A survey of computational frameworks for solving the acoustic inverse problem in three-dimensional photoacoustic computed tomography
---
paper_title: Handbook of mathematical methods in imaging

paper_content:

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.
---
paper_title: Photoacoustic imaging and spectroscopy

paper_content:

Photoacoustics Photoacoustic Monopole Radiation: Waves from Objects with Symmetry in One, Two, and Three Dimensions, Gerald. J. Diebold Photoacoustic Doppler Effect and Flow Sensing, Hui Fang and Lihong V. Wang Modeling Photoacoustic Propagation in Tissue Using k-Space Techniques, Ben Cox and Paul C. Beard Photoacoustic Image Reconstruction: Algorithm, Resolution, and View Universal Back-Projection Algorithm for Photoacoustic Tomography, Minghua Xu and Lihong V. Wang Analysis of Spatial Resolution in Photoacoustic Tomography, Minghua Xu and Lihong V. Wang Limited View Thermoacoustic Tomography, Yuan Xu, Lihong V. Wang, Gaik Ambartsoumian, and Peter Kuchment Photoacoustic Image Reconstruction: Mathematical Perspectives Recovering a Function from Its Spherical Mean Values in Two and Three Dimensions, David Finch and Rakesh On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography, Mark Agranovsky, Peter Kuchment, and Leonid Kunyansky Photoacoustic and Thermoacoustic Tomography: Consistency Conditions and the Partial Scan Problem, Sarah K. Patch Photoacoustic I Part mage Reconstruction: Advanced Algorithms Time Reversal in Photoacoustic or Thermoacoustic Tomography, Yuan Xu and Lihong V. Wang Quantitative Photoacoustic Imaging: Measurement of Absolute Chromophore Concentrations for Physiological and Molecular Imaging, Paul C. Beard, Jan G. Laufer, Ben Cox, and Simon R. Arridge Image Reconstruction in Optoacoustic Tomography Accounting for Frequency-Dependent Attenuation, Patrick La Riviere, Jin Zhang, and Mark A. Anastasio Half-Time Image Reconstruction in Photoacoustic Tomography, Mark A. Anastasio, Jin Zhang, Xiaochuan Pan, and Lihong V. Wang Adaptive and Robust Methods for Thermoacoustic and Photoacoustic Tomography, Bin Guo and Jian Li Ultrasonic Detectors for Photoacoustic Tomography Photoacoustic Imaging with a Double-Ring Sensor, Roy G.M. Kolkman, Ton G. van Leeuwen, and Wiendelt Steenbergen Photoacoustic Tomography Based on Ring-Shaped Virtual Point Ultrasonic Detector, Xinmai Yang and Lihong V. Wang 3D Photoacoustic Scanner Based on an Optical Ultrasound-Mapping System for Imaging Superficial Vascular Anatomy In Vivo, Paul C. Beard, Edward Z. Zhang, and Jan G. Laufer High-Frequency Optoacoustic Transducers for Ultrasonic and Photoacoustic Imaging, Shai Ashkenazi, Yang Hou, Sheng-Wen Huang, Takashi Buma, and Matt O'Donnell Interferometry-Based Optoacoustic Tomography, Hagyong Kihm, Stefan A. Carp, and Vasan Venugopalan Photoacoustic Tomography with Integrating Area and Line Detectors, Gunther Paltauf, Robert Nuster, Markus Haltmeier, and Peter Burgholzer Focused Scanning Photoacoustic Part Tomography with Laser Excitation Dark-Field Confocal Photoacoustic Microscopy, Hao F. Zhang, Konstantin Maslov, and Lihong V. Wang Deep-Penetrating Reflection-Mode Photoacoustic Imaging, Kwang Hyun Song and Lihong V. Wang Array-Based Photoacoustic Tomography with Laser Excitation Combined Ultrasound and Photoacoustic System for Real-Time High-Contrast Imaging Using a Linear Array Transducer, Michael Jaeger and Martin Frenz Fast Photoacoustic Imaging System Based on 320-Element Linear Transducer Array, Da Xing Design and Characterization of an Array-Based Photoacoustic Tomographic System for Small Animal Imaging, John Gamelin, Andres Aguirre, Anastasios Maurudis, Fei Huang, Diego Castillo, Lihong V. Wang, and Quing Zhu T hermoacoustic Tomography with Microwave Excitation Thermoacoustic Computed Tomography of the Breast, Robert A. Kruger Microwave-Induced Acoustic (Thermoacoustic) Tomography, Lihong V. Wang Functional and Molecular Photoacoustic Tomography Functional and Molecular Photoacoustic Tomography of Small-Animal Brains, Xueding Wang and Lihong V. Wang Photoacoustic Imaging of Gene Expression in Small Animals In Vivo, Roger J. Zemp, Li Li, and Lihong V. Wang Photoacoustic Tomography w Part ith Nanoparticles Gold and Silver Nanoparticles as Contrast Agents for Optoacoustic Tomography, Alexander A. Oraevsky Photoacoustic Flow Measurements with Gold Nanoparticles, Pai-Chi Li, Chen-Wei Wei, Sheng-Wen Huang, Chao-Kang Liao, and Churng-Ren Wang Photoacoustic Imaging and Therapy Utilizing Molecular Specific Plasmonic Nanoparticles, Stanislav Emelianov, Srivalleesha Mallidi, Timothy Larson, and Konstantin Sokolov Photoacoustic Tomography of Breasts, Joints, and Vessels Optoacoustic Tomography of the Breast, Alexander A. Oraevsky Photoacoustic Mammography with a Flat Detection Geometry, Srirang Manohar, Ton G. van Leeuwen, Joost M. Klaase, Frank M. van den Engh, and Wiendelt Steenbergen Photoacoustic Tomography: A New Imaging Technology for Inflammatory Arthritis, Xueding Wang, David L. Chamberland, Paul L. Carson, and J. Brian Fowlkes Intravascular Photoacoustic Imaging of Atherosclerosis, Shriram Sethuraman, Bo Wang, Richard Smalling, and Stanislav Emelianov Photoacoustic Depth Determination and Imaging of Port Wine Stain Birthmarks, John A. Viator, Roy G.M. Kolkman, and Wiendelt Steenbergen Multimodal Imaging Thermoacoustic Reconstruction in Acoustically Heterogeneous Media with the Aid of Ultrasound Tomography, Xing Jin and Lihong V. Wang Role of Photoacoustic and Ultrasound Imaging in Photothermal Therapy, Jignesh Shah, Suhyun Park, Salavat Aglyamov, and Stanislav Emelianov Index
---
paper_title: Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications

