WO2009070120A1 - Régularisation lp de représentations éparses appliquée à des procédés de détermination de structures en biologie moléculaire/chimie structurale - Google Patents

Régularisation lp de représentations éparses appliquée à des procédés de détermination de structures en biologie moléculaire/chimie structurale Download PDF

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WO2009070120A1
WO2009070120A1 PCT/SE2008/051382 SE2008051382W WO2009070120A1 WO 2009070120 A1 WO2009070120 A1 WO 2009070120A1 SE 2008051382 W SE2008051382 W SE 2008051382W WO 2009070120 A1 WO2009070120 A1 WO 2009070120A1
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data
regularization
defining
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given
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Ozan ÖKTEM
Hans RULLGÅRD
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Sidec Technologies Ab
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    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01J—ELECTRIC DISCHARGE TUBES OR DISCHARGE LAMPS
    • H01J37/00—Discharge tubes with provision for introducing objects or material to be exposed to the discharge, e.g. for the purpose of examination or processing thereof
    • H01J37/26—Electron or ion microscopes; Electron or ion diffraction tubes
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16B—BIOINFORMATICS, i.e. INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR GENETIC OR PROTEIN-RELATED DATA PROCESSING IN COMPUTATIONAL MOLECULAR BIOLOGY
    • G16B15/00—ICT specially adapted for analysing two-dimensional or three-dimensional molecular structures, e.g. structural or functional relations or structure alignment
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06V—IMAGE OR VIDEO RECOGNITION OR UNDERSTANDING
    • G06V20/00—Scenes; Scene-specific elements
    • G06V20/60—Type of objects
    • G06V20/69—Microscopic objects, e.g. biological cells or cellular parts
    • G06V20/693—Acquisition
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C20/00—Chemoinformatics, i.e. ICT specially adapted for the handling of physicochemical or structural data of chemical particles, elements, compounds or mixtures
    • G16C20/20—Identification of molecular entities, parts thereof or of chemical compositions
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06V—IMAGE OR VIDEO RECOGNITION OR UNDERSTANDING
    • G06V2201/00—Indexing scheme relating to image or video recognition or understanding
    • G06V2201/12—Acquisition of 3D measurements of objects
    • G06V2201/122—Computational image acquisition in electron microscopy
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01J—ELECTRIC DISCHARGE TUBES OR DISCHARGE LAMPS
    • H01J2237/00—Discharge tubes exposing object to beam, e.g. for analysis treatment, etching, imaging
    • H01J2237/22—Treatment of data
    • H01J2237/226—Image reconstruction

