Abstract: We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a “non-lifting” relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is basis pursuit, which implies that the phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.

Abstract: Imaging systems' performance at low light intensity is affected by shot noise, which becomes increasingly strong as the power of the light source decreases. In this Letter, we experimentally demonstrate the use of deep neural networks to recover objects illuminated with weak light and demonstrate better performance than with the classical Gerchberg-Saxton phase retrieval algorithm for equivalent signal over noise ratio. The prior contained in the training image set can be leveraged by the deep neural network to detect features with a signal over noise ratio close to one. We apply this principle to a phase retrieval problem and show successful recovery of the object's most salient features with as little as one photon per detector pixel on average in the illumination beam. We also show that the phase reconstruction is significantly improved by training the neural network with an initial estimate of the object, as opposed to training it with the raw intensity measurement.

Abstract: Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, $$y_k = \left| \varvec{a}_k^* \varvec{x} \right| $$ for $$k = 1, \ldots , m$$ , is it possible to recover $$\varvec{x} \in \mathbb C^n$$ (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretic explanations. In this paper, we take a step toward bridging this gap. We prove that when the measurement vectors $$\varvec{a}_k$$ ’s are generic (i.i.d. complex Gaussian) and numerous enough ( $$m \ge C n \log ^3 n$$ ), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal $$\varvec{x}$$ , up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.

Abstract: Holography encodes the three-dimensional (3D) information of a sample in the form of an intensity-only recording. However, to decode the original sample image from its hologram(s), autofocusing and phase recovery are needed, which are in general cumbersome and time-consuming to perform digitally. Here we demonstrate a convolutional neural network (CNN)-based approach that simultaneously performs autofocusing and phase recovery to significantly extend the depth of field (DOF) and the reconstruction speed in holographic imaging. For this, a CNN is trained by using pairs of randomly defocused back-propagated holograms and their corresponding in-focus phase-recovered images. After this training phase, the CNN takes a single back-propagated hologram of a 3D sample as input to rapidly achieve phase recovery and reconstruct an in-focus image of the sample over a significantly extended DOF. This deep-learning-based DOF extension method is non-iterative and significantly improves the algorithm time complexity of holographic image reconstruction from O(nm) to O(1), where n refers to the number of individual object points or particles within the sample volume, and m represents the focusing search space within which each object point or particle needs to be individually focused. These results highlight some of the unique opportunities created by data-enabled statistical image reconstruction methods powered by machine learning, and we believe that the presented approach can be broadly applicable to computationally extend the DOF of other imaging modalities.

Abstract: It is well known that in-line digital holography (DH) makes use of the full pixel count in forming the holographic imaging. But it usually requires phase-shifting or phase retrieval techniques to remove the zero-order and twin-image terms, resulting in the so-called two-step reconstruction process, i.e., phase recovery and focusing. Here, we propose a one-step end-to-end learning-based method for in-line holography reconstruction, namely, the eHoloNet, which can reconstruct the object wavefront directly from a single-shot in-line digital hologram. In addition, the proposed learning-based DH technique has strong robustness to the change of optical path difference between reference beam and object light and does not require the reference beam to be a plane or spherical wave.

Abstract: Holographic reconstruction is troubled by the phase-conjugate wave front arising from Hermitian symmetry of the complex field. The so-called twin image obfuscates the reconstruction in solving the inverse problem. Here we quantitatively reveal how and how much the twin image affects the reconstruction and propose a compressive sensing (CS) approach to reconstruct a hologram completely free from the twin image. Using the canonical basis, the incoherence condition of CS is naturally satisfied by the Fourier transformation associated with wave propagation. With the propagation kernel function related to the distance, the object wave diffracts into a sharp pattern while the phase-conjugate wave diffracts into a diffuse pattern. An iterative algorithm using a total variation sparsity constraint could filter out the diffuse conjugated signal and overcome the inherent physical symmetry of holographic reconstruction. The feasibility is verified by simulation and experimental results, as well as a comparative study to an existing phase retrieval method.

Abstract: The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise.

