical errors often manifest themselves as blur, therefore these methods are also known as deblurring algorithms. All optical systems are limited by the quality of their optical components and especially microscopes are strongly limited by their depth of field. Standard optical systems use incoherent illumination, which for the scope of this paper means a spatially extended lightsource which can have a broad light spectrum.
Coherent imaging in contrast requires spatial and frequency coherence of the illumination. It has growing applications, especially in microscopy, spectral astronomy and microwave imaging systems. Coherent microscopy is an important field of research and we will show in this paper how this field can benefit from image restoration. We will show that the effective depth of field of an existing system can be extended with the correction of wavefront deformations. It is important to note that the intensity distribution of an aberrated image lacks phase information, which would be needed for straightforward aberration correction, therefore the coherent image restoration is an algorithmic challenge. The foundation of these algorithms is the physical observation that focused images have a planar wavefront and that this property can be used to restore an ideal image from a disturbed image. The algorithms are inspired by phase retrieval algorithms such as the Hybrid-input-output algorithm , however they solve an entirely different problem, namely image restoration. From the distorted image and the known phase aberrations, the sharp in-focus image is recovered.
The main contribution of this paper are several novel algorithms for coherent image restoration which achieve good results even on noisy real images.
These algorithms are a large improvement over our previous work , in which we presented the WFC-GS (Wavefront correction - Gerchberg Saxton) and WFC-HIO (Wavefront correction - hybrid-input-output) algorithms. The novel algorithms, which are based on projections onto convex sets, are the wavefront correction - average projection (WFC-AP), the wavefront correction - relaxed average alternating reflections (WFC-RAAR) and the wavefront correction - Haugazeau-like average alternating reflections (WFC-HAAR) algorithms. Based on fast iterative shrinkage thresholding, the novel WFC-FISTA algorithm is presented. Table 1 gives an overview. We show with several direct comparisons in Section 5 how our novel algorithms are superior to previously presented basic algorithms and to other related work in coherent restoration. Furthermore we present extended results in application on real microscopic images of biological and medical samples.
In addition, we give a more in-depth introduction and perspectives on the relation between wavefront correction, phase retrieval and non-coherent image deconvolution. We also give insights into the convergence and convexity of projective image restoration with wavefront correction and demonstrate that coherent imaging can be easily implemented even with a simple, low cost LED illumination. The results show that for coherent microscopy even very faulty imaging settings can be compensated with wavefront correction.
Incoherent image model
The imaging model for incoherent systems is defined by
One can regard the problem of inverting the influence of this blur kernel as a deconvolution. Therefore this subcategory of image restoration algorithms is known as deconvolution algorithms. All general incoherent image deconvolution methods rely on the blur model in Eq. 1.
Widely used are so called non-blind deconvolution algorithms, which require prior knowledge of the blur kernel B. In contrast, if the kernel is estimated from the observed image itself, the algorithms are called blind deconvolution algorithms.
Image priors are important to the success of modern blind and non-blind deconvolution algorithms such as [3, 4]. Most algorithms incorporate a-priori knowledge about the image in terms of gradient sparsity, which is derived from natural image statistics . Total variation is arguably the most common way to model this prior, see [6, 7] and more recent work in . It is based on the assumption of locally smooth areas and it takes advantage of the L1 norm. Different optimization strategies can be used for solving the resulting optimization problem, such as variable splitting resulting in algorithms like FISTA [9,10].
Coherent image model
In contrast to the intensity based incoherent image formation model, in coherent imaging the image formation model is based on amplitude. In the following a description of image formation as a linear filter is used. It is based on the scalar diffraction theory of Sommerfeld and more precisely the approximations of Fresnel diffraction. Detailed derivations can be found in  and .
