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Hagai Kirshner

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Korea Advanced Institute of Science and Technology

Jorge Martin-Gutierrez

Universidad de La Laguna

University of Amsterdam

Sakarya University

Seyed Mostafa Mousavi Kahaki

FDA

Holy Angel University

Ain Shams University

Gujarat Technological University

Lyudmila S Mihaylova

The University of Sheffield

Mohamad Ivan Fanany

University of Indonesia

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Papers by Hagai Kirshner

Adaptive Reconstruction Along Mobile Sensing Paths

2018 IEEE Statistical Signal Processing Workshop (SSP), 2018

We address the problem of reconstructing missing parts of mobile sensing signals. Such a case occ... more We address the problem of reconstructing missing parts of mobile sensing signals. Such a case occurs when sensor information is occluded or not transmitted over finite periods of time. As the sampling rate along each path is essentially unlimited, we consider the asymptotic case of having continuous-time information from the sensor. We embed the partially available signal in a functional space of smooth and finite-energy functions, while adapting the parameters of the space to the signal at hand. We then analytically solve a specifically designed error measure and obtain a minimum-norm reconstruction for the missing parts. We demonstrate the proposed algorithm for both simulated- and real data.

On The Representation Error Of Digitized Signals

Publication in the conference proceedings of EUSIPCO, Antalya, Turkey, 2005

On interpolation of differentially structured images

2007 15th European Signal Processing Conference, 2007

A vector space approach to image reconstruction is derived and introduced. The continuous-domain ... more A vector space approach to image reconstruction is derived and introduced. The continuous-domain image is assumed to belong to a reproducing kernel Hilbert space and the sampling process is shown to correspond to an appropriate orthogonal projection. The values at the interpolating grid are shown to correspond to a set of inner product calculations, giving rise to a minimax solution for an ℓ2 approximation problem. A tight upper bound on the ensued error is then derived and demonstrated. Examples of image resizing show that the proposed method yields better results than presently available methods, including the cubic B-spline method, in terms of SNR.

On signal representation from generalized samples: Minmax approximation with constraints

2006 14th European Signal Processing Conference, 2006

Signal representation plays a major role in many DSP applications. In this paper we consider the ... more Signal representation plays a major role in many DSP applications. In this paper we consider the task of calculating representation coefficients of an analog signal, where the only available data is its samples based on practical non-ideal acquisition devices. We adopt a minmax approach and further incorporate regularity constrains on the original continuous-time signal. These constraints stem from the nature of applications where smooth signals serve as input data. The ensued solution is shown to consist of an orthogonal projection within a Sobolev space. Illustrating examples are given, utilizing this constrained minmax approach. Our conclusion is that this new approach to signal representation could improve presently available systems, especially in non-ideal situations.

On spectral estimation and Fourier transform approximation from sampled data

2009 17th European Signal Processing Conference, 2009

A method for spectral estimation of a continuous-domain signal, given by its sampled version only... more A method for spectral estimation of a continuous-domain signal, given by its sampled version only, is introduced. Unlike the discrete Fourier transform (DFT), the proposed approach reduces aliasing effects. The proposed approach relies on finite-duration Sobolev functions, for which the ideal sampling process is characterized by means of an inner product operation. The point-wise evaluation of the Fourier transform is based on a Sobolev type inner product too, allowing for a minimax approximation approach to be derived and utilized. Experimental results show that the proposed approach is a preferred alternative over the DFT in cases where spectral analysis of sampled signals is required.

Models for Fluorescence Microscopy in ImageJ

The current version allows for five different PSF models: The Gaussian model simulates a blurring... more The current version allows for five different PSF models: The Gaussian model simulates a blurring effect by setting three different variance values. These values characterize the width of the PSF at various axial positions along the z-stack. The defocus model is simulated in the Fourier domain by a modulated Gaussian where the sinc modulation depends on the axial defocusing distance. Another defocusing model is due to Koehler, which also uses the Gaussian function in the Fourier domain. In this model, however, the sinc modulation is replaced by a linear term for the variance.