Abstract: Dual-comb interferometry utilizes two optical frequency combs to map the optical field's spectrum to a radio-frequency signal without using moving parts, allowing improved speed and accuracy. However, the method is compounded by the complexity and demanding stability associated with operating multiple laser frequency combs. To overcome these challenges, we demonstrate simultaneous generation of multiple frequency combs from a single optical microresonator and a single continuous-wave laser. Similar to space-division multiplexing, we generate several dissipative Kerr soliton states - circulating solitonic pulses driven by a continuous-wave laser - in different spatial (or polarization) modes of a $\mathrm{MgF_2}$ microresonator. Up to three distinct combs are produced simultaneously, featuring excellent mutual coherence and substantial repetition rate differences, useful for fast acquisition and efficient rejection of soliton intermodulation products. Dual-comb spectroscopy with amplitude and phase retrieval, as well as optical sampling of a breathing soliton, is realised with the free-running system. Compatibility with photonic-integrated resonators could enable the deployment of dual- and triple-comb-based methods to applications where they remained impractical with current technology.

Abstract: The phase retrieval problem asks to recover a natural signal $y_0 \in \mathbb{R}^n$ from $m$ quadratic observations, where $m$ is to be minimized. As is common in many imaging problems, natural signals are considered sparse with respect to a known basis, and the generic sparsity prior is enforced via $\ell_1$ regularization. While successful in the realm of linear inverse problems, such $\ell_1$ methods have encountered possibly fundamental limitations, as no computationally efficient algorithm for phase retrieval of a $k$-sparse signal has been proven to succeed with fewer than $O(k^2\log n)$ generic measurements, exceeding the theoretical optimum of $O(k \log n)$. In this paper, we propose a novel framework for phase retrieval by 1) modeling natural signals as being in the range of a deep generative neural network $G : \mathbb{R}^k \rightarrow \mathbb{R}^n$ and 2) enforcing this prior directly by optimizing an empirical risk objective over the domain of the generator. Our formulation has provably favorable global geometry for gradient methods, as soon as $m = O(kd^2\log n)$, where $d$ is the depth of the network. Specifically, when suitable deterministic conditions on the generator and measurement matrix are met, we construct a descent direction for any point outside of a small neighborhood around the unique global minimizer and its negative multiple, and show that such conditions hold with high probability under Gaussian ensembles of multilayer fully-connected generator networks and measurement matrices. This formulation for structured phase retrieval thus has two advantages over sparsity based methods: 1) deep generative priors can more tightly represent natural signals and 2) information theoretically optimal sample complexity. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms sparse phase retrieval methods.

Abstract: Super-resolution fluorescence microscopy provides unprecedented insight into cellular and subcellular structures However, going ‘beyond the diffraction barrier’ comes at a price, since most far-field super-resolution imaging techniques trade temporal for spatial super-resolution We propose the combination of a novel label-free white light quantitative phase imaging with fluorescence to provide high-speed imaging and spatial super-resolution The non-iterative phase retrieval relies on the acquisition of single images at each z-location and thus enables straightforward 3D phase imaging using a classical microscope We realized multi-plane imaging using a customized prism for the simultaneous acquisition of eight planes This allowed us to not only image live cells in 3D at up to 200 Hz, but also to integrate fluorescence super-resolution optical fluctuation imaging within the same optical instrument The 4D microscope platform unifies the sensitivity and high temporal resolution of phase imaging with the specificity and high spatial resolution of fluorescence microscopy By combining the sensitivity and high temporal resolution of phase imaging with the specificity and high spatial resolution of fluorescence microscopy, a 4D microscope is demonstrated that visualizes in three dimensions the fast cellular processes in living cells at up to 200 Hz

Abstract: Traditional digital holographic imaging algorithms need multiple iterations to obtain focused reconstructed image, which is time-consuming. In terms of phase retrieval, there is also the problem of phase compensation in addition to focusing task. Here, a new method is proposed for fast digital focus, where we use U-type convolutional neural network (U-net) to recover the original phase of microscopic samples. Generated data sets are used to simulate different degrees of defocused image, and verify that the U-net can restore the original phase to a great extent and realize phase compensation at the same time. We apply this method in the construction of real-time off-axis digital holographic microscope and obtain great breakthroughs in imaging speed.