Let U be the undisturbed complex amplitude distribution of a wavefield, then
This paper is organized as follows. Firstly, necessary information about the related work, the phase retrieval problem and projections onto convex sets (POCS) algorithms is given in Section 2. The main contribution of this paper starts with an introduction of the concept of wavefront correction (Section 3). In Section 4, using the similarities to phase retrieval algorithms, a generalized mathematical formulation of the wavefront correction as a problem of projections onto non-convex sets is developed, leading to the novel WFC-AP, WFC-RAAR and WFC-HAAR algorithms. The question how wavefront correction must be formulated to apply FISTA is addressed in Section 4.1 together with the consequences of this formulation of wavefront correction and our novel algorithm WFC-FISTA. Then we present an extensive evaluation on simulated and real microscopy images in Section 5 and a conclusion in Section 6.
2. Related work
2.1. Coherent restoration
Coherent restoration techniques have many application, as can be seen by the very diverse areas of application contexts used in related work. In  the aberrated image of a laser illuminated target is to be corrected to improve the image quality in coherence tomography. A novel correlation technique is used to estimate the phase error, which is corrected by Fourier space multiplication with the inverted estimated phase error. The same restoration technique is used in  with prior phase error optimization based on a sharpness criterion. However the scale and application is entirely different, as the goal is to compensate the effects of atmospheric turbulence on an aerial image. With the convolution theorem it follows, that a multiplication in Fourier space with a compensating phase is equivalent to a convolution with the Fourier transformed phase modulation. In this context it is called filtering with the inverse PSF(point spread function), which is also known as spatial filtering. This technique of spatial filtering is the state of the art , which  explicitly confirms.  also proposes different approach for the reconstruction of a digital hologram using a concept called cepstrum, however this approach puts very strong conditions on the object-wave modulation and on the intensity of the reference wave. In  spatial filtering is extended to a multi step approach to correct for the phase effects caused by off-axis tilt, curvature of the microscopic lens and other factors. Furthermore  gives good perspectives on the physical motivation and how this algorithmic spatial filtering can be replaced by a purely optical compensation for spherical phase aberrations. A related approach is presented in  where a quasi-physically setup is used to compensate the phase curvature induced by a lens in holographic microscopy.
Another more recent application of this technique is used in  for 3D tracking of bacteria. Coherent illumination is used and this work employs a strong denoising algorithm on the microscopic images, because this restoration algorithm seems to be very noise sensitive.
Using this spatial filtering technique, the phase of the aberrated image is not included in the calculation of the corrected image. Admittedly the phase cannot captured directly by an image sensor and is therefore not immediately available. This is not physically correct and ignores that the phase of an image is a substantial information, therefore this is not an appropriate technique for high quality coherent image restorations.
This weakness is confirmed by the strong remaining restoration artifacts in the result images of [13, 14]. We solve this inherent problem with the wavefront correction algorithm (WFC), which iteratively recovers both sharp image and phase. A direct comparison in Section 5 with PSNR values of results with the method from [13, 14] and the novel WFC-HAAR algorithm confirms this.
In  we presented a first basic algorithms capable of coherent image restoration by wavefront correction. Although already delivering good results on simulated images, the algorithms  are not robust, as can be seen from the imperfect restorations of noisy real images. As outlined in the contribution, in this paper largely improved algorithms are presented (for comparisons see results in Section 5).
2.2. Phase retrieval
In this section an overview over the phase retrieval problem and algorithms is given.
Phase retrieval algorithms solve the task of restoring the unknown phase of a wavefront from the measured intensity distributions in the Fourier plane and the image plane with optional spatial constraints in the image plane. Although a different problem is solved and no aberrations are considered, phase retrieval is closely related to wavefront correction, because both restore missing information in a coherent imaging system from measurements. While phase retrieval targets the phase or phase aberration, the wavefront correction algorithms recover the sharp image.
For a detailed survey of phase retrieval algorithms see , for a performance comparison of various algorithms see . The first phase retrieval algorithm was presented by Gerchberg and Saxton in , the Gerchberg-Saxton (GS) algorithm.
