Table of Contents
Overview
Definition and Characteristics
Inverse problems are mathematical challenges focused on deducing unknown parameters, objects, or changes in medium properties from indirectly related data. The goal is to reconstruct an image or model of the object based on this data, a process often termed imaging in this context.[1.1] Central to inverse problems is the Measurement Operator (MO), which maps the parameters of interest to the observed measurements.[5.1] Often, this MO acts as a smoothing operator, leading to an unbounded inverse, thereby limiting access to only the low-frequency components of the object from noisy measurements.[39.1] Understanding the MO's properties, particularly its injectivity and stability, is crucial for effectively analyzing inverse problems.[5.1] Inverse problems differ fundamentally from direct problems in mathematical modeling. While direct problems involve calculating responses from known data and parameters, inverse problems aim to infer unknown parameters from observed effects, often in partially understood systems.[21.1] Typically, direct problems are well-posed, possessing existence, uniqueness, and stability of solutions, as defined by Hadamard.[20.1] In contrast, inverse problems are often ill-posed, complicating solution processes.[20.1] Despite these challenges, inverse problems are significant in various scientific fields, providing insights into parameters that cannot be directly observed, thus representing some of the most important mathematical problems in science and mathematics.[6.1] These problems arise from the need to gain information about unknown objects from indirect measurements and are prevalent in fields such as medical imaging, industrial process monitoring, ozone layer tomography, and financial market modeling.[2.1] A defining characteristic of inverse problems is understanding the relationship between observed data and the unknown parameters to be estimated, underscoring their significance across multiple applications in science and engineering.[2.1][2.1]Importance in Various Fields
Inverse problems are crucial in various fields, such as geophysics, medical imaging, and scientific research, due to their capacity to derive valuable insights from indirect measurements. In geophysics, these problems involve reconstructing subsurface properties from surface data, which is vital for applications like carbon control, monitoring, and earthquake detection. These challenges are often ill-posed, nonunique, and nonlinear, requiring sophisticated computational techniques and algorithms for effective resolution.[13.1] In medical imaging, filtered back projection (FBP) has been the standard for over four decades, known for its speed and reliability in producing high-quality images across clinical applications.[15.1] This analytic reconstruction algorithm addresses the limitations of conventional back projection by applying a convolution filter to remove blurring and uses simultaneous equations of ray sums at different angles to compute attenuation coefficients.[15.1] While FBP remains prevalent, iterative reconstruction (IR) has gained attention for its ability to reduce image noise, especially with lower radiation doses.[18.1] IR advancements have led to its consideration as a potential replacement for FBP, now available in commercial solutions from all vendors.[18.1] These developments in reconstruction techniques underscore their importance in enhancing diagnostic accuracy and clinical outcomes in medical imaging.[16.1] The theoretical framework of inverse problems is essential for addressing challenges like ill-posedness, uncertainty, nonlinearity, and under-sampling, common in many scientific fields.[28.1] This framework is crucial for training models to learn physical properties from indirect measurements or develop predictive models replicating past observations.[29.1] Recent advancements, particularly integrating deep learning with physics-driven designs, are revolutionizing how inverse problems are tackled, enabling accurate extraction of physical properties from complex observations.[32.1] The historical development of inverse problem theory, especially in geophysics, includes significant milestones like Gauss's method of least squares, foundational for modern subsurface imaging and resource detection methodologies.[34.1] These advancements highlight the ongoing significance of inverse problems in understanding complex systems and improving predictive accuracy in scientific research.History
Early Developments
The study of inverse problems in geosciences has its roots in the foundational work of Backus and Gilbert during the late 1960s and early 1970s, which laid the groundwork for understanding how to estimate parameters and construct models of the Earth that align with observational data. This early research emphasized the importance of creating models that not only fit the data but also share properties with the actual Earth, reflecting a significant advancement in the field.[67.1] As the field progressed, researchers encountered challenges due to the inherent noise and incompleteness of real geophysical observations. These imperfections often hindered the ability to accurately constrain the quantities of interest, complicating the interpretation of subsurface data.[68.1] In response to these challenges, Monte Carlo techniques emerged as a successful approach for addressing geophysical inverse problems, with their application expanding steadily over the past 15 years.[68.1]Key Milestones in Inverse Problem Theory
Inverse problems have a rich history, particularly within the geophysical sciences, where they first emerged as a means to study wave or chemical component dispersion and subsurface studies, often in contexts where direct measurement is limited.[43.1] The evolution of inverse problems has been significantly influenced by advancements in computational power, which have enabled the development of modern techniques that address the complexities of these problems.[46.1] One of the earliest examples of inverse problems can be traced back to geophysics, where researchers sought to determine the physical properties of the Earth's subsurface, such as density and velocity, based on indirect observations like seismic waves and gravity measurements.[52.1] This foundational work laid the groundwork for the formalization of inverse problems as equations in vector spaces, addressing critical issues such as ill-posedness, which remains a central challenge in the field.[51.1] The journal "Inverse Problems," established as a peer-reviewed platform for interdisciplinary research, has played a crucial role in disseminating theoretical, experimental, and mathematical advancements related to inverse problems.[44.1] This journal has highlighted the importance of utilizing realistic mathematical models that can be shown to be consistent with measured data, thereby facilitating accurate physical interpretations of solutions.[47.1] Key breakthroughs in the field include the introduction of computational methods for geophysical inverse problems, such as gravity inversion and seismic impedance inversion, which have significantly advanced the ability to analyze complex data sets.[48.1] Additionally, the integration of artificial intelligence and machine learning techniques has transformed the approach to solving inverse problems, allowing for greater flexibility and effectiveness in handling incomplete or noisy data.[56.1]Mathematical Foundations
Well-posed vs. Ill-posed Problems