$Figure 2: Gibson & Lanni 3D PSF model. Shown here is a non-symmetric z-stack due to refractive indices mismatch.$

Bi-plane Calibration in Super-resolution Microscopy

In the localisation stage, the two stacks are processed separately. Local maxima are assumed to o... more In the localisation stage, the two stacks are processed separately. Local maxima are assumed to originate from fluorophore beads, and a threshold value is then used to keep the prominent ones only [2]. Each one of these local maxima is then fitted with a PSF model, yielding the lateral and the axial coordinates of each fluorophore bead. The fitting error is further compared with a threshold value for obtaining reliable localisation results. The algorithm fits the local maxima with the Gibson and Lanni PSF [3]. This model considers aberrations that are due to refractive indices mismatches at the sample cover slip immersion interfaces, as well as for the thickness of these layers. In particular, axial stage displacements correspond to different values of the immersion layer thickness. The output of the localisation stage is two lists of 3D beads locations.

ON THE REPRESENTATION ERROR OF DIGITIZED SIGNALS ( MonPmOR 7 ) Author ( s )

Although the origin of many sources of information is analogue or continuous−time, due to practic... more Although the origin of many sources of information is analogue or continuous−time, due to practical considerations signal representation is usually based on discrete−time basis functions. These basis functions are in many cases sampled versions of harmonic signals (Fourier), Gabor, wavelets and the like. In this paper, we examine the accuracy of this approach, where the calculation of the representation coefficients is performed using digital inner product rather than analogue. By interpreting the sampling process as a bounded linear operator, we analyze the difference between the analogue domain inner product and the result one may get in the digital domain. We consider both the one−dimensional and two−dimensional cases, and several applicable examples are given.

Approximating Representation Coefficients From Non Ideal Samples

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

Many sources of information are of analogue or continuous-time nature. However, digital signal pr... more Many sources of information are of analogue or continuous-time nature. However, digital signal processing applications rely on discrete data. We consider the problem of approximating L2 inner products, i.e., representation coefficients of a continuous-time signal, while having the possibly non-ideal signal samples, as the only available data. By adopting a generalized sampling scheme, a minimax solution is suggested. We then compare our approach with the piecewiseconstant approximation method, commonly used for this problem.

From Analog Information to Digital Databases — Does it Keep Everything Intact?

Information Systems Development

ABSTRACT Information systems encapsulate digital data although the origin of many sources of info... more ABSTRACT Information systems encapsulate digital data although the origin of many sources of information is analog or continuous. Digital signal representation has been widely used since Shannon formulated his sampling theorem. Still, several questions regarding digital information processing remain unsolved. One of the most relevant up-to-date goals is data storage and mining. The purpose of this work is to analyze the relation between continuous signals, or data sources, and their digitized representation. We concentrate on the operation widely used to compare and find similarities in vector representation, the inner-product. Applying the sampling scheme to continuous information sources that do not satisfy Shannon conditions (non band-limited), but with higher enough sampling rate, is assumed to yield only small approximation errors. In this work it is shown, however, that this assumption should be made with much prudence. In some cases, the result is likely to differ from that of the original continuous signals. We provide an analytic estimation to this error of digitization, and several applications are considered, including medical records. Our results provide a quantitative tool for calculating sampling errors, thus affording a useful means in evaluating the ongoing process worldwide of digitizing analog information sources.

A stochastic model-based approach to image and texture interpolation

2009 16th IEEE International Conference on Image Processing (ICIP), 2009

We introduce an exponential-based consistent approach to image scaling. Our model stems from Sobo... more We introduce an exponential-based consistent approach to image scaling. Our model stems from Sobolev reproducing kernels, motivated by their role in continuousdomain stochastic autoregressive processes. The proposed approach imposes consistency and applies the minimum-norm criterion for determining the scaled image. We show by experimental results that the proposed approach provides images that are visually better than other consistent solutions. We also observe that the proposed exponential kernels yield better interpolation results than polynomial B-spline models. Our conclusion is that the proposed Sobolev-based image modeling could be instrumental and a preferred alternative in major image processing tasks.