Using the Fourier transform, a light distribution in the pupil plane can be transformed into the focal plane, see Fig. 1. The Gerchberg-Saxton algorithm restores the phase of the image by iterative cyclic application of the object-plane constraint, which ensures that the measured intensity equals the intensity with novel phase, and the pupil plane constraint, which ensures that the current iteratives and the measurements intensity in the pupil plane match.
A disadvantage of this basic algorithm is its slow convergence. This was greatly improved by Fienup in . His Hybrid-Input-Output (HIO) algorithm is widely used today.
It is shown in , that the GS algorithm (error reduction approach) converges in the weak sense, which means that each following step reduces the RMS (root mean squared) error. Moreover an equivalent formulation in terms of steepest gradient descent is possible.
The Gerchberg-Saxton algorithm can be seen as a projective algorithm , which corresponds to the Alternating Projections algorithm or von-Neumann algorithm . Similarly the Fienup Hybrid-Input-Output algorithm can be seen as a non-convex POCS algorithm, equivalent to the Douglas-Rachford algorithm, as shown by . For results of the application and comparison of different projective algorithms for phase retrieval see .
We follow the definition of FISTA in  with pL, the proximal map of function g, defined as:
3. Wavefront correction
3.1. Physical introduction
This section starts with a motivation of wavefront correction in optical terms, which leads to the formulation of our previously developed WFC-GS algorithm in . Then the novel algorithms are presented.
In a coherent system, the light distribution in the focal plane is the Fourier transform of the light distribution of the pupil plane, the Fourier-space. Thus assuming that the image is captured in perfect focus and assuming that there are no aberrations, the image will be sharp and we can switch between this focal image plane and the pupil plane by applying the Fourier transform and the inverse Fourier transform, as in phase retrieval algorithms (see Section 2.2 and Fig. 1). If the image is not captured in perfect focus or if any other aberrations occur, the image is the Fourier transform of the ideal wave distorted by the additional aberration. This wavefront deformation can be seen as caused by an optical path difference across the aperture. If there are no aberrations, then the wavefront deformation is zero and thus the image is sharp.
Now, if the image is out of focus, an additional spherical phase delay occurs. In Fig. 2 a defocused imaging system with spherical defocus wavefront deformation is depicted. The colored graph in the left corners shows the path delay across x and y, the horizontal and vertical distances from the center of the lens perpendicular to the optical axis.
For coherent light incident on a convex thin lens with radius r (with standard Fresnel approximation ) the spherical phase delay is11].
The spherical wavefront deformation due to wavefront propagation can be described as 
Hence the effect of a lens is equivalent to a spherical wavefront deformation caused by defocus. In focus both compensate each other producing a sharp image with planar phase.
As a consequence of the linearity of an aberration free imaging system, we know that in an ideal coherent optical system the image of a real object is a real function with planar phase. We enforce a zero angle phase vector by calculating the magnitude, effectively setting the amplitude to real and positive values. Note that in a restored image, positive or negative amplitude values would show the same intensity.
We describe these wavefront deformation using orthogonal Zernike polynomials . They parametrize the deformation in terms of distance from the center of a unit disk and are traditionally used to describe wavefront aberrations . This is because they form the basis of a vector space, which means that any aberration can be described as a combination of Zernike polynomials.
Given an image disturbed by wavefront aberrations and if we had complete information about the light distribution in the pupil plane of the imaging system in phase and amplitude, adding the compensating wavefront and applying the inverse Fourier transform would be all we needed to do to recover a sharp image. However practical image restoration is not that simple, because in most applications, such as in a typical microscopic setting, we can only measure the intensity distribution and have no knowledge about the phase distribution of the distorted image. Because of this lacking pupil plane information, phase retrieval algorithms are also not applicable.
The task is to develop an image restoration algorithm, based solely on the knowledge of the intensity distribution of the distorted image and some knowledge of the wavefront deformation.