In the study of inverse problems, a critical distinction is made between well-posed and ill-posed problems. A well-posed problem is characterized by the existence of a unique solution that depends continuously on the input data, meaning that small changes in the input lead to small changes in the output. This property is essential for ensuring the stability and reliability of the solutions derived from such problems.[107.1] Conversely, ill-posed problems lack one or more of these properties, often resulting in solutions that are highly sensitive to perturbations in the input data. This sensitivity can lead to significant challenges in obtaining reliable solutions, as small errors in measurements can be magnified during the inversion process.[90.1] Regularization techniques are commonly employed to address the issues associated with ill-posed problems. These methods transform an ill-posed problem into a family of well-posed problems, allowing for the computation of regularized solutions that serve as approximations to the desired solution of the inverse problem.[103.1] By incorporating additional information or imposing constraints on the solution space, regularization helps stabilize the solution process, preventing overfitting and guiding optimization efforts.[119.1] This approach is particularly important in fields such as geophysics, where inverse problems frequently arise and are often ill-posed due to the complexity of the underlying physical models.[91.1] The challenges posed by ill-posedness are further compounded by the presence of nonlinearity in many inverse problems. Nonlinear inverse problems often require specialized solution methods, as traditional linear approaches may not suffice. The nonlinearity can stem from various factors, including multiple scattering effects within the domain of interest, which complicates the inversion process.[102.1] As a result, dedicated algorithms and optimization techniques, such as the Gauss-Newton method, are often utilized to effectively tackle these nonlinearities.[101.1]Regularization Techniques
Inverse problems are often characterized by their ill-posed nature, which necessitates the use of regularization techniques to ensure stable and meaningful solutions. These problems arise in various fields, including physics, geophysics, and medicine, where they are typically unstable and nonlinear, making them particularly challenging to solve.[93.1] Regularization methods are essential in addressing the issues of nonuniqueness and sensitivity to data noise that are inherent in ill-posed problems.[96.1] Regularization techniques are essential for addressing the challenges posed by ill-posed inverse problems, which often arise when a linear operator lacks a bounded inverse in the appropriate Banach or Hilbert spaces, despite being injective.[92.1] Most interesting inverse problems require regularization to stabilize the solution process, and this topic encompasses fundamental elements of functional analysis, regularization theory, and optimization.[94.1] A well-known example of such problems is the Cauchy problem for elliptic partial differential equations, which illustrates the application of regularization techniques in fields like computerized tomography.[95.1] In seismic imaging, the challenges posed by noise and uncertainty are significant due to the ill-posed nature of the inverse problem, which is influenced by bandwidth and aperture limitations. These factors, along with the presence of noise and linearization errors, can lead to multiple data-satisfying solutions, making the solutions unstable with respect to small perturbations in prior information and data noise.[98.1] To effectively address these challenges, uncertainty quantification (UQ) is employed, providing a probabilistic description of the solution's nonuniqueness and sensitivity to data noise. By framing seismic imaging within a Bayesian context, researchers can systematically study uncertainty by solving for the model posterior distribution.[99.1] Recent advancements in computational techniques, particularly through the integration of machine learning (ML) and deep learning (DL), have significantly enhanced the analysis of seismic data, allowing researchers to identify and categorize patterns and anomalies more effectively.[116.1] Hybrid models that combine analytical methods with deep learning approaches have been introduced to address generalization issues while retaining the efficacy of deep learning models.[111.1] These innovations have enabled the creation of detailed 3D models of Earth's interior, particularly in response to unusual seismic wave arrivals, which have led to discoveries such as an overturned slab in the Mediterranean.[117.1] Consequently, these developments have transformed the methodologies employed by engineers and researchers in the field of geophysics, particularly in the context of inverse problems.Applications
Medical Imaging
Recent advancements in machine learning, particularly deep neural networks, have significantly enhanced the ability to address inverse problems in medical imaging. These techniques are increasingly utilized to solve various challenges in computational imaging, leading to improved reconstruction methods and categorization of imaging issues.[131.1] Current research primarily focuses on underdetermined inverse problems, which are crucial in the medical imaging domain. These problems aim to produce high-resolution images while minimizing data acquisition time and costs.[132.1] Deep learning solutions have emerged as robust mechanisms for tackling these challenges, facilitating precise and early disease detection through sophisticated algorithms that optimize iterative processes.[134.1] Additionally, diffusion models have been identified as effective tools for solving inverse problems, particularly in MRI and CT reconstruction. Recent studies have explored methodologies to enhance these models, including manifold constraints and pre-trained models, to accelerate reconstruction and improve image quality.[133.1] The application of deep learning in inverse problems within biomedical image analysis represents a significant frontier in disease detection, utilizing computational methodologies and mathematical modeling to unravel complex data embedded within medical images.[158.1] Inverse problem imaging involves inferring raw information or images from observed data, essential for deducing unknown properties of biological structures or tissues.[159.1] Traditional methods face limitations due to model assumptions, computational complexity, and noise interference, which can adversely affect image quality when addressing complex inverse problems.[159.1] As advancements continue, these methodologies are expected to enhance the speed and accuracy of image reconstruction, thereby improving diagnostic capabilities in clinical settings.[158.1]Geophysics and Remote Sensing
Inverse problems play a crucial role in geophysics and remote sensing, driven by scientific curiosity, georesource exploration, and environmental concerns, which prompt the geoscientific community to undertake complex subsurface investigations.[150.1] These problems require the integration of geophysical and hydrogeological data to develop a conceptual Earth model, enhancing the understanding of subsurface structures and resources.[152.1] This approach not only aids in resource exploration but also in creating geologically realistic priors and generating accurate realizations through sampling.[152.1] By combining geological concepts with geophysical data, researchers can address complex subsurface phenomena, contributing to resource management and _environmental sustainability.[150.1] Recent advancements in the theory and applications of inverse problems have significantly expanded their use across scientific and engineering disciplines.[128.1] In geophysics, they are applied in non-destructive testing, oil and gas exploration, and tomography, which are essential for assessing subsurface conditions and resources.[129.1] The integration of geophysics with remote sensing technologies enhances data analysis and interpretation, improving the effectiveness of environmental studies.[151.1] The application of inverse problems in geophysics is evident in enhancing subsurface mapping for multidisciplinary purposes, such as resource exploration and _heritage conservation.[149.1] By integrating geological concepts with geophysical data, researchers can create geologically realistic priors and generate accurate realizations that reflect subsurface complexities, improving model accuracy and facilitating better decision-making in resource management and _environmental protection.[152.1]In this section:
Recent Advancements
New Modeling Approaches
Recent advancements in inverse problems have underscored the potential of diffusion models as innovative approaches. These models are celebrated for generating high-quality samples and have opened new avenues for solving inverse problems, particularly in image restoration and reconstruction, by treating them as unsupervised priors.[173.1] While current research predominantly targets natural image restoration, the application of diffusion models in scientific inverse problems, such as those in medical imaging, remains largely unexplored.[171.1] However, studies have reviewed their principles and application in MRI and CT reconstruction tasks, suggesting a promising direction for future research.[172.1] In medical imaging, diffusion models have been applied to MRI and CT reconstruction, effectively addressing challenges associated with reconstructing images from incomplete or noisy data, thus enhancing medical diagnostics.[172.1] Frameworks like InverseBench have been introduced to evaluate the performance of diffusion models across various scientific inverse problems, highlighting their versatility and potential for broader applications.[171.1] Another significant approach involves integrating data-driven models with domain-specific knowledge to create a mathematically coherent foundation for solving inverse problems. This integration allows for the incorporation of physical-analytical models alongside deep learning techniques, enhancing solution robustness.[190.1] For example, using the approximate distribution learned by Generative Adversarial Networks (GANs) as a prior in Bayesian updates has proven effective in efficiently solving large-scale Bayesian inverse problems.[191.1] Moreover, mathematical modeling has evolved significantly, particularly in biomedical image analysis. Advanced techniques such as the Deep Learning Solution for Inverse Problems in Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) have been developed, utilizing deep learning to detect diseases from complex data embedded within medical images, showcasing the practical implications of these innovative modeling approaches.[198.1]Computational Methods and Techniques
Recent advancements in computational methods for solving inverse problems have significantly enhanced the ability to recover functions, signals, and images from incomplete or noisy data. These advancements include the integration of machine learning techniques, which have proven effective in various applications, particularly in the fields of imaging and signal processing. One notable development is the use of diffusion models, which leverage diffusion sampling steps to induce data priors while employing measurement guidance gradients to ensure data consistency. This approach has shown promise in addressing general inverse problems, although approximations are often necessary when utilizing unconditionally trained diffusion models.[168.1] Additionally, the application of artificial intelligence (AI) has led to the emergence of several methodologies for electromagnetic inverse problems, characterized by high reliability and computational efficiency. These methodologies include three-step learning-by-examples, system-by-design, and deep learning frameworks.[175.1] The role of deep learning in solving inverse problems has been particularly transformative. Recent research indicates that deep neural networks can effectively tackle a wide variety of inverse problems in computational imaging. This body of work has resulted in a taxonomy that categorizes different problems and reconstruction methods, highlighting the central themes in this emerging area.[176.1] Furthermore, deep learning strategies have been classified into three main categories: Direct Mapping, Data Consistency Optimizer, and Deep Regularizer, each demonstrating varying degrees of robustness across different types of inverse problems.[177.1] Inverse problem imaging involves the process of inferring raw information or images from observed data. Traditional methods face limitations due to model assumptions, computational complexity, and noise interference, which can adversely affect image quality when dealing with complex inverse problems. However, there is an emerging body of literature that highlights the challenges and opportunities presented by modern AI techniques, particularly deep learning. This literature indicates that deep learning has been very successful for a variety of image and signal recovery and restoration tasks, including denoising, magnetic resonance (MR) reconstruction, and nano-scale imaging.[178.1]In this section:
Challenges And Future Directions
Stability and Uniqueness of Solutions
Solving inverse problems presents several challenges, primarily due to their ill-posedness and the presence of noise in the observed data.[206.1] These problems entail the difficulty of deducing input parameters from observed output data, which can lead to non-unique solutions.[208.1] The non-uniqueness of solutions can result in multiple plausible interpretations of the data, complicating the decision-making process.[206.1] Additionally, the noise in the data can obscure the true signal, making it challenging to identify the correct solution, as there may be many potential solutions that differ significantly from one another while still fitting the observed data.[211.1] Thus, the inherent complexities of inverse problems necessitate innovative problem-solving strategies to effectively address these issues.[208.1] Inverse and ill-posed problems are prevalent in numerous real-world applications, including medical imaging techniques such as MRI, ultrasound, and optical imaging, as well as in oil prospecting and nondestructive testing of materials.[209.1] These problems often lack the properties defined by Hadamard for a problem to be considered "well-posed," which complicates the interpretation of the resulting data.[209.1] Regularization methods are essential tools in addressing these challenges, as they introduce prior knowledge to stabilize solutions and facilitate the approximation of ill-posed (pseudo-)inverses.[235.1] Recent trends indicate a shift in interest from linear to nonlinear regularization methods, even for linear inverse problems, reflecting ongoing advancements in this field.[234.1] The field of inverse problems, while relatively new, has seen a significant amount of research, yet it remains immature with numerous opportunities for future exploration.[205.1] Recent advancements include the application of deep learning methods to various linear inverse problems, particularly through the development of structured neural network architectures that integrate traditional knowledge.[204.1] However, challenges persist, such as those posed by massive least squares problems, where the size of the forward process can exceed the storage capabilities of computer memory or where data may not be available all at once.[207.1] Addressing these challenges is crucial for advancing the methodologies in this field and identifying potential future directions for research.[205.1]References
[1] Overview of Inverse Problems | SpringerLink — In inverse problems, the goal is to find objects, sources, or changes in medium properties from indirectly related data. The solution is usually given as an image, and as such the word imaging is often a descriptor for an inverse problem.