On Optimal Derivative DSP Operators for Sampled Data

2007 15th International Conference on Digital Signal Processing, 2007

ABSTRACT Motivated by partial differential equation (PDE) models in signal processing, an l2 appr... more ABSTRACT Motivated by partial differential equation (PDE) models in signal processing, an l2 approach to derivative calculation is introduced based on sampled data. This approach utilizes regularity constraints on the continuous-domain signal that are already embedded in the PDE model. In particular, the continuous-domain input signal is assumed to belong to a reproducing kernel Hilbert space and the sampling process (ideal or non-ideal) is shown to correspond to an appropriate orthogonal projection. The values of the derivative function are shown to correspond to a set of inner product calculations, giving rise to a minimax solution for an h approximation problem. Several matrix operators are then demonstrated for 1D and 2D cases, found to be superior to the backward-forward difference approach.

Can localization microscopy benefit from approximation theory?

2013 IEEE 10th International Symposium on Biomedical Imaging, 2013

We introduce a general and computationally efficient approach to 3-D localization microscopy. The... more We introduce a general and computationally efficient approach to 3-D localization microscopy. The main idea is to construct a continuous-domain representation of the PSF by expanding it in a polynomial B-spline basis. This allows us to fit the PSF to the data with sub-pixel accuracy. Since the basis functions are compactly supported, the evaluation of the PSF is computationally efficient. Furthermore, our approach can accommodate for any 3-D PSF design, and it does not require a calibration curve for the axial position. We further introduce a computationally efficient implementation of the least-squares criterion and demonstrate its potential use for fast and accurate reconstruction of super-resolution data.

On the role of exponential functions in image interpolation

ABSTRACT A reproducing-kernel Hilbert space approach to image interpolation is introduced. In par... more ABSTRACT A reproducing-kernel Hilbert space approach to image interpolation is introduced. In particular, the reproducing kernels of Sobolev spaces are shown to be exponential functions. These functions, in turn, give rise to alternative interpolation kernels that outperform presently available designs. Both theoretical and experimental results are presented.

Continuous localization using sparsity constraints for high-density super-resolution microscopy

2013 IEEE 10th International Symposium on Biomedical Imaging, 2013

Super-resolution localization microscopy relies on sparse activation of photo-switchable probes. ... more Super-resolution localization microscopy relies on sparse activation of photo-switchable probes. Such activation, however, introduces limited temporal resolution. High-density imaging overcomes this limitation by allowing several neighboring probes to be activated simultaneously. In this work, we propose an algorithm that incorporates a continuous-domain sparsity prior into the high-density localization problem. We use a Taylor approximation of the PSF, and rely on a fast proximal gradient optimization procedure. Unlike currently available methods that use discrete-domain sparsity priors, our approach does not restrict the estimated locations to a pre-defined sampling grid. Experimental results of simulated and real data demonstrate significant improvement over these methods in terms of accuracy, molecular identification and computational complexity.

FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data

Scientific Reports, 2014

Super resolution microscopy such as STORM and (F)PALM is now a well known method for biological s... more Super resolution microscopy such as STORM and (F)PALM is now a well known method for biological studies at the nanometer scale. However, conventional imaging schemes based on sparse activation of photo-switchable fluorescent probes have inherently slow temporal resolution which is a serious limitation when investigating live-cell dynamics. Here, we present an algorithm for high-density super-resolution microscopy which combines a sparsity-promoting formulation with a Taylor series approximation of the PSF. Our algorithm is designed to provide unbiased localization on continuous space and high recall rates for high-density imaging, and to have orders-of-magnitude shorter run times compared to previous high-density algorithms. We validated our algorithm on both simulated and experimental data, and demonstrated live-cell imaging with temporal resolution of 2.5 seconds by recovering fast ER dynamics.

New Bounds on Representation Errors in Signal Processing Systems

Most signal processing systems are based on discrete-time signals although the origin of many sou... more Most signal processing systems are based on discrete-time signals although the origin of many sources of information is analog. In this work we consider the task of signal representation by a set of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator, an upper bound on the representation error is derived and demonstrated. Based on our theorems, one can then determine the maximum representation error induced by the sampling process. We further propose a new approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum representation error. Our results are applicable to signal processing systems where analog signals are represented by their s...