We can use two constraints: The first constraint is that the measured amplitude distribution, which is the square root of the intensity distribution is preserved. The second constraint is the requirement for a real and positive amplitude distribution in the virtual ideal image plane where all aberrations are removed. The application of the phase aberration function ps or its inverse introduces a phase shift on the corresponding complex amplitudes. It is defined analogously to the specific defocus phase shift pd in Eq. (12).
3.2. WFC-GS algorithm
To reach an algorithmic description of the algorithm we start with the virtual ideal imaging plane. It is introduced as virtual focal plane f, which can be reached from pupil plane A via the inverse Fourier transform . The following equations define our Gerchberg-Saxton (GS) inspired version of the Wavefront Correction Algorithm (WFC-GS) in iteration step i, illustrated by Fig. 3:Eq. (15) and the planar wavefront 8condition is enforced by replacing all values by their magnitude and constraining the values to real and positive values in Eq. (16). Back in the image plane with inverse steps in Eqs. (17) and (18), the object magnitude constraint is applied in Eq. (19). This concludes iteration i. In the last step of the algorithm the object magnitude is set to the initial magnitude with application of Eq. (19). Thus oi is in fact the complex combination of the initial magnitude from the square root of the measured intensity ║o0(n)║ and phase information gained during the course of the algorithm. Therefore we can conclude that wavefront correction algorithm also conducts phase retrieval.
The aim of wavefront correction is to restore the sharp image, which resides in the virtual focal plane. Hence after the convergence criterion is reached, image plane value oi from the last iteration must be transfered to this focal plane. This is achieved by applying the Fourier transform to reach Fourier space, then the inverse wavefront deformation and finally an inverse Fourier transform into the focal plane. These steps are equivalent to applying Eqs. (14) and (15).
4. Projection algorithms
In the previous section the WFC-GS algorithm was introduced within the domain and notation of optical imaging. In this section an alternative view is sought. Similar to phase retrieval for the WFC-algorithm the fulfilling of constraints can be described in the framework of projections onto convex sets. The question of actual convexity of WFC will be discussed later. In the most general sense, the enforcement of a constraint projects an element from the sets of possible data points onto the set of valid data points, i.e. points which do not violate the constraint. While in the general non-convex case, multi-valuedness of the projectors is an issue, this can be resolved by arbitrarily choosing a valid data point if the projection result is multivalued .
With the projective formulation of wavefront correction in terms of the POCS (projections onto convex sets), the Gerchberg-Saxton algorithm can now be identified as the von-Neumann algorithm . The update formula for wavefront correction with this most basic POCS algorithm is already given (see Eq. (20)), we will continue to call this algorithm the WFC-GS algorithm in the following.
The WFC-GS algorithm includes two constraints, the focal plane constraint and the image plane constraint. We denote the application of the image plane constraint of Eq. (19) as projection Po and the application of focal plane constraint including necessary transformations in Eqs. (14)–(18) as projection Pf.
The WFC-GS algorithm can now be written as
The Hybrid-Input-Output (HIO) algorithm  has a greatly improved convergence speed over the Gerchberg-Saxton phase retrieval algorithm. In  we applied the concept of wavefront correction. Because the object support constraint of the HIO algorithm is not applicable for image restoration, it is modeled instead with a characteristic function and can then be eliminated using . The WFC-HIO algorithm  reads:2]. For a more general proof see , for the HIO algorithm see .
This projective formulation allows us to use more advanced POCS algorithms introduced in the Appendix to derive novel wavefront correction algorithms. To express the constraint applications as projections, we always identify one of the projections as the image plane constraint projection Po and the other as the focal plane constraint projection Pf.
We apply the averaged projection (AP) algorithm  to wavefront correction. This leads to the WFC-AP algorithm with update formula
The RAAR (Relaxed averaged alternating reflections) algorithm by Luke  is a POCS algorithm with improved convergence. The WFC-RAAR algorithm is defined by this update formula:
A recent contribution by Bauschke et al.  is a strongly convergent POCS algorithm based on concepts by Haugazeau , the HAAR algorithm. The significance of this algorithm lies in its convergence properties, which are discussed in the Appendix.