Definitions

  • the present invention relates to a method for solving the structure determination problem in molecular biology/structural chemistry.
  • This is often an ill-posed inverse problems that is quite challenging, both from an experimen- tal and algorithmic point of view.
  • the idea of the invention is to first identify a basis/frame where the molecular structure of interest (or a feature thereof) is a-priori known to have a sparse representation.
  • the inverse problem that is associated with the experimental data, and which must be solved in order to recover the molecular structure of interest is solved by ap- plying variational regularization with a regularizing functional based on the ⁇ p -norm applied to the coefficients in this basis/frame.
  • Such inverse problems occurs in particular when one uses the transmission electron microscope in a tomographic setting (electron tomography) in order to recover the molecular structure of a biological sample.
  • Problem 1 Recover the three dimensional structure of an individual molecule (e.g. a protein or a macromolecular assembly) at highest possible resolution in its natural environment, which could be in-situ (the cellular environment) or in-vitro (aqueous environment).
  • an individual molecule e.g. a protein or a macromolecular assembly
  • an inverse problem is well-posed (i.e. not ill-posed) if it has a unique solution for all (admissible) data and the solution depends contin- uously on the data.
  • the last criterion simply ensures that the problem is stable, i.e. small errors in the data are not amplified. If one has non-attainable data then there are no solutions, i.e. we have non-existence. Next, there can be solutions but they are not unique, i.e. we have non-uniqueness. Finally, even if there is a unique solution, the process of inversion is not continuous, so the solution does not depend continuously on the data and we have instability. If a problem is ill-posed, non-existence is usually not the main concern.
  • the performance of a regularization methods when applied to a particu- lar inverse problem depends strongly on how much a-priori knowledge about the specimen that is utilized.
  • prior knowledge can either be encoded in the definition of the regularization functional or as side conditions to the optimization problem.
  • the regularization functional enforces uniqueness by selecting a unique element among the least squares solutions and it also acts as a stabiliser by enforcing a smoothing. It is e.g. well known that 2-norms on the gradient yields smooth and continuous solutions and 1-norms on the gradient (i.e. TV-regularization) allow for non-smooth solutions.
  • Sparse £i-regularization for point enhancement is then defined as variational regularization with the ⁇ -norm applied to the coefficients in the ⁇ i ⁇ expansion.
  • This idea can also be applied to a feature of F, e.g. its gradient.
  • the a-priori assumption would be that the gradient of F admits a sparse representation in a certain orthonormal basis or a frame and this latter case is often referred to as sparse ⁇ -regularization for region enhancement.
  • the gradient of F is sparse already in the voxel representation, the resulting regularization method is known as TV-regularization.
  • phantom that accurately represents such a sample, and then try a large number of "data compression" representations of this phantom and/or its gradient. If the phantom is a good representative for the type of specimen that is to be studied, then a fair a priori assumption would be that a sparse representation of the phantom and/or its gradient would also be a sparse representation of the unknown specimen. The next step would then be to apply ⁇ i-regularization on that representation for point and/or region enhancement.
  • This invention proposes for the first time the usage of ⁇ p -regularization in order to solve the structure determination problem in the fully three dimensional setting.
  • each feature is defined as a mapping A x : 2£ ⁇ W x acting on / that yields a scalar (real- or complex) valued function and W x is some metric space defining all possible values of the features in question; - for each feature A(Z) 5 constructing a basis/frame ⁇ ) ⁇ 3 C W 1 in which it is known a-priori that the feature A(Z) admits a sparse representation;
  • the method may further comprise the steps of:
  • a data ensemble comprising the formulation of the inverse problem, defining the forward operator modeling the experiment wherein the inverse problem and forward operator has been derived using the scientific knowledge about how input data is related to output data in the experiment, and a model of the proba- bilistic distribution of the data;
  • a phantom generator may be provided, denning the function / for a typical biological specimen using the scientific knowledge about how input data is related to output data in the experiment;
  • a data compression library may be provided, defining a sparse representation of a phantom / and/or a feature Ai(J) thereof;
  • a computer readable storage media containing the data ensemble may be provided.
  • a signal relating to the data ensemble for transportation in a communication network is provided.
  • the present invention finds advantageous applications within structural determination in molecular biology and/or structural chemistry, i.e. using wave imaging technologies and especially for direct or semi-direct imaging technologies such as EM, NMR, X-ray crystallography, and FEL.
  • Figure 1 illustrates schematically a general method according to the present invention
  • Figure 2 illustrates schematically a system for implementing the present invention
  • Figure 3 illustrates schematically a device for using the present invention
  • reference numeral 200 generally denotes a system that acquires data from an experiment that replicates the inverse problem in question 201 with a data acquisition device (not shown) connected 203, 205 to a processing device 202 directly or indirectly through a network 204.
  • Data from the experiment device is transmitted to the processing device 202 that also may be arranged to control the configuration and operation of the data acquisition in the experiment 201.
  • the data acquisition device 201 are known to the person skilled in the art in the relevant scientific discipline for the inverse problem in question and therefore not described further in this document.
  • Data obtained in the processing device may be transmitted to other devices such as an external computer 206 connected 205 to the same network 204 or on a separate network and with data transmitted using other techniques (e.g. using a USB based memory, disk, or similar).