Abstract: This paper deals with finding an $n$ -dimensional solution $\boldsymbol {x}$ to a system of quadratic equations of the form $y_i=|\langle \boldsymbol {a}_i,\boldsymbol {x}\rangle |^2$ for $1\leq i \leq m$ , which is also known as the generalized phase retrieval problem. For this NP-hard problem, a novel approach is developed for minimizing the amplitude-based least-squares empirical loss, which starts with a weighted maximal correlation initialization obtainable through a few power or Lanczos iterations, followed by successive refinements based on a sequence of iteratively reweighted gradient iterations. The two stages (initialization and gradient flow) distinguish themselves from prior contributions by the inclusion of a fresh (re)weighting regularization procedure. For certain random measurement models, the novel scheme is shown to be able to recover the true solution $\boldsymbol {x}$ in time proportional to reading the data $\lbrace (\boldsymbol {a}_i;y_i)\rbrace _{1\leq i \leq m}$ . This holds with high probability and without extra assumption on the signal vector $\boldsymbol {x}$ to be recovered, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns in the signal vector, namely $m>cn$ . Empirically, the upshots of this contribution are: first, (almost) $\text{100}{\%}$ perfect signal recovery in the high-dimensional (say $n\geq 2000$ ) regime given only an information-theoretic limit number of noiseless equations, namely $m=2n-1$ , in the real Gaussian case; and second, (nearly) optimal statistical accuracy in the presence of additive noise of bounded support. Finally, substantial numerical tests using both synthetic data and real images corroborate markedly improved recovery performance and computational efficiency of the novel scheme relative to the state-of-the-art approaches.

Abstract: Phase retrieval, or the process of recovering phase information in reciprocal space to reconstruct images from measured intensity alone, is the underlying basis to a variety of imaging applications including coherent diffraction imaging (CDI). Typical phase retrieval algorithms are iterative in nature, and hence, are time-consuming and computationally expensive, precluding real-time imaging. Furthermore, iterative phase retrieval algorithms struggle to converge to the correct solution especially in the presence of strong phase structures. In this work, we demonstrate the training and testing of CDI NN, a pair of deep deconvolutional networks trained to predict structure and phase in real space of a 2D object from its corresponding far-field diffraction intensities alone. Once trained, CDI NN can invert a diffraction pattern to an image within a few milliseconds of compute time on a standard desktop machine, opening the door to real-time imaging.

Abstract: We consider a phase retrieval problem, where we want to reconstruct a $n$ -dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable initialization procedure, the classical algorithm of alternating projections (Gerchberg–Saxton) succeeds with high probability when $m\geq Cn$ , for some $C>0$ . We conjecture that this result is still true when no special initialization procedure is used, and present numerical experiments that support this conjecture.

Abstract: Phase retrieval plays an important role in vast industrial and scientific applications. We consider a noisy phase retrieval problem in which the magnitudes of the Fourier transform (or a general linear transform) of an underling object are corrupted by Poisson noise, since any optical sensors detect photons, and the number of detected photons follows the Poisson distribution. We propose a variational model for phase retrieval based on a total variation regularization as an image prior and maximum a posteriori estimation of a Poisson noise model, which is referred to as “TV-PoiPR”. We also propose an efficient numerical algorithm based on an alternating direction method of multipliers and establish its convergence. Extensive experiments for coded diffraction, holographic, and ptychographic patterns are conducted using both real- and complex-valued images to demonstrate the effectiveness of our proposed methods.

Abstract: We introduce a novel deep-learning inspired formulation of the phase retrieval problem, which asks to recover a signal y0 ∈ ℝn from m quadratic observations, under structural assumptions on the underlying signal. As is common in many imaging problems, previous methodologies have considered natural signals as being sparse with respect to a known basis, resulting in the decision to enforce a generic sparsity prior. However, these methods for phase retrieval have encountered possibly fundamental limitations, as no computationally efficient algorithm for sparse phase retrieval has been proven to succeed with fewer than O(k2 log n) generic measurements, which is larger than the theoretical optimum of O(k log n). In this paper, we sidestep this issue by considering a prior that a natural signal is in the range of a generative neural network G : ℝk → ℝn. We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as m = O(k), under the model of a multilayer fully-connected neural network with random weights. Specifically, we show that there exists a descent direction outside of a small neighborhood around the true k-dimensional latent code and a negative multiple thereof. This formulation for structured phase retrieval thus benefits from two effects: generative priors can more tightly represent natural signals than sparsity priors, and this empirical risk formulation can exploit those generative priors at an information theoretically optimal sample complexity, unlike for a sparsity prior. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms both sparse and general phase retrieval methods.

Abstract: This paper proposes a novel method to substantially reduce motion-introduced phase error in phase-shifting profilometry. We first estimate the motion of an object from the difference between two subsequent 3D frames. After that, by leveraging the projector’s pinhole model, we can determine the motion-induced phase shift error from the estimated motion. A generic phase-shifting algorithm considering phase shift error is then utilized to compute the phase. Experiments demonstrated that proposed algorithm effectively improved the measurement quality by compensating for the phase shift error introduced by rigid and nonrigid motion for a standard single-projector and single-camera digital fringe projection system.