[2] PDF — Introduction to Inverse Problems Inverse problems arise from the need to gain information about an unknown object of inter-est from given indirect measurements. Inverse problems have several applications varying from medical imaging and industrial process monitoring to ozone layer tomography and modelling of nancial markets. The common feature for inverse problems is the need to understand
[5] PDF — What is an Inverse Problem Three essential ingredients de ne an inverse problem in this book. The central element is the Measurement Operator (MO), which maps objects of interest, called parameters, to information collected about these objects, called measurements or data. The main objective of inverse problem theory is to analyze such a MO, primarily its injectivity and stability properties
[6] Inverse problem - Wikipedia — It is the inverse of a forward problem, which starts with the causes and then calculates the effects. Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe.
[13] Solving geophysical inverse problems with scientific machine learning — Specifically, geophysical inverse problems seek to determine various Earth properties critical for geophysical exploration, carbon control, monitoring, and earthquake detection. These problems pose unique challenges: the parameters of interest are often high-dimensional, and the mapping from parameters to observables is computationally demanding.
[15] Filtered back projection | Radiology Reference Article - Radiopaedia.org — Filtered back projection is an analytic reconstruction algorithm designed to overcome the limitations of conventional back projection; it applies a convolution filter to remove blurring. It was the primary method in cross-sectional imaging reconstruction. It utilizes simultaneous equations of ray sums taken at differing angles of a sine wave to compute the values of attenuation coefficients
[16] Filtered BackProjection (FBP) Illustrated Guide for Radiologic ... — Filtered back projection (FBP) preceded filtered back projection in the CT industry and therefore it became the industry standard for speed and image texture in CT imaging. Iterative reconstruction does have advantages at reducing the noise in the image, which is present when lower radiation doses are used.
[18] Image reconstruction: Part 1 - understanding filtered back projection ... — Iterative Reconstruction (IR) is at present an adjunct to standard Filtered Back Projection (FBP) reconstruction, but could become a replacement for it. Due to its potential for scanning at lower radiation doses, IR has received a lot of attention in the medical literature and all vendors offer commercial solutions.
[20] Fractal‐Based Methods and Inverse Problems for Differential Equations ... — There is a fundamental difference between the direct and the inverse problem; often the direct problem is well-posed while the corresponding inverse problem is ill-posed. Hadamard [ 7 ] introduced the concept of well-posed problem to describe a mathematical model that has the properties of existence, uniqueness, and stability of the solution.
[21] PDF — There are two ways of dealing with such a mathematical formulation. Direct Problems: A direct problem consists in calculating the response d from the data of the solicitations X and the parameters p. This is shown in the diagram 1. Inverse Problems: In an inverse problem, either we are partially unaware of the system G, or we ignore some
[28] The imperative of physics-based modeling and inverse theory in ... - Nature — Inverse theory provides a crucial perspective for addressing the challenges of ill-posedness, uncertainty, nonlinearity and under-sampling. ... many of today's scientific grand challenges suffer
[29] Solving inverse problems in building physics: An overview of guidelines ... — Inverse problem theory can be summed up as the science of training models using measurements. The target of such a training is either to learn physical properties of a system by indirect measurements, or setting up a predictive model that can reproduce past observations. ... These scientific challenges are gaining visibility due to the
[32] Physics-driven learning for inverse problems in quantum ... - Nature — The integration of deep learning techniques and physics-driven designs is reforming the way we address inverse problems, in which accurate physical properties are extracted from complex observations.
[34] Several Inverse Problems in Geophysics - Purdue University — Several Inverse Problems in Geophysics Several historical examples of inverse problems are now given. 18 th century. Gauss developed the method of least squares and applied it to a number of problems including geodetic mapping, estimation of orbital parameters of the asteroid Ceres, and problems in magnetism. ... After its development in the
[39] PDF — In many inverse problems, the measurement operator, which maps objects of interest to available measurements, is a smoothing (regularizing) operator. Its inverse is therefore unbounded and as a consequence, only the low-frequency component of the object of interest is accessible from inevitably noisy measurements.
[43] Inverse Problem - an overview | ScienceDirect Topics — 9.07.2 Brief History of LIM. Inverse problems first surfaced in the geophysical sciences where they are still commonly used for studying wave or chemical component dispersion, and especially in subsurface studies where direct measurement is often limited. The translation of the geophysical inverse problem into the estimation of the flows in a
[44] Inverse Problems - Wikipedia — Inverse Problems is a peer-reviewed, broad-based interdisciplinary journal for pure and applied mathematicians and physicists produced by IOP Publishing.It combines theoretical, experimental and mathematical papers on inverse problems with numerical and practical approaches to their solution. The journal has a specialized relevance to workers in geophysics, optics, radar, acoustics
[46] PDF — Inverse problems in geophysics and oceanography do have a very long history although the recent terminologies and methods depend directly upon the availability of massive computer power. An outstanding early example is the well-known problem of
[47] Overview of Inverse Problems - SpringerLink — An important feature in inverse problems is to utilize a realistic mathematical model whose numerical or exact solution can be shown to be consistent with measured data and to use the model to make the correct physical interpretation of the inverse problems solution. The mathematical structure utilized to obtain the solution is also related to
[48] Editorial: Advances in geophysical inverse problems - Frontiers — Conclusion In this Research Topic, we focus on computational methods for geophysical inverse problems, the topic includes gravity inversion, statistical inversion for fault parameters, nano-scale imaging of shale, seismic impedance inversion, EM inversion and geophysical joint inversion.
[51] Inverse Problems: Basics, Theory and Applications in Geophysics ... — Beginning with four examples of inverse problems, the opening chapter establishes core concepts, such as formalizing these problems as equations in vector spaces and addressing the key issue of ill-posedness. Chapter Two then moves on to the discretization of inverse problems, which is a prerequisite for solving them on computers.