  • the processing device 202, 300 is shown in detail in Figure 3, wherein
  • a processing unit 301 handles the reconstruction procedure and interaction with the data acquisition device and user.
  • the processing device 300 further comprises a volatile (e.g. RAM) 302 and/or non volatile memory (e.g. a hard disk or flash disk) 303, an interface unit 304.
  • the processing device 300 may further comprise a data acquisition unit 305 and communication unit io 306, each with a respective connecting interface. All units in the processing device can communicate with each other directly or indirectly through the processing unit 301.
  • the processing unit 301 processes data, controls data acquisition, and handles interface commands using appropriate software, data and analysis results may be stored in the memory unit(s) 302, 303.
  • the inter- i5 face unit 304 interacts with interface equipment (not shown), such as input devices (e.g. keyboard and mouse) and display devices.
  • the data acquisition unit 305 interacts with and receives data from the data acquisition device 201.
  • the communication unit 306 communicates with other devices via for instance a network (e.g. Ethernet). Experimental data can also be stored
  • the analysis method according to the present invention is usually realized as computer software stored in the memory 302, 303 and run in the processing unit 301.
  • the analysis software can be implemented as a computer program product
  • a removable computer readable media e.g. diskette, CD- ROM (Compact Disk-Read Only Memory), DVD (Digital Video Disk), flash or similar removable memory media (e.g. compactflash, SD sectire digital, memorystick, miniSD, MMC multimediacard, smartmedia, transflash, XD), HD-DVD (High Definition DVD), or Bluray DVD, USB (Universal Serial Bus
  • a computer network e.g. Internet, a Local Area Network (LAN), or similar networks.
  • the same type of media may be used for dis- tributing results from the measurements of the data acquisition device for post analysis at some other computational/processing device.
  • the processing device may be a stand alone device or a device built into the data acquisition device. It may alternatively be a device with- out possibilities to control the data acquisition device and used for running image analysis, e.g. the external computer 206.
  • Actual experimental data may be pre-processed as to represent data originating from the inverse problem in question.
  • the type of pre-processing is determined by the inverse problem and the experimental setup. In a formal sense, unperturbed data are assumed to lie in the range of the forward operator. Other types of pre processing may also be done on the experimental data.
  • the present invention is a new method to process experimental data to get better solutions of inverse problems. This algorithm is useful for many problems, including electron microscopy tomography.
  • the present invention is an algorithm and virtual machine to provide high-quality solutions (reconstructions) of data from a inverse problem.
  • Such data are acquired from a large range of structure determination problems within structural chemistry and biology including different experimental modalities, different data acquisition geometries and stochastic models for the er- rors in the data. It can be adapted to each of these situations by altering the choices for the forward operator and the choices for selecting and updating the corresponding regularization parameters.
  • the algorithm substantially includes the following steps:
  • T(/) : E[data[/] provided that the expectation value of data[/] exists.
  • the inverse problem is the problem to estimate / G SC from the data, i.e. from a single sample of data[/].
  • the forward problem is to generate a sample in Jf of data[/] when / E S£ is given.
  • data[/] will be the sum of a random element Or[J], whose distribution depends on T(f), and an independent random element E with a fixed distribution.
  • Another common approach is to consider the measured data g itself as an estimator of T(f), which is different from the setting in Definition 3 where the data g is assumed to be a sample of the random variable data[/] with expectation value T(/). The inverse problem is then reduced to the problem of solving the operator equation
  • a reconstruction method will refer to a method that claims to (approximately) solve the inverse problem in Definition 3 (or (I)). Formally it is defined by a reconstruction operator 7Z ⁇ : J ⁇ ⁇ SC and the vector ⁇ G 1" is the parameters of the reconstruction method (it could e.g. be some stepsize or number of iterations).
  • a regularization method is a reconstruction method whereas there are reconstruction methods that are not regularization methods.
  • T we choose to consider.
  • a basic assumption for a reasonable theory is to assume that _3T and Jf are Banach spaces and T: 3£ ⁇ jrff is continuous and weakly sequentially closed.
  • Definition 9 Consider the inverse problem given in Definition 3 (or (I)) and assume that it has least squares solutions. Also, let p E SC be a fixed element which we call the prior and we are given a fixed functional S( - ; p) : JT ⁇ ] — oo, oo] (that depends on the prior p). Then p G SC is a ⁇ S( • ; p) -minimizing least squares solution if it is a least squares solution that minimizes S( • ; /?), i.e.
  • T is weakly closed in SC .
  • Theorem 11 provides us with the much needed concept of a unique solution to the inverse problem in Definition 3 (or (I)). We are now ready to introduce the notion of a regularization method in a rigorous way and discuss what is i5 commonly included in a mathematical analysis of such a method.
  • each H ⁇ is a reconstruction operator and the above definition formalizes the fact that a reconstruction method that is a regularization must 5 be stable (1Z ⁇ are continuous) and it must have a convergence feature in the sense that the regularized solutions converge (in the norm) to a least squares solution of the inverse problem when the error ⁇ in the data goes to zero and the regularization parameter is chosen according to the associated parameter choice rule.
  • the parameter choice rule must now also include limiting as ⁇ ⁇ 0.
  • Definition 14 Let ⁇ : R + X Jf ⁇ ]0, ⁇ j[x . . . x]0, X Q [ be a parameter choiceo rule as in Definition 12. If ⁇ does not depend on g ⁇ but only on ⁇ , then we say that ⁇ is an a-priori parameter choice rule. Otherwise, we say that ⁇ is an a-posteriori parameter choice rule.
  • Variational regularization methods are reconstruction methods that can be formulated as solving an optimization problem which is defined by a regularization functional
  • V Jf? x / ⁇ R + .