As a Haugazeau based algorithm, the WFC-HAAR algorithm readsAppendix, Eq. (42)).
The HAAR algorithm is strongly convergent, but relies on the assumption that both projections are projections onto convex sets. For the wavefront correction problem, we have projections Po and Pf, the image plane constraint and the focal plane constraint. It is easy to see that the magnitude projection of the focal plane constraint is convex, because the set of positive real images is convex. For the image plane constraint, that we implemented by enforcing the measured amplitude and leaving the phase as it is, the situation is not so simple. We observe that the phase retrieval problem exhibits the same constraint and thus we can use the arguments made in  for phase retrieval. If we visualize this projection onto a given amplitude in a plane of complex numbers, the projection set can be visualized by a circle. The radius is defined by that amplitude and the points on the circle are defined by different phase angles. As seen in Fig. 4 the connecting line between two points of this sets is not inside the set, which means that this set is not convex. Therefore image restoration with wavefront correction in projective formulation is not convex and the WFC-HAAR algorithm is not strongly convergent. Nevertheless we can benefit from the very good convergence properties as will be seen in the results.
4.1. The novel WFC-FISTA algorithm
Our literature review of incoherent deconvolution algorithm has shown that many algorithms gain their performance from appropriate priors. In the following section we will derive a formulation of WFC suitable for proximal algorithms and show how to apply the FISTA optimization algorithm. For the application of proximal algorithms, we need to see the problem of wavefront correction from a optimization perspective. We regard the mean squared error between a blurred version of the optimized amplitude and the input amplitude.
Let F be the matrix that applies the Fourier transform by left multiplication and o the image plane light distribution with dimensions m × n. Thus the aperture image Ap is formed by
Furthermore to revert to the image plane the inverse Fourier transform is applied. Hence, the transformation matrix W from focal to image plane reads:29].
Because Wf can be expressed as a left multiplication with an invertible matrix, a linear transform, and an absolute value projection, it is clear that f is convex. Thus with this different view of wavefront correction we gained convexity, which means that the requirements on f for FISTA are satisfied.
For the minimization of the quadratic term, FISTA requires the gradient of f. The derivative of the linear part of Wf is easy to calculate and constant, but the absolute value function complicates it. When convergence is reached the image should be real and positive. If the image is already real and positive, than the values remain unchanged by the absolute value function. Hence the function does not need to be applied and can be neglected. In the other case, we can use an approximation for the derivative by calculating the derivative without absolute value function influence and then project it onto the real values by applying the real part function Re. Hence in the next optimization step, only real values can occur.
Thus the gradient of f (x) can be calculated very fast and easily, as if the absolute value function had no impact on it and Wf were only a matrix multiplication with W:29]. We see that although the physical background and arguments differ, similar optimization problems must be solved.
With the formulation of W as derived above, the allocation of matrix W for an image of size n × n requires n4 elements. Moreover matrix multiplication is of higher computational complexity than fast Fourier transform. Thus in the implementation of the algorithm, the steps to create it are applied separately on any given image x with the fast Fourier transform.
An overview of the algorithms used and developed in this paper is shown in Table 1.
5.1. Application on simulated images
The wavefront correction algorithm works by removing the blur from an aberrated image, as shown in Fig. 5. In this section we work on image with simulated aberrations. Real images follow in the next section. Images with simulated aberrations are created by first applying the Fourier transform, adding the appropriate wavefront deformation (described in Eq. (13)) in Fourier-space and then transforming it into the image plane. An illustration of the transformations involved is shown in Fig. 2. There are two criteria that define a good restoration algorithm, speed and robustness.
The restored image is the output of all our algorithms and is therefore used for evaluation. We will compare the algorithms by their peak signal to noise ratio (PSNR) calculated from the ratio between the difference between original image and restoration from defocused image as noise and the original image as signal. Clearly visible in Fig. 5 is the large difference between the related work method [13,14] and wavefront correction (23.9dB to 49.6dB), thus  and  are not included in further evaluation.