[52] Inverse Problems In Geophysics: Maths, Techniques & Application — Inverse problems in geophysics involve determining the physical properties of the Earth's subsurface (like density, velocity, or conductivity) based on indirect observations, such as seismic waves, gravity measurements, or magnetic fields. Unlike forward problems, where you start with a known model and predict the data (e.g., how seismic waves propagate through a given structure), inverse
[56] BIAN: A Deep Learning Method to Solve Inverse Problems Using Only ... — methods less effective in handling complex and incomplete data. In contrast, AI demonstrates considerable flexibility and effectiveness in solving inverse problems. AI can seamlessly integrate data-driven and model-driven approaches by incorporating numerical information into objective functions, constraints, and optimization algorithms.
[67] Trans-dimensional inverse problems, model comparison and the evidence ... — 1 Introduction. The study of inverse problems has a long history in the geosciences, dating back to the pioneering work of Backus,Gilbert (1967, 1968 , 1970).Over the past 30 years there has been a strong focus on estimating parameters, that is, building models of the Earth which satisfy data and are in some sense 'close' to the real Earth, or have properties in common with it.
[68] Monte Carlo Methods in Geophysical Inverse Problems — Real geophysical observations are often noisy and incomplete and always imperfectly constrain the quantities of interest. Monte Carlo techniques are one of a number of approaches that have been applied with success to geophysical inverse problems. Over the past 15 years the range of problems to which they have been applied has grown steadily.
[90] An Introduction to the Mathematical Theory of Inverse Problems — One of the main challenges in inverse problems is that they are often ill-posed; so that small errors in the "input" are magnified when one attempts to determine the inverse solution. To me, one of the big takeaways from the book was how useful functional analysis is in inverse problems, both from an analysis point of view and an applied
[91] Inverse Problems: Basics, Theory and Applications in Geophysics ... — Inverse Problems: Basics, Theory and Applications in Geophysics | SpringerLink Inverse Problems This second edition includes an expanded and up-to-date treatment of nonlinear problems of inverse gravimetry and seismic tomography This textbook is an introduction to the subject of inverse problems with an emphasis on practical solution methods and applications from geophysics. Containing up-to-date methods, this book will provide readers with the tools necessary to compute regularized solutions of inverse problems. Chapter Two then moves on to the discretization of inverse problems, which is a prerequisite for solving them on computers. inverse problems geophysical applications inverse problems numerical analysis inverse problems gravimetry inverse gravimetry problem Regularization inverse problems Regularization of Linear Inverse Problems Regularization of Nonlinear Inverse Problems Book Title: Inverse Problems
[92] Functional Analysis in Action: Inverse Problems | Numerik — In the language of functional analysis such problems arise, when the linear operator fails to have a bounded inverse in the appropriate Banach or Hilbert spaces - despite its injectivity. ... It is the purpose of this lecture course to step into the breach and to investigate possible pitfalls when dealing with ill-posed problems, ways to
[93] PDF — differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of
[94] ECE598ID — Inverse Problems and Learning - University of Illinois ... — Most interesting inverse problems are ill-posed and need to be regularized. This course will cover the fundamentals of inverse problems theory including elements from functional analysis, regularization theory, and optimization. ... A good part of the course will be on the major machine learning and data driven techniques. In particular
[95] Functional Analysis in Action: Inverse Problems | Numerik — Well-known examples include the Cauchy problem for elliptic partial differential equations and many notable auxiliary problems which arise in computerized tomography techniques. In the language of functional analysis such problems arise, when the linear operator fails to have a bounded inverse in the appropriate Banach or Hilbert spaces
[96] PDF — Due to the ill-posedness of the underlying inverse problems, all the functional reconstruction methods involve some form of regularization which enables stable reconstruction. These methods are called regularization techniques (see for instance ). ... respect to the image sequence exploits recent algorithms from convex analysis to minimize
[98] Seismic Tomography and the Assessment of Uncertainty — In most practical seismic tomography applications, the inverse problem is under- or mixed-determined, so multiple data-satisfying solutions exist, and solutions (e.g., maximum likelihood in a linearized least squares formulation) tend to be unstable with respect to small perturbations in prior information and data noise in the absence of
[99] Uncertainty quantification in imaging and automatic horizon tracking—a ... — In inverse problems, uncertainty quantification (UQ) deals with a probabilistic description of the solution nonuniqueness and data noise sensitivity. Setting seismic imaging into a Bayesian framework allows for a principled way of studying uncertainty by solving for the model posterior distribution.
[101] On optimization techniques for solving nonlinear inverse problems — This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential equations. Such problems are often solved by utilizing a Gauss-Newton method in which the forward model constraints are implicitly incorporated.
[102] PDF — known that solving ISPs is difficulty and challenging due to the large number of unknowns, ill-posedness, and nonlinearity . The nonlinearity with respect to the unknown constitutive parameters in the domain of interest (DoI) is due to multiple scattering effects inside the DoI. The traditional model-based inversion methods are usually
[103] PDF — Regularization methods replace an ill-posed problem by a family of well-posed problems, their solution, called regularized solutions, are used as approximations to the desired solution of the inverse problem. These methods always involve some parameter measuring the close-ness of the regularized and the original (unregularized) inverse problem
[107] What is an inverse problem - GitHub Pages — 10 Lectures on Inverse Problems and Imaging =========================================== Welcome to Inverse Problems and Imaging The inverse problem consists of reconstructing the idealized image from the measured one. Here, K is called the forward operator; u is the image or parameter and f∈F are the measurements. The solution depends continuously on the data, i.e., there is a constant C<∞ such that ‖u−u′‖≤C‖f−f′‖ where K(u)\=f and K(u′)\=f′. This modified operator arises when the original inverse problem is ill-posed and is replaced by a modified inverse problem K~(u)\=f which is well-posed. ax.plot(u1,3-u1,'k',label\=r'$Ku=f$') The idea is to replace the original equation K(u)\=f by a minimization problem. To study well-posedness of the problem, we consider noisy measurements fδ(x)\=f(x)+δsin(kx/δ) for fixed arbitrary k and small δ>0.
[111] Deep learning approaches to inverse problems in imaging: Past, present ... — Hybrid models combining analytical and deep learning approaches have been introduced to solve such generalization issues while retaining the efficacy of deep learning models. In this work, we review deep learning and hybrid methods for solving imaging inverse problems, focusing on image and video super-resolution and image restoration.