  • V is given as the square of some norm on Jf, but it can also be a log-likelihood function that depends on assumptions regarding the probabilistic distribution of data[/].
  • the closely associated ⁇ -regularization for region enhancement is based on the assumption that the absolute value of gradient of elements in X are contained in C p .
  • € p -regularization for region enhancement of the inverse problem is defined as solving (3) (or (4)) with
  • Definition 15 Let & be an abstract normed space with norm
  • W x is some normed space containing the range of A 1 .
  • a (/true) admits a sparse representation in a known basis/frame C W 1 , i.e.
  • the first part of the algorithm consists of a phantom generator which is simply a set of tools for constructing elements / ph G SC (i.e. phantoms) which have similar sparsity properties as the element / true G SC that is to be reconstructed in the inverse problem in Definition 3 (or in (I)) given measured data g G Jrff.
  • Figure 1 shows a flowchart of the algorithm specified in sectin .
  • the system includes an data acquisition device connected to a computer.
  • the computer is programmed with the algorithm outlined in subsection and it is connected to a output device, e.g. a display, a file or a printer.
  • the result of the invention is a computer file that contains the solution to the multicomponcnt inverse problem.
  • the reference numerals in Figure 1 have the following meaning:
  • Reference numeral "1" The actual measured data from the experiment delivered by the data acquisition device to the computer.
  • the concrete operation of the data acquisition device and its connection to the computer is not shown and should be understood by the persons skilled in the art for the inverse problem in question.
  • Reference numeral "2" The pre-processing step that transforms the measured data into a form suitable for the inverse problem.
  • the concrete definition of this transform should be understood by the persons skilled in the art for the inverse problem in question.
  • Reference numeral "3” The stored pre-processed data g G J ⁇ that acts as input data for the inverse problem.
  • Reference numeral "4" Document describing the stochastic model for the data.
  • Reference numeral "5" The internal representation of the stochastic model for the pre-processed data.
  • Reference numeral "6" The internal representation of the of parameter selection rule for determining the regular ization parameters.
  • Reference numeral "7" The internal representation of the data discrep- ancy measure.
  • Reference numeral "8" The internal representation of the forward operator T.
  • Reference numeral “9” The document defining the features of the specimen in question.
  • Reference numeral “10” The document defining the data compression libraries that are of interest.
  • Reference numeral "12" The document describing the phantom gener- ator.
  • Reference numeral "14” The internal representation of the data compression library 2).
  • Reference numeral "16” The internal representation of the phantom generator.
  • Reference numeral "19" The document describing the tolerance or penalization based sparse ⁇ p -regularization method.
  • Reference numeral "21" The internal representation of the process that defines the regulai ⁇ zation functional according to the scheme in (5).
  • Reference numeral "22" The regularization functional ⁇ S (or 5 smooth ) defined as in (5).
  • Reference numeral "24" Document describing the forward operator.
  • Reference numeral "25" The internal representation of the process of defining the regularization problem.
  • Reference numeral "26" The internal representation of the process of solving the regularization problem.
  • ET reconstruction problem in electron tomography
  • sample preparation is to achieve this while preserving the structure of the specimen.
  • Sample preparation techniques are rather elaborate and depend on the type of specimen. However, from a simplistic point of view one can say that in-vitro specimens are flash-frozen in a millisecond (cryo-EM). In-situ specimens are chemically fixed, cryo-sectioneds and immuno-labeled, and can also be treated in a similar way to in-vitro specimens.
  • Data collection in ET is done by mounting the specimen on a holder (goniometer) that allows one to change its positioning relative to the optical axis of the TEM.
  • a holder goniometer
  • the specimen is radiated with an0 electron beam and the resulting data, referred to as a micrograph, is recorded by a detector.
  • each fixed orientation of the specimen yields one micrograph and the procedure is then repeated for a set of different positions.
  • the most common data acquisition geometry is single axis tilting where the specimen plane is only allowed to rotate around a fixed single axis, called the5 tilt axis, which is orthogonal to the optical axis of the TEM.
  • the rotation angle is called the tilt angle and the angular range is commonly in [—60°, 60°].
  • the scattering features of the specimen are in this case given by the Coulomb potential and the electron- specimen interaction is modeled by the scalar Schr ⁇ dinger equation.
  • the picture is completed by adding a description of the effects of the optics and the detector of the TEM, both modeled as convolution operators.
  • the basic assumption of perfect coherent imaging must be relaxed.
  • Inelas- tic scattering introduces partial incoherence which is accounted for within a coherent framework by introducing an imaginary part to the scattering potential, called the absorption potential.
  • the incoherence that stems from incoherent illumination is modeled by modifying the convolution kernel that describes the effect of the optics.
  • the resulting model for the image formation can only account for the amplitude contrast, so the main contrast mechanism, namely phase contrast, is not adequately captured by this model for image formation.
  • the scattering potential and intensity In order to precisely state the forward operator we introduce the scattering potential / : R 3 ⁇ C that defines the structure of the specimen. Following [10], the scattering potential is given as
  • PSF det is the detector point spread function
  • gain ⁇ is the detector gain
  • ⁇ j j C ⁇ x is the set denning the (i, j):th pixel.
  • ⁇ is a measure whose action on a subset ⁇ C ⁇ 1 is a Poisson distributed stochastic variable with expected value
  • Dose( ⁇ ) is the incoming dose (in number of electrons hitting the specimen per area unit).
  • Ei j is a stochastic variable representing the noise introduced by the detector.
  • T The forward operator in ET, denoted by T, is defined as the expected value of data[/]( ⁇ ) ij -, i.e.
  • T(f) ( ⁇ ) tJ gain, Dose(u;) / ⁇ PSF det ⁇ l(f) ( ⁇ , - ) ⁇ z) dz + ⁇ h3
  • T(J)( ⁇ ) tJ gain ⁇