The restoration quality with a given number of iterations is shown in Fig. 6(a). All tested wavefront correction algorithms achieve good results, the WFC-HAAR algorithm shows the best convergence, confirming the theoretical advantages. The WFC-RAAR and WFC-HIO algorithm have slower convergence and the WFC-AP algorithm is the slowest of the presented projection algorithms. The WFC-FISTA algorithm converges very fast and does not improve in PSNR from then on. This behavior is caused by the regularization, which inhibits this algorithm from reaching an exact restoration. However this can become advantageous as the results on noisy data below show.
In the next step we visually and quantitatively by PSNR compare the results of algorithm with a fixed very low number of iterations, the results are shown in Fig. 7. This very low number of iterations should make it easy to spot differences. However even though the PSNR values differ much, the visual difference in restoration quality is not as prominent as expected. The previous result is still confirmed, the WFC-FISTA algorithm shows a slightly better restoration and the WFC-AP algorithm a slightly worse restoration.
To simulate the effects of real data, we conduct two tests: First, restoring an image with a spherical wavefront that deviates from the true deformation by 2% in radius. Second, we add Gaussian noise of different strength to the image, after restoration the result is still compared to the original noise free image.
The results for both tests (Figs. 6(b) and 8) are very similar. This confirms that with our approach the actual robustness of the algorithms can be measured. The diagrams show a large difference in how well the algorithms are able to restore imperfect data. For low noise levels the projective algorithm can be used with good results. The highest noise sensitivity has the WFC-HIO algorithm and the slowest algorithm (WFC-AP) is the most robust of the projective algorithms. On this noisy input data we see a clear advantage in robustness for the WFC-RAAR algorithm over the WFC-HIO algorithm in previous works . The WFC-HAAR algorithm has the fastest convergence, however it is more complex to implement.
For noisy input data or in case the optical aberrations are not precisely known, the WFC-FISTA is superior over the projective algorithms due to the noise canceling property of the total variation regularizer. With a 25dB noise input image, the restoration result of the WFC-FISTA reaches a PSNR of 29.2dB. This means that the result image is a nearly accurate representation and less noisy than the input data.
5.2. Application on real images
With the next experiments, we want to explore the capabilities and limitations of the algorithm on real images acquired with an microscope imaging setting and LED illumination. An observation target is bright field illuminated with partially spatial and frequency coherent light. Depending on color of the observation target, we use a red 627nm or a green 530nm Luxeon LED at a distance of approx. 30cm from the target with a 0.6mm aperture in front of the LED as shown in Fig. 9. In practical application the coherence condition is dependent on the size of the specimen. With the help of a microscopic lens, the light is captured by a digital image sensor. The focus is adjustable to be able to produce both sharp and out of focus images.
In Fig. 10 a strongly defocused image of a USAF test target is restored with the previous work algorithms  and with the new algorithms. Because it is difficult to calculate PSNR values on real, acquired images, we use qualitative comparison of visual quality for natural image. Fig. 10(c) is the best result of previous work , a slight improvement with less noise in the restoration can be seen in the result of WFC-AP in Fig. 10(d), the result of WFC-HAAR seems to be similarly noisy. In contrast a large improvement can be seen with the WFC-FISTA algorithm. This confirms the results on simulated images. The regularization removes the restoration artifacts while the image content is restored.
Small objects such as the small blood vessels streaks in Fig. 11 are not visible anymore in a strongly defocused image. Only the ringing artifacts hint to their existence. They can still be restored, because the information about the objects structure is preserved in these strong ringing artifacts, see Fig. 11. The restoration also shows the advantages of the novel WFC-FISTA algorithm over the prior WFC-GS and the more stable projective algorithm WFC-AP.
Cell counting is an important task in medial imaging. In Fig. 12 a pigeon blood smear is shown. It is very hard to make an accurate cell count without guessing on the input image, while with the restoration even the shape of the blood cells can be measured accurately.