Abstract: Object motion can introduce phase error and thus measurement error for phase-shifting profilometry. This paper proposes a generic motion error compensation method based on our finding that the dominant motion-introduced phase error doubles the frequency of the projected fringe frequency, and the Hilbert transform shifts the phase of a fringe pattern by π/2. We apply a Hilbert transform to phase-shifted fringe patterns to generate another set of fringe patterns, calculate one phase map using the original fringe patterns and another phase map using Hilbert transformed fringe patterns, and then use the average of these two phase maps for three-dimensional reconstruction. Both simulation and experiments demonstrated that the proposed method can substantially reduce motion-introduced measurement error.

Abstract: We demonstrate a three-dimensional (3D) optical diffraction tomographic technique with multi-frequency combination (MFC-ODT) for the 3D quantitative phase imaging of unlabeled specimens. Three sets of through-focus intensity images are captured under an annular aperture and two circular apertures with different coherence parameters. The 3D phase optical transfer functions (POTF) corresponding to different illumination apertures are combined to obtain a synthesized frequency response, achieving high-quality, low-noise 3D reconstructions with imaging resolution up to the incoherent diffraction limit. Besides, the expression of 3D POTF for arbitrary illumination pupils is derived and analyzed, and the 3D imaging performance of annular illumination is explored. It is shown that the phase-contrast washout effect in high-NA circular apertures can be effectively addressed by introducing a complementary annular aperture, which strongly boosts the phase contrast and improves the imaging resolution. By incorporating high-NA illumination as well as high-NA detection, MFC-ODT can achieve a theoretical transverse resolution up to 200 nm and an axial resolution of 645 nm. To test the feasibility of the proposed MFC-ODT technique, the 3D refractive index reconstruction results are based on a simulated 3D resolution target and experimental investigations of micro polystyrene bead and unstained biological samples are presented. Due to its capability for high-resolution 3D phase imaging as well as the compatibility with a widely available commercial microscope, the MFC-ODT is expected to find versatile applications in biological and biomedical research.

Abstract: Phase retrieval algorithms have become an important component in many modern computational imaging systems. For instance, in the context of ptychography and speckle correlation imaging, they enable imaging past the diffraction limit and through scattering media, respectively. Unfortunately, traditional phase retrieval algorithms struggle in the presence of noise. Progress has been made recently on more robust algorithms using signal priors, but at the expense of limiting the range of supported measurement models (e.g., to Gaussian or coded diffraction patterns). In this work we leverage the regularization-by-denoising framework and a convolutional neural network denoiser to create prDeep, a new phase retrieval algorithm that is both robust and broadly applicable. We test and validate prDeep in simulation to demonstrate that it is robust to noise and can handle a variety of system models. A MatConvNet implementation of prDeep is available at this https URL.

Abstract: In phase retrieval, we want to recover an unknown signal $${{\varvec{x}}}\in {{\mathbb {C}}}^d$$ from n quadratic measurements of the form $$y_i = |\langle {{\varvec{a}}}_i,{{\varvec{x}}}\rangle |^2+w_i$$ , where $${{\varvec{a}}}_i\in {{\mathbb {C}}}^d$$ are known sensing vectors and $$w_i$$ is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator $${\hat{{{\varvec{x}}}}}({{\varvec{y}}})$$ that is positively correlated with the signal $${{\varvec{x}}}$$ ? We consider the case of Gaussian vectors $${{\varvec{a}}}_i$$ . We prove that—in the high-dimensional limit—a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For $$n\le d-o(d)$$ , no estimator can do significantly better than random and achieve a strictly positive correlation. For $$n\ge d+o(d)$$ , a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theoretic arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper bound and lower bound generalize beyond phase retrieval to measurements $$y_i$$ produced according to a generalized linear model. As a by-product of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.

Abstract: Phase retrieval is an important tool to unveil wavefront of light, especially in high performance microscopy such as Fourier ptychographic microscopy (FPM). In general phase-retrieval methods, the resolution and the number of measurements are in a trade-off relationship. Inspired by FPM, we devise what we believe is a novel microscopic phase-retrieval method, termed single-shot FPM (SSFPM). In our approach, the imaging performance exceeds the trade-off relationship in that it enables phase retrieval for high resolution with a single measurement. By placing the lens array at the Fourier plane of the objective lens, multiple intensity profiles required for the FPM algorithm are collected in a single shot. To achieve enough redundancy of data for satisfying convergence condition of FPM, the specimen is simultaneously illuminated by multiple light-emitting diodes. SSFPM reconstructs quantitative phase profile and enhances the resolution sacrificed by applying lens-array imaging. We demonstrate the performance of SSFPM with numerical simulation and experiments. The prototype achieves lateral resolution of 3.10 μm over a field of view of 0.34 mm2. Without an interferometer or scanning devices, SSFPM can reconstruct high resolution of a complex profile with a single shot.