On Sampling Invariant Signal Representation Or: On the effect of sampling on algebraic relations

Most signal processing systems are based on discrete-time signals although the origin of many sou... more Most signal processing systems are based on discrete-time signals although the origin of many sources of information is analog. In this work we consider the task of signal representation by a set of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator, an upper bound on the representation error is derived and demonstrated. Based on our theorems, one can then determine the maximum representation error induced by the sampling process. We further propose a new approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum representation error. Our results are applicable to signal processing systems where analog signals are represented by their s...

$e20: Vector relations for optimal approximation of (f,g)z2. g is such that (f@),2= (fp )w2 and @2 is any functions within Span {u(t-nT)}. Setting pg, to be go, namely the orthogonal projection of g onto Spanfu(t-nT)\ would result in the minimum possible upper bound on the approximation error.$

On Sampling Invariant Signal Representation

Most signal processing systems are based on discrete-time signals although the origin of manysour... more Most signal processing systems are based on discrete-time signals although the origin of manysources of information is analog. In this work ,we consider ,the task of signal ,representation by a set ,of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator, an upper bound on the representation error is derived and demonstrated. Based on our theorems, one can then determine the maximumrepr esentation error induced bythe sampling process. We further propose a new approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum representation error. Our results are applicable to signal processing systems where analog signals are represented bytheir ...

3‐D PSF fitting for fluorescence microscopy: implementation and localization application

Journal of Microscopy, 2012

SummaryLocalization microscopy relies on computationally efficient Gaussian approximations of the... more SummaryLocalization microscopy relies on computationally efficient Gaussian approximations of the point spread function for the calculation of fluorophore positions. Theoretical predictions show that under specific experimental conditions, localization accuracy is significantly improved when the localization is performed using a more realistic model. Here, we show how this can be achieved by considering three‐dimensional (3‐D) point spread function models for the wide field microscope. We introduce a least‐squares point spread function fitting framework that utilizes the Gibson and Lanni model and propose a computationally efficient way for evaluating its derivative functions. We demonstrate the usefulness of the proposed approach with algorithms for particle localization and defocus estimation, both implemented as plugins for ImageJ.

Adaptive Reconstruction Along Mobile Sensing Paths

2018 IEEE Statistical Signal Processing Workshop (SSP), 2018

We address the problem of reconstructing missing parts of mobile sensing signals. Such a case occ... more We address the problem of reconstructing missing parts of mobile sensing signals. Such a case occurs when sensor information is occluded or not transmitted over finite periods of time. As the sampling rate along each path is essentially unlimited, we consider the asymptotic case of having continuous-time information from the sensor. We embed the partially available signal in a functional space of smooth and finite-energy functions, while adapting the parameters of the space to the signal at hand. We then analytically solve a specifically designed error measure and obtain a minimum-norm reconstruction for the missing parts. We demonstrate the proposed algorithm for both simulated- and real data.

On The Representation Error Of Digitized Signals

Publication in the conference proceedings of EUSIPCO, Antalya, Turkey, 2005

On interpolation of differentially structured images

2007 15th European Signal Processing Conference, 2007

A vector space approach to image reconstruction is derived and introduced. The continuous-domain ... more A vector space approach to image reconstruction is derived and introduced. The continuous-domain image is assumed to belong to a reproducing kernel Hilbert space and the sampling process is shown to correspond to an appropriate orthogonal projection. The values at the interpolating grid are shown to correspond to a set of inner product calculations, giving rise to a minimax solution for an ℓ2 approximation problem. A tight upper bound on the ensued error is then derived and demonstrated. Examples of image resizing show that the proposed method yields better results than presently available methods, including the cubic B-spline method, in terms of SNR.