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Abstract

La présente invention porte sur une solution pour résoudre le problème de détermination de structure en biologie moléculaire/chimie structurale. Ceci est souvent un problème inverse mal posé qui est très stimulant, à la fois d'un point de vue expérimental et algorithmique. L'idée de l'invention est d'identifier tout d'abord une base/un cadre où la structure moléculaire d'intérêt (ou une caractéristique de celle-ci) est a priori connue comme présentant une représentation éparse. Ensuite, le problème inverse qui est associé aux données expérimentales, et qui doit être résolu pour récupérer la structure moléculaire d'intérêt, est résolu par application d'une régularisation à variation avec un élément fonctionnel de régularisation basé sur la norme Z7 appliquée aux coefficients dans cette base/ce cadre. De tels problèmes inverses se produisent en particulier lorsque l'on utilise le microscope électronique à transmission dans un montage tomographique (tomographie à faisceau d'électrons) pour récupérer la structure moléculaire d'un échantillon biologique.
PCT/SE2008/051382 2007-11-30 2008-12-01 Régularisation lp de représentations éparses appliquée à des procédés de détermination de structures en biologie moléculaire/chimie structurale WO2009070120A1 (fr)

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Cited By (2)

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CN111680762A (zh) * 2018-11-27 2020-09-18 成都工业学院 中药材适生地的分类方法及装置
CN116167948A (zh) * 2023-04-21 2023-05-26 合肥综合性国家科学中心人工智能研究院(安徽省人工智能实验室) 一种基于空变点扩散函数的光声图像复原方法及系统

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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111680762A (zh) * 2018-11-27 2020-09-18 成都工业学院 中药材适生地的分类方法及装置
CN111680762B (zh) * 2018-11-27 2023-08-04 成都大学 中药材适生地的分类方法及装置
CN116167948A (zh) * 2023-04-21 2023-05-26 合肥综合性国家科学中心人工智能研究院(安徽省人工智能实验室) 一种基于空变点扩散函数的光声图像复原方法及系统
CN116167948B (zh) * 2023-04-21 2023-07-18 合肥综合性国家科学中心人工智能研究院(安徽省人工智能实验室) 一种基于空变点扩散函数的光声图像复原方法及系统

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