[116] (PDF) Advanced Seismic Data Analysis: Comparative study of Machine ... — This study delves into the application of machine learning (ML) and deep learning (DL) techniques for the analysis of seismic data, aiming to identify and categorize patterns and anomalies within
[117] The New Age of Seismology: Breakthroughs in Technology and Data-Driven ... — For example, strange seismic wave arrivals have recently led to the discovery of an overturned slab in the Mediterranean . With increased computational power, researchers can now create detailed 3D models of Earth's interior using seismic data . a) Global Seismographic Network as of 2021 with stations colored by primary sensor type.
[119] Regularization Methods for Solving Inverse Problems: A ... - Springer — Given the tendency of ill-posed inverse problems to produce unreliable solutions due to noise or inaccuracies in measurements, regularization techniques act as a stabilizing force. By incorporating additional information or imposing constraints on the solution space, regularization helps prevent overfitting and guides the optimization process
[128] Inverse Problems: Theory and Application to Science and Engineering ... — In the recent years, theory and applications of inverse problems have undergone a tremendous growth. They can be formulated in many mathematical areas and analyzed by different theoretical and computational techniques. This special issue aims to highlight recent research, development, and applications of inverse problems in science and engineering.
[129] PDF — Applications of inverse problems include medical imaging, non-destructive testing, oil and gas exploration, cryptography and forensics, tomography and process control/monitoring, to mention only a few.
[131] Deep Learning Techniques for Inverse Problems in Imaging - arXiv.org — Deep Learning Techniques for Inverse Problems in Imaging Gregory Ongie, Ajil Jalaly, Christopher A. Metzler z Richard G. Baraniukx, Alexandros G. Dimakis {, Rebecca Willett k April 2020 Abstract Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging.
[132] Deep learning-based solvability of underdetermined inverse problems in ... — Recently, with the significant developments in deep learning techniques, solving underdetermined inverse problems has become one of the major concerns in the medical imaging domain, where underdetermined problems are motivated by the willingness to provide high resolution medical images with as little data as possible, by optimizing data collection in terms of minimal acquisition time, cost
[133] Diffusion Models for Inverse Problems in Medical Imaging — In this chapter, we review the principles of diffusion models and study how they can be used to solve inverse problems that arise in medical imaging, focusing on MRI and CT reconstruction tasks. Chung H, Ye JC (2022) Score-based diffusion models for accelerated MRI. Chung H, Sim B, Ryu D, Ye JC (2022) Improving diffusion models for inverse problems using manifold constraints. Chung H, Sim B, Ye JC (2022) Come-closer-diffuse-faster: accelerating conditional diffusion models for inverse problems through stochastic contraction. Chung H, Ryu D, Mccann MT, Klasky ML, Ye JC (2023) Solving 3d inverse problems using pre-trained 2d diffusion models. Lee S, Chung H, Park M, Park J, Ryu WS, Ye JC (2023) Improving 3D imaging with pre-trained perpendicular 2D diffusion models.
[134] Deep learning solutions for inverse problems in advanced biomedical ... — Inverse problems contribute to uncovering subtle abnormalities by employing iterative optimization techniques and sophisticated algorithms, enabling precise and early disease detection. Deep learning (DL) solutions have emerged as robust mechanisms for addressing inverse problems in biomedical image analysis, especially in disease recognition.
[149] Cross-Gradient Joint Inversion of DC Resistivity and Gravity Gradient ... — This approach has proven effective in enhancing subsurface mapping for multi-disciplinary purposes, including resource exploration, heritage conservation, and risk mitigation for the built environment.
[150] Geological realism in hydrogeological and geophysical inverse modeling ... — Scientific curiosity, exploration of georesources and environmental concerns are pushing the geoscientific research community toward subsurface investigations of ever-increasing complexity. This review explores various approaches to formulate and solve inverse problems in ways that effectively integrate geological concepts with geophysical and hydrogeological data. Modern geostatistical
[151] Integrating Geophysics With Remote Sensing For Environmental Studies — Explore the benefits of combining geophysics and remote sensing technologies to enhance environmental studies and improve data analysis and interpretation.
[152] Geological realism in hydrogeological and geophysical inverse modeling ... — Section 2 formulates the inverse problem as the integration of the information offered by geophysical and hydrogeological data, their relationship, and an underlying conceptual Earth model. Section 3 describes approaches to create geologically realistic priors and how to generate geologically realistic realizations by sampling this prior.
[158] Deep learning solutions for inverse problems in advanced biomedical ... — Inverse problems in biomedical image analysis represent a significant frontier in disease detection, leveraging computational methodologies and mathematical modelling to unravel complex data embedded within medical images. These problems include deducing the unknown properties of biological structures or tissues from the observed imaging data
[159] PDF — Application of Deep Learning in Inverse Problem Imaging. Inverse problem imaging involves the process of inferring raw information or images from observed data . Traditional methods are limited by model assumptions, computational complexity, and noise interference when dealing of image quality and with complex inverse problems.
[168] Diffusion State-Guided Projected Gradient for Inverse Problems — Recent advancements in diffusion models have been effective in learning data priors for solving inverse problems. They leverage diffusion sampling steps for inducing a data prior while using a measurement guidance gradient at each step to impose data consistency. For general inverse problems, approximations are needed when an unconditionally trained diffusion model is used since the
[171] InverseBench: Benchmarking Plug-and-Play Diffusion Models for Inverse ... — However, current studies primarily focus on natural image restoration, leaving the performance of these algorithms in scientific inverse problems largely unexplored. To address this gap, we introduce InverseBench, a framework that evaluates diffusion models across five distinct scientific inverse problems. These problems present unique
[172] Diffusion Models for Inverse Problems in Medical Imaging — In this chapter, we review the principles of diffusion models and study how they can be used to solve inverse problems that arise in medical imaging, focusing on MRI and CT reconstruction tasks. Chung H, Ye JC (2022) Score-based diffusion models for accelerated MRI. Chung H, Sim B, Ryu D, Ye JC (2022) Improving diffusion models for inverse problems using manifold constraints. Chung H, Sim B, Ye JC (2022) Come-closer-diffuse-faster: accelerating conditional diffusion models for inverse problems through stochastic contraction. Chung H, Ryu D, Mccann MT, Klasky ML, Ye JC (2023) Solving 3d inverse problems using pre-trained 2d diffusion models. Lee S, Chung H, Park M, Park J, Ryu WS, Ye JC (2023) Improving 3D imaging with pre-trained perpendicular 2D diffusion models.