On signal representation from generalized samples: Minmax approximation with constraints

2006 14th European Signal Processing Conference, 2006

Signal representation plays a major role in many DSP applications. In this paper we consider the ... more Signal representation plays a major role in many DSP applications. In this paper we consider the task of calculating representation coefficients of an analog signal, where the only available data is its samples based on practical non-ideal acquisition devices. We adopt a minmax approach and further incorporate regularity constrains on the original continuous-time signal. These constraints stem from the nature of applications where smooth signals serve as input data. The ensued solution is shown to consist of an orthogonal projection within a Sobolev space. Illustrating examples are given, utilizing this constrained minmax approach. Our conclusion is that this new approach to signal representation could improve presently available systems, especially in non-ideal situations.

On spectral estimation and Fourier transform approximation from sampled data

2009 17th European Signal Processing Conference, 2009

A method for spectral estimation of a continuous-domain signal, given by its sampled version only... more A method for spectral estimation of a continuous-domain signal, given by its sampled version only, is introduced. Unlike the discrete Fourier transform (DFT), the proposed approach reduces aliasing effects. The proposed approach relies on finite-duration Sobolev functions, for which the ideal sampling process is characterized by means of an inner product operation. The point-wise evaluation of the Fourier transform is based on a Sobolev type inner product too, allowing for a minimax approximation approach to be derived and utilized. Experimental results show that the proposed approach is a preferred alternative over the DFT in cases where spectral analysis of sampled signals is required.

Models for Fluorescence Microscopy in ImageJ

The current version allows for five different PSF models: The Gaussian model simulates a blurring... more The current version allows for five different PSF models: The Gaussian model simulates a blurring effect by setting three different variance values. These values characterize the width of the PSF at various axial positions along the z-stack. The defocus model is simulated in the Fourier domain by a modulated Gaussian where the sinc modulation depends on the axial defocusing distance. Another defocusing model is due to Koehler, which also uses the Gaussian function in the Fourier domain. In this model, however, the sinc modulation is replaced by a linear term for the variance.

$Figure 2: Gibson & Lanni 3D PSF model. Shown here is a non-symmetric z-stack due to refractive indices mismatch.$

Bi-plane Calibration in Super-resolution Microscopy

In the localisation stage, the two stacks are processed separately. Local maxima are assumed to o... more In the localisation stage, the two stacks are processed separately. Local maxima are assumed to originate from fluorophore beads, and a threshold value is then used to keep the prominent ones only [2]. Each one of these local maxima is then fitted with a PSF model, yielding the lateral and the axial coordinates of each fluorophore bead. The fitting error is further compared with a threshold value for obtaining reliable localisation results. The algorithm fits the local maxima with the Gibson and Lanni PSF [3]. This model considers aberrations that are due to refractive indices mismatches at the sample cover slip immersion interfaces, as well as for the thickness of these layers. In particular, axial stage displacements correspond to different values of the immersion layer thickness. The output of the localisation stage is two lists of 3D beads locations.

ON THE REPRESENTATION ERROR OF DIGITIZED SIGNALS ( MonPmOR 7 ) Author ( s )

Although the origin of many sources of information is analogue or continuous−time, due to practic... more Although the origin of many sources of information is analogue or continuous−time, due to practical considerations signal representation is usually based on discrete−time basis functions. These basis functions are in many cases sampled versions of harmonic signals (Fourier), Gabor, wavelets and the like. In this paper, we examine the accuracy of this approach, where the calculation of the representation coefficients is performed using digital inner product rather than analogue. By interpreting the sampling process as a bounded linear operator, we analyze the difference between the analogue domain inner product and the result one may get in the digital domain. We consider both the one−dimensional and two−dimensional cases, and several applicable examples are given.

Approximating Representation Coefficients From Non Ideal Samples

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

Many sources of information are of analogue or continuous-time nature. However, digital signal pr... more Many sources of information are of analogue or continuous-time nature. However, digital signal processing applications rely on discrete data. We consider the problem of approximating L2 inner products, i.e., representation coefficients of a continuous-time signal, while having the possibly non-ideal signal samples, as the only available data. By adopting a generalized sampling scheme, a minimax solution is suggested. We then compare our approach with the piecewiseconstant approximation method, commonly used for this problem.

From Analog Information to Digital Databases — Does it Keep Everything Intact?