Abstract: A feature-based phase retrieval wavefront sensing approach using machine learning is proposed in contrast to the conventional intensity-based approaches. Specifically, the Tchebichef moments which are orthogonal in the discrete domain of the image coordinate space are introduced to represent the features of the point spread functions (PSFs) at the in-focus and defocus image planes. The back-propagation artificial neural network, which is one of most wide applied machine learning tool, is utilized to establish the nonlinear mapping between the Tchebichef moment features and the corresponding aberration coefficients of the optical system. The Tchebichef moments can effectively characterize the intensity distribution of the PSFs. Once well trained, the neural network can directly output the aberration coefficients of the optical system to a good precision with these image features serving as the input. Adequate experiments are implemented to demonstrate the effectiveness and accuracy of proposed approach. This work presents a feasible and easy-implemented way to improve the efficiency and robustness of the phase retrieval wavefront sensing.

Abstract: A novel phase modulation method for holographic data storage with phase-retrieval reference beam locking is proposed and incorporated into an amplitude-encoding collinear holographic storage system Unlike the conventional phase retrieval method, the proposed method locks the data page and the corresponding phase-retrieval interference beam together at the same location with a sequential recording process, which eliminates piezoelectric elements, phase shift arrays and extra interference beams, making the system more compact and phase retrieval easier To evaluate our proposed phase modulation method, we recorded and then recovered data pages with multilevel phase modulation using two spatial light modulators experimentally For 4-level, 8-level, and 16-level phase modulation, we achieved the bit error rate (BER) of 03%, 15% and 66% respectively To further improve data storage density, an orthogonal reference encoding multiplexing method at the same position of medium is also proposed and validated experimentally We increased the code rate of pure 3/16 amplitude encoding method from 05 up to 10 and 15 using 4-level and 8-level phase modulation respectively

Abstract: An optical setup and corresponding reconstruction algorithm are proposed to realize single-shot Fourier ptychography (FP). Multiple angle-varied object waves are generated by placing a Dammann grating at a certain distance behind the object, and the generated image array of low resolution corresponding to different diffraction orders formed on the detector plane is recorded simultaneously in a single exposure. The amplitude, as well as the phase information of the object, can be properly reconstructed with a common FP algorithm from the recorded image array. This method eliminates the requirement for the angular scanning of common FP, and the total acquisition time is dramatically reduced. The feasibility of this proposed method was demonstrated both numerically and experimentally. The proposed method has the advantages of fast data acquisition and corresponding high temporal resolution, making it very suitable for applications in which high imaging speed is required.

Abstract: Phase retrieval, or the process of recovering phase information in reciprocal space to reconstruct images from measured intensity alone, is the underlying basis to a variety of imaging applications including coherent diffraction imaging (CDI). Typical phase retrieval algorithms are iterative in nature, and hence, are time-consuming and computationally expensive, making real-time imaging a challenge. Furthermore, iterative phase retrieval algorithms struggle to converge to the correct solution especially in the presence of strong phase structures. In this work, we demonstrate the training and testing of CDI NN, a pair of deep deconvolutional networks trained to predict structure and phase in real space of a 2D object from its corresponding far-field diffraction intensities alone. Once trained, CDI NN can invert a diffraction pattern to an image within a few milliseconds of compute time on a standard desktop machine, opening the door to real-time imaging.

Abstract: This work demonstrates experimental approaches to characterize a single multimode fiber imaging system without a reference beam. Spatial light modulation is performed with a digital micro-mirror device that enables high-speed binary amplitude modulation. Intensity-only images are recorded by the camera and processed by a Bayesian inference based algorithm to retrieve the phase of the output optical field as well as the transmission matrix of the fiber. The calculated transmission matrix is validated by three standards: prediction accuracy, transmission imaging, and focus generation. Also, it is found that information on mode count and eigenchannels can be extracted from the transmission matrix by singular value decomposition. This paves the way for a more compact and cheaper single multimode fiber imaging system for many demanding imaging tasks.