[173] [2410.00083] A Survey on Diffusion Models for Inverse Problems - arXiv.org — Change to arXiv's privacy policy The arXiv Privacy Policy has changed. cs arXiv:2410.00083 arXiv author ID Help pages A Survey on Diffusion Models for Inverse Problems This has unlocked exciting new possibilities for solving inverse problems, especially in image restoration and reconstruction, by treating diffusion models as unsupervised priors. This survey provides a comprehensive overview of methods that utilize pre-trained diffusion models to solve inverse problems without requiring further training. This work aims to be a valuable resource for those interested in learning about the intersection of diffusion models and inverse problems. Cite as: arXiv:2410.00083 [cs.LG] (or arXiv:2410.00083v1 [cs.LG] for this version) From: Giannis Daras [view email] cs Bibliographic and Citation Tools Bibliographic Explorer Toggle Connected Papers Toggle
[175] AI‐Driven Approaches for Solving Electromagnetic Inverse Problems ... — Abstract This chapter provides an overview of artificial intelligence‐driven methods for solving electromagnetic (EM) inverse problems (IPs) with high reliability, robustness, and computational efficiency. Several methodologies are detailed and discussed, including the recent developments within the so‐called (i) three‐step learning‐by‐examples, (ii) system‐by‐design, and (iii
[176] Deep Learning Techniques for Inverse Problems in Imaging — Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging. We explore the central prevailing themes of this emerging area and present a taxonomy that can be used to categorize different problems and reconstruction methods.
[177] Survey of Deep Learning Methods for Inverse Problems — In this paper we investigate a variety of deep learning strategies for solving inverse problems. We classify existing deep learning solutions for inverse problems into three categories of Direct Mapping, Data Consistency Optimizer, and Deep Regularizer. We choose a sample of each inverse problem type, so as to compare the robustness of the three categories, and report a statistical analysis of
[178] Tutorial on Modern AI for Inverse Problems - USC Center on AI ... — This tutorial discusses the challenges and opportunities of using modern AI for inverse problems and scientific applications more broadly. In particular I will discuss an emerging literature on deep learning for inverse problems that have been very successful for a variety of image and signal recovery and restoration tasks ranging from denoising and MR reconstruction to nano-scale imaging
[190] Solving inverse problems using data-driven models — Abstract Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical-analytical models.
[191] Solution of physics-based Bayesian inverse problems with deep ... — Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems.
[198] Deep learning solutions for inverse problems in advanced biomedical ... — Inverse problems in biomedical image analysis represent a significant frontier in disease detection, leveraging computational methodologies and mathematical modelling to unravel complex data embedded within medical images. This study develops a DL Solution for Inverse Problems in the Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) technique. The DLSIP-ABIADD technique exploits the DL approach to solve inverse problems and detect the presence of diseases on biomedical images. This study develops a DL Solution for Inverse Problems in the Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) technique. The DLSIP-ABIADD technique exploits the DL approach to solve the inverse problem and detect the presence of diseases on biomedical images. The DLSIP-ABIADD technique exploits the DL approach to solve the inverse problem and detect the presence of diseases on biomedical images.
[204] Deep Learning Methods for Solving Linear Inverse Problems: Research ... — We review how deep learning methods are used in solving different linear inverse problems, and explore the structured neural network architectures that incorporate knowledge used in traditional methods. Furthermore, we identify open challenges and potential future directions along this research line. READ FULL TEXT
[205] Advances and applications in inverse reinforcement learning: a ... — Though it is a relatively new area of research, a significant amount of work has been done. However, the field is not yet mature and has many opportunities and a wide scope for future research. The following sections will summarize the future directions in this field from the techniques, problem domains, datasets, and application perspectives.
[206] What is: Inverse Problem - Understanding Inverse Problems — Solving inverse problems presents several challenges, primarily due to their ill-posedness and the presence of noise in the observed data. The non-uniqueness of solutions can lead to multiple plausible interpretations of the data, complicating the decision-making process. Additionally, computational limitations can hinder the ability to solve
[207] PDF — In this talk, we discuss modern challenges in inverse problems and introduce novel approaches to overcome such challenges. For instance, w e discuss massive least squares problems, where the size of the forward proces s exceeds the storage capab ilities of computer memory or the data i s simply not available all at once.
[208] Mastering Inverse Problems: A Comprehensive Guide for Understanding — These problems entail the challenge of deducing input parameters from observed output data, posing significant challenges due to their ill-posed nature. The unique feature of defining inverse problems lies in their ability to reverse the traditional input-output relationship, opening avenues for innovative problem-solving strategies.
[209] PDF — Introduction: Inverse and ill-posed problems Inverse and ill-posed problems arise in many real-world applications including medical microwave, optical and ultrasound imaging, MRT, MRI, oil prospecting and shape reconstruction, nondestructive testing of materials and detection of explosives, seeing through the walls and constructing of new
[211] PDF — noise in the data. Inverse problems are frequently ill-posed in the sense that there are many z's, which may be far from one another, such that F(z; ) dhas the same magnitude as the data's noise. This causes considerable challenges when solving the inverse problem. 2.1. Optimization problem. We focus on the optimization problem min z2Rm
[230] Solving inverse problems using data-driven models — It offers a rich set of tools for incorporating data into the recovery of the model parameter, so it is a natural framework to consider when data-driven approaches from machine learning are to be used for solving ill-posed inverse problems.