Information Systems Development

ABSTRACT Information systems encapsulate digital data although the origin of many sources of info... more ABSTRACT Information systems encapsulate digital data although the origin of many sources of information is analog or continuous. Digital signal representation has been widely used since Shannon formulated his sampling theorem. Still, several questions regarding digital information processing remain unsolved. One of the most relevant up-to-date goals is data storage and mining. The purpose of this work is to analyze the relation between continuous signals, or data sources, and their digitized representation. We concentrate on the operation widely used to compare and find similarities in vector representation, the inner-product. Applying the sampling scheme to continuous information sources that do not satisfy Shannon conditions (non band-limited), but with higher enough sampling rate, is assumed to yield only small approximation errors. In this work it is shown, however, that this assumption should be made with much prudence. In some cases, the result is likely to differ from that of the original continuous signals. We provide an analytic estimation to this error of digitization, and several applications are considered, including medical records. Our results provide a quantitative tool for calculating sampling errors, thus affording a useful means in evaluating the ongoing process worldwide of digitizing analog information sources.

A stochastic model-based approach to image and texture interpolation

2009 16th IEEE International Conference on Image Processing (ICIP), 2009

We introduce an exponential-based consistent approach to image scaling. Our model stems from Sobo... more We introduce an exponential-based consistent approach to image scaling. Our model stems from Sobolev reproducing kernels, motivated by their role in continuousdomain stochastic autoregressive processes. The proposed approach imposes consistency and applies the minimum-norm criterion for determining the scaled image. We show by experimental results that the proposed approach provides images that are visually better than other consistent solutions. We also observe that the proposed exponential kernels yield better interpolation results than polynomial B-spline models. Our conclusion is that the proposed Sobolev-based image modeling could be instrumental and a preferred alternative in major image processing tasks.

On Optimal Derivative DSP Operators for Sampled Data

2007 15th International Conference on Digital Signal Processing, 2007

ABSTRACT Motivated by partial differential equation (PDE) models in signal processing, an l2 appr... more ABSTRACT Motivated by partial differential equation (PDE) models in signal processing, an l2 approach to derivative calculation is introduced based on sampled data. This approach utilizes regularity constraints on the continuous-domain signal that are already embedded in the PDE model. In particular, the continuous-domain input signal is assumed to belong to a reproducing kernel Hilbert space and the sampling process (ideal or non-ideal) is shown to correspond to an appropriate orthogonal projection. The values of the derivative function are shown to correspond to a set of inner product calculations, giving rise to a minimax solution for an h approximation problem. Several matrix operators are then demonstrated for 1D and 2D cases, found to be superior to the backward-forward difference approach.

Can localization microscopy benefit from approximation theory?

2013 IEEE 10th International Symposium on Biomedical Imaging, 2013

We introduce a general and computationally efficient approach to 3-D localization microscopy. The... more We introduce a general and computationally efficient approach to 3-D localization microscopy. The main idea is to construct a continuous-domain representation of the PSF by expanding it in a polynomial B-spline basis. This allows us to fit the PSF to the data with sub-pixel accuracy. Since the basis functions are compactly supported, the evaluation of the PSF is computationally efficient. Furthermore, our approach can accommodate for any 3-D PSF design, and it does not require a calibration curve for the axial position. We further introduce a computationally efficient implementation of the least-squares criterion and demonstrate its potential use for fast and accurate reconstruction of super-resolution data.

On the role of exponential functions in image interpolation

ABSTRACT A reproducing-kernel Hilbert space approach to image interpolation is introduced. In par... more ABSTRACT A reproducing-kernel Hilbert space approach to image interpolation is introduced. In particular, the reproducing kernels of Sobolev spaces are shown to be exponential functions. These functions, in turn, give rise to alternative interpolation kernels that outperform presently available designs. Both theoretical and experimental results are presented.