An example of the limits of the WFC-FISTA algorithm is the restoration of a human brain tissue in Fig. 13. The image shows a seemingly noisy input image and we can see how the restoration image quality is limited by the noise. Even with longer exposure times, we were unable to remove this unsteady background, which is not formed by image sensor noise. We reason that this is due to the light scattering in a three dimensional non homogeneous medium. The black dots on the sample are cores of transparent blood cells. This is not an ideal imaging situation for wavefront correction. Another point is that even though the cores of the surrounding cells and the structure of the neuron are clearly sharper in the right image, some parts of the neuron remain unsharp. From the three dimensional structure of the neuron only one layer can be focused using the WFC-algorithm. A completely sharp image would require a spatially variant defocus estimate.
In this paper novel methods for Wavefront correction are presented, which show faster convergence and are more robust than earlier methods from  and substantially better results than related work (see Fig. 5, [13,14]).
The algorithms introduced in this paper can be summarized in two classes, projective and proximal algorithms. For the projective algorithms, experiments on simulated images have shown that the WFC-HAAR is the fastest algorithm reaching the best result with the lowest number of iterations. The WFC-AP algorithm is very easy to implement and is at the same time most stable against noise and faulty aberration estimates, however it is also the algorithm which requires the most iterations. After extensive evaluation on real images we would strongly recommend this algorithm, as the good restoration quality outweighs the slight decrease in convergence speed compared to other novel and previous works projective algorithms. Due to the structure of the algorithm, a GPU implementation of this algorithm is easily possible. Our implementation reaches a very fast runtime of 3.6ms per iteration for an image of size 512 × 512.
The WFC-FISTA algorithm with total variation regularization delivers superior results on images with imperfections, such as noise and on real images of biological samples. Evaluation and especially comparison with non-regularization methods is difficult because the regularization also has a denoising effect on the sharp image, which negatively impacts the PSNR scores. Visual comparison on real images shows that the results of the WFC-FISTA are superior to the projective algorithms, showing less noise and clearer restoration of edges. Unfortunately the WFC-FISTA algorithm is very slow and therefore less suited for real time applications. Our CPU implementation on a 4GHz processor requires 0.20s per iteration with an image of size 512 × 512.
If real time application is required we recommend the WFC-AP algorithm, otherwise the WFC-FISTA algorithm is clearly the best choice.
The presented good restoration results on real images show that coherent image restoration with wavefront correction can largely extend the depth of field of microscopes. The presented methods are not only capable of removing defocus, but also other strong optical aberrations can be compensated very nicely. We believe the success of coherent restoration lies in the nature of coherent imaging, where in contrast to incoherent smeared-blur defects, the coherent ringing-blur defects retain the structural information of the undisturbed object. This makes imperfect optics with larger numerical aperture and size to viable components for coherent microscopy, making it superior to incoherent imaging, where blur cannot be compensated to such an extent.
Future work will be focused on the restoration of images from 3D objects with spatially variant parameters and on automatic image based estimation of optical aberration parameters.
Appendix: Projections onto convex sets
Projections onto convex sets (POCS) are a common approach to solving feasibility problems. Given a Hilbert space H with norm ║…║ and inner dot product, let a and b be two points in S, then the set S is convex if
For t ∈ H and S ⊂ H, let the result of a projection PS onto set S be defined as25] algorithm reaches a new iterative by alternating application of projections onto these sets: 27].
In  Haugazeau presents a strongly convergent algorithm for finding the intersection of two convex sets. Based on his works Bauschke et. al. define the Haugazeau-like average alternating reflections algorithm (HAAR) .
Let X be a real Hilbert space and let (x, y, z) ∈ X3 be a tuple, with the following property33].
In  Haugazeau defines the sequence (yn)n∈N as32] Bauschke et al. improve on this recursive algorithm and establish the HAAR algorithm: With operator T defined as
German Research Foundation (DFG) Cluster of Excellence FUTURE OCEAN (CP1331, CP1525).
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