[231] PDF — Key Take-away ü Physics-informed ML exploits the underlying laws of physics to define an appropriate Inductive Bias (e.g., ML architecture, Loss function) for the solving the inverse problem ü This leads to improvement in model transparency, learning speed, data efficiency, and generalization performance
[234] [1801.09922] Modern Regularization Methods for Inverse Problems - arXiv.org — Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of
[235] Regularization Methods for Solving Inverse Problems: A ... - Springer — Regularization Methods for Solving Inverse Problems: A Comprehensive Review This work conducts a comprehensive review of regularization methods aimed at stabilizing solutions to inverse problems. Focusing on techniques such as Tikhonov regularization, machine learning-based regularization, and Bayesian regularization, we explore their mathematical foundations, numerical implementations, and applications in diverse fields. A fast iterative regularization method for ill-posed problems Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, Cham (2023) Download references Author information Authors and Affiliations You can also search for this author in Corresponding author © 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG About this paper Regularization Methods for Solving Inverse Problems: A Comprehensive Review. Share this paper Provided by the Springer Nature SharedIt content-sharing initiative
[244] PDF — Given this background, we next turn our attention to linear inverse problems. By linear inverse problems we really mean problems whose variational forms can put into some type of linear least squares structure.
[245] What is an Inverse Problem? - Inverse Problems — Inverse problems have a wide range of applications, such as making clear a blurred photo, medical imaging, oil drilling, and echolocation (SONAR, bats, and dolphins). A common characteristic is that we attempt to infer causes from measured effects. A forward, or direct problem has known causes that produce effects or results defined by the mathematical model. Because the measured data are
[246] Inverse Problems - Mathematical Association of America — Inverse Problems – Mathematical Association of America About the MAA Mathematics Magazine MAA American Mathematics Competitions Policies Community Events Algebra Algebraic Geometry Analysis Applied Mathematics Computational Algebraic Geometry Constructive Mathematics Differential Calculus Field Theory and Polynomials Group Theory History of Mathematics Information Theory Mathematical Education Mathematical Logic Mathematical Modeling Mathematical Physics Mathematics for Teachers Number Theory Probability Theory Recreational Mathematics An Introduction to the Mathematical Theory of Inverse Problems -------------------------------------------------------------- Andreas Kirsch successfully wrote this book not only for mathematics students but also physics and engineering students. Recovery Methodologies: Regularization and Sampling --------------------------------------------------- Recovery questions arise in applied mathematics when one needs to recover something- a function, a signal, or an image – from partial or incomplete information. About the MAA
[249] Deep learning solutions for inverse problems in advanced biomedical ... — Inverse problems in biomedical image analysis represent a significant frontier in disease detection, leveraging computational methodologies and mathematical modelling to unravel complex data embedded within medical images. This study develops a DL Solution for Inverse Problems in the Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) technique. The DLSIP-ABIADD technique exploits the DL approach to solve inverse problems and detect the presence of diseases on biomedical images. This study develops a DL Solution for Inverse Problems in the Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) technique. The DLSIP-ABIADD technique exploits the DL approach to solve the inverse problem and detect the presence of diseases on biomedical images. The DLSIP-ABIADD technique exploits the DL approach to solve the inverse problem and detect the presence of diseases on biomedical images.
[252] Deep Learning Techniques for Inverse Problems in Imaging - arXiv.org — Subsequently, one can train a network that takes in measurements y and reconstructs the image x, i.e. learns an inverse mapping. A. Fessler, “Deep dictionary-transform learning for image reconstruction,” in IEEE International Symposium on Biomedical Imaging (ISBI) ,2018, pp. Y. Chun, “Training deep learning based denoisers without ground truth data,” in Advances in Neural Information Processing Systems , 2018, pp. Jeong, “Compressed sensing MRI reconstruction using a generative adversarial network with a cyclic loss,” IEEE Transactions on Medical Imaging , vol. C. Hansen, “On instabilities of deep learn-ing in image reconstruction-does ai come at a cost?” arXiv preprint arXiv:1902.05300 ,2019. G. Dimakis, “Compressed sensing with deep image prior and learned regularization,” arXiv preprint arXiv:1806.06438 , 2018.
[263] Inverse problems as statistics - IOPscience — This paper discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results. It shows how statistical measures of performance apply to techniques used in practical inverse problems, such as regularization, maximum penalized likelihood, Bayes estimation and the
[264] Inverse Problems as Statistics | Department of Statistics — Standard statistical concepts, questions, and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency, and various forms of optimality apply to inverse problems. This article discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of
[265] A comprehensive analysis of the impacts of Image Resolution and ... — To assess the quality of reconstructed images, it is essential to measure the similarity and difference of the constructed image with the phantom to assess the accuracy of the method.
[266] A Systematic Review of Medical Image Quality Assessment — High-quality medical images are crucial for accurate diagnosis, treatment planning, and disease monitoring . With the continuous advancement of imaging technologies, robust MIQA methodologies are essential to ensure the reliability and efficacy of these technologies in clinical practice.
[267] Geological realism in hydrogeological and geophysical inverse modeling ... — Section 2 formulates the inverse problem as the integration of the information offered by geophysical ... The concept of a training image can be seen as a vehicle to convey the prior conceptual geological knowledge that is to be combined with other sources of ... For example, geophysical amplitude data can be used as it may reveal
[268] PDF — underdetermined geophysical inverse problem, there are an infinite number of models that can fit the geophysical data to the desired degree: the problem is non-unique. Additional information is essential for a unique solution. Incorporating previous geological knowledge, and combining several complimentary types of geophysical data collected
[269] SE - Integration of automatic implicit geological modelling in ... — Abstract. We propose and evaluate methods for the integration of automatic implicit geological modelling into the geophysical (potential field) inversion process. The objective is to enforce structural geological realism and to consider geological observations in a level set inversion, which inverts for the location of the boundaries between rock units. We propose two approaches. In the first