Continuous localization using sparsity constraints for high-density super-resolution microscopy

2013 IEEE 10th International Symposium on Biomedical Imaging, 2013

Super-resolution localization microscopy relies on sparse activation of photo-switchable probes. ... more Super-resolution localization microscopy relies on sparse activation of photo-switchable probes. Such activation, however, introduces limited temporal resolution. High-density imaging overcomes this limitation by allowing several neighboring probes to be activated simultaneously. In this work, we propose an algorithm that incorporates a continuous-domain sparsity prior into the high-density localization problem. We use a Taylor approximation of the PSF, and rely on a fast proximal gradient optimization procedure. Unlike currently available methods that use discrete-domain sparsity priors, our approach does not restrict the estimated locations to a pre-defined sampling grid. Experimental results of simulated and real data demonstrate significant improvement over these methods in terms of accuracy, molecular identification and computational complexity.

FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data

Scientific Reports, 2014

Super resolution microscopy such as STORM and (F)PALM is now a well known method for biological s... more Super resolution microscopy such as STORM and (F)PALM is now a well known method for biological studies at the nanometer scale. However, conventional imaging schemes based on sparse activation of photo-switchable fluorescent probes have inherently slow temporal resolution which is a serious limitation when investigating live-cell dynamics. Here, we present an algorithm for high-density super-resolution microscopy which combines a sparsity-promoting formulation with a Taylor series approximation of the PSF. Our algorithm is designed to provide unbiased localization on continuous space and high recall rates for high-density imaging, and to have orders-of-magnitude shorter run times compared to previous high-density algorithms. We validated our algorithm on both simulated and experimental data, and demonstrated live-cell imaging with temporal resolution of 2.5 seconds by recovering fast ER dynamics.

New Bounds on Representation Errors in Signal Processing Systems

Most signal processing systems are based on discrete-time signals although the origin of many sou... more Most signal processing systems are based on discrete-time signals although the origin of many sources of information is analog. In this work we consider the task of signal representation by a set of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator, an upper bound on the representation error is derived and demonstrated. Based on our theorems, one can then determine the maximum representation error induced by the sampling process. We further propose a new approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum representation error. Our results are applicable to signal processing systems where analog signals are represented by their s...

On Sampling Invariant Signal Representation Or: On the effect of sampling on algebraic relations

Most signal processing systems are based on discrete-time signals although the origin of many sou... more Most signal processing systems are based on discrete-time signals although the origin of many sources of information is analog. In this work we consider the task of signal representation by a set of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator, an upper bound on the representation error is derived and demonstrated. Based on our theorems, one can then determine the maximum representation error induced by the sampling process. We further propose a new approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum representation error. Our results are applicable to signal processing systems where analog signals are represented by their s...

$e20: Vector relations for optimal approximation of (f,g)z2. g is such that (f@),2= (fp )w2 and @2 is any functions within Span {u(t-nT)}. Setting pg, to be go, namely the orthogonal projection of g onto Spanfu(t-nT)\ would result in the minimum possible upper bound on the approximation error.$

On Sampling Invariant Signal Representation

Most signal processing systems are based on discrete-time signals although the origin of manysour... more Most signal processing systems are based on discrete-time signals although the origin of manysources of information is analog. In this work ,we consider ,the task of signal ,representation by a set ,of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator, an upper bound on the representation error is derived and demonstrated. Based on our theorems, one can then determine the maximumrepr esentation error induced bythe sampling process. We further propose a new approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum representation error. Our results are applicable to signal processing systems where analog signals are represented bytheir ...

3‐D PSF fitting for fluorescence microscopy: implementation and localization application

Journal of Microscopy, 2012

SummaryLocalization microscopy relies on computationally efficient Gaussian approximations of the... more SummaryLocalization microscopy relies on computationally efficient Gaussian approximations of the point spread function for the calculation of fluorophore positions. Theoretical predictions show that under specific experimental conditions, localization accuracy is significantly improved when the localization is performed using a more realistic model. Here, we show how this can be achieved by considering three‐dimensional (3‐D) point spread function models for the wide field microscope. We introduce a least‐squares point spread function fitting framework that utilizes the Gibson and Lanni model and propose a computationally efficient way for evaluating its derivative functions. We demonstrate the usefulness of the proposed approach with algorithms for particle localization and defocus estimation, both implemented as plugins for ImageJ.