paper_content:

Preface. Acknowledgments. Special Functions Of Preface. Acknowledgements. Special Functions of the Fractional Calculus. Gamma Function. Mittag-Leffler Function. Wright Function. Fractional Derivatives and Integrals. The Name of the Game. Grunwald-Letnikov Fractional Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Sequential Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of Fractional Derivatives. Mellin Transforms of Fractional Derivatives. Existence and Uniqueness Theorems. Linear Fractional Differential Equations. Fractional Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of Solution. Dependence of a Solution on Initial Conditions. The Laplace Transform Method. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. Fractional Green's Function. Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General Case: n-term Equation. Other Methods for the Solution of Fractional-order Equations. The Mellin Transform Method. Power Series Method. Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. Numerical Evaluation of Fractional Derivatives. Approximation of Fractional Derivatives. The "Short-Memory" Principle. Order of Approximation. Computation of Coefficients. Higher-order Approximations. Numerical Solution of Fractional Differential Equations. Initial Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions. The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations. Fractional-Order Systems and Controllers. Fractional-Order Systems and Fractional-Order Controllers. Example. On Viscoelasticity. Bode's Analysis of Feedback Amplifiers. Fractional Capacitor Theory. Electrical Circuits. Electroanalytical Chemistry. Electrode-Electrolyte Interface. Fractional Multipoles. Biology. Fractional Diffusion Equations. Control Theory. Fitting of Experimental Data. The "Fractional-Order" Physics? Bibliography. Tables of Fractional Derivatives. Index.
---
paper_title: Handbook of mathematical methods in imaging

paper_content:

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.
---
paper_title: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light

paper_content:

Historical introduction 1. Basic properties of the electromagnetic field 2. Electromagnetic potentials and polarization 3. Foundations of geometrical optics 4. Geometrical theory of optical imaging 5. Geometrical theory of aberrations 6. Image-forming instruments 7. Elements of the theory of interference and interferometers 8. Elements of the theory of diffraction 9. The diffraction theory of aberrations 10. Interference and diffraction with partially coherent light 11. Rigorous diffraction theory 12. Diffraction of light by ultrasonic waves 13. Scattering from inhomogeneous media 14. Optics of metals 15. Optics of crystals 16. Appendices Author index Subject index.
---
paper_title: Handbook of mathematical methods in imaging

paper_content:

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.
---
paper_title: Fixed-Point Algorithms for Inverse Problems in Science and Engineering

paper_content:

"Fixed-Point Algorithms for Inverse Problems in Science and Engineering" presents some ofthe most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems. This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry. Topics presented include: Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory. Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods. Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas. Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.
---
paper_title: Emission Tomography: The Fundamentals of PET and SPECT

paper_content:

PET and SPECT are two of todays most important medical-imaging methods, providing images that reveal subtle information about physiological processes in humans and animals. "Emission Tomography: The Fundamentals of PET and SPECT" explains the physics and engineering principles of these important functional-imaging methods. The technology of emission tomography is covered in detail, including historical origins, scientific and mathematical foundations, imaging systems and their components, image reconstruction and analysis, simulation techniques, and clinical and laboratory applications. The book describes the state of the art of emission tomography, including all facets of conventional SPECT and PET, as well as contemporary topics, such as iterative image reconstruction, small-animal imaging, and PET/CT systems. This book is intended as a textbook and reference resource for graduate students, researchers, medical physicists, biomedical engineers, and professional engineers and physicists in the medical-imaging industry. Thorough tutorials of fundamental and advanced topics are presented by dozens of the leading researchers in PET and SPECT. SPECT has long been a mainstay of clinical imaging, and PET is now one of the worlds fastest growing medical imaging techniques, owing to its dramatic contributions to cancer imaging and other applications. "Emission Tomography: The Fundamentals of PET and SPECT" is an essential resource for understanding the technology of SPECT and PET, the most widely used forms of molecular imaging. The features are: contains thorough tutorial treatments, coupled with coverage of advanced topics; three of the four holders of the prestigious Institute of Electrical and Electronics Engineers Medical Imaging Scientist Award are chapter contributors; and includes color artwork.
---
paper_title: Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity

paper_content:

This book presents the state of the art in sparse and multiscale image and signal processing, covering linear multiscale transforms, such as wavelet, ridgelet, or curvelet transforms, and non-linear multiscale transforms based on the median and mathematical morphology operators. Recent concepts of sparsity and morphological diversity are described and exploited for various problems such as denoising, inverse problem regularization, sparse signal decomposition, blind source separation, and compressed sensing. This book weds theory and practice in examining applications in areas such as astronomy, biology, physics, digital media, and forensics. A final chapter explores a paradigm shift in signal processing, showing that previous limits to information sampling and extraction can be overcome in very significant ways. Matlab and IDL code accompany these methods and applications to reproduce the experiments and illustrate the reasoning and methodology of the research available for download at the associated Web site.
---
paper_title: Gradient methods for minimizing composite objective function

paper_content:

In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both in convex and nonconvex cases, can be solved with efficiency typical for the good part of the objective. For convex problems of the above structure, we consider primal and dual variants of the gradient method (converge as O (1/k)), and an accelerated multistep version with convergence rate O (1/k2), where k isthe iteration counter. For all methods, we suggest some efficient "line search" procedures and show that the additional computational work necessary for estimating the unknown problem class parameters can only multiply the complexity of each iteration by a small constant factor. We present also the results of preliminary computational experiments, which confirm the superiority of the accelerated scheme.
---
paper_title: Iterative Methods for Solving Linear Systems

paper_content:

List of Algorithms Preface 1. Introduction. Brief Overview of the State of the Art Notation Review of Relevant Linear Algebra Part I. Krylov Subspace Approximations. 2. Some Iteration Methods. Simple Iteration Orthomin(1) and Steepest Descent Orthomin(2) and CG Orthodir, MINRES, and GMRES Derivation of MINRES and CG from the Lanczos Algorithm 3. Error Bounds for CG, MINRES, and GMRES. Hermitian Problems-CG and MINRES Non-Hermitian Problems-GMRES 4. Effects of Finite Precision Arithmetic. Some Numerical Examples The Lanczos Algorithm A Hypothetical MINRES/CG Implementation A Matrix Completion Problem Orthogonal Polynomials 5. BiCG and Related Methods. The Two-Sided Lanczos Algorithm The Biconjugate Gradient Algorithm The Quasi-Minimal Residual Algorithm Relation Between BiCG and QMR The Conjugate Gradient Squared Algorithm The BiCGSTAB Algorithm Which Method Should I Use? 6. Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result Implications 7. Miscellaneous Issues. Symmetrizing the Problem Error Estimation and Stopping Criteria Attainable Accuracy Multiple Right-Hand Sides and Block Methods Computer Implementation Part II. Preconditioners. 8. Overview and Preconditioned Algorithms. 9. Two Example Problems. The Diffusion Equation The Transport Equation 10. Comparison of Preconditioners. Jacobi, Gauss--Seidel, SOR The Perron--Frobenius Theorem Comparison of Regular Splittings Regular Splittings Used with the CG Algorithm Optimal Diagonal and Block Diagonal Preconditioners 11. Incomplete Decompositions. Incomplete Cholesky Decomposition Modified Incomplete Cholesky Decomposition 12. Multigrid and Domain Decomposition Methods. Multigrid Methods Basic Ideas of Domain Decomposition Methods.
---
paper_title: Photons Plus Ultrasound: Imaging and Sensing

paper_content:

Imaging and sensing based on fusing the compelling features of optical and ultrasonic waves is the fastest growing area of research in biomedical optics. The annual SPIE conference on this topic, co-chaired by both of us, has been doubling in size approximately every three years since 2003 (Fig. 1). As of 2009, this conference has become the largest at SPIE Photonics West. Hybrid modalities such as photoacoustic or optoacoustic tomography can provide deep tissue penetration, high ultrasonic resolution, and speckle-free optical contrast. Applications include in vivo functional and molecular imaging of cancer, neurophysiology, and vascular disease in both animals and humans. Major challenges include development of quantitative imaging, improvement of contrast and resolution, and commercialization of the technology. We look forward to seeing significant preclinical and clinical impact from this emerging technology.
---

Title: A Survey of Computational Frameworks for Solving the Acoustic Inverse Problem in Three-Dimensional Photoacoustic Computed Tomography

Section 1: Introduction to PACT
Description 1: This section introduces Photoacoustic Computed Tomography (PACT), its principles, and its biomedical applications.

Section 2: Canonical forward models in their continuous forms
Description 2: This section describes two types of 3D canonical forward models related to the measurement of propagated pressure wavefields in PACT.

Section 3: Wave-equation based PACT forward model in its continuous form
Description 3: This section discusses the wave-equation-based forward model for PACT, detailing the mathematical framework and physical assumptions.

Section 4: Integral geometry-based forward model
Description 4: This section presents the integral geometry-based forward model, including special cases and common approximations.

Section 5: Review of semidiscrete and discrete forward modeling
Description 5: This section reviews the transition from continuous to discrete forward models in PACT and discusses C-D and D-D operators.

Section 6: D-D forward models for use with homogeneous media
Description 6: This section details specific examples and methods for constructing discrete-domain forward models assuming homogeneous acoustics in the media.

Section 7: D-D forward models for use with heterogeneous acoustic media
Description 7: This section reviews the methods for developing forward models for acoustically heterogeneous media, including finite difference, finite element, and spectral methods.

Section 8: Incorporation of transducer responses in D-D forward models
Description 8: This section explains how transducer responses are incorporated into discrete forward models to improve accuracy.

Section 9: Image reconstruction approaches
Description 9: This section provides an overview of conventional analytic and time-reversal image reconstruction methods applied in PACT.

Section 10: Optimization-based image reconstruction
Description 10: This section focuses on the development and implementation of optimization-based image reconstruction methods for PACT, including regularization strategies.

Section 11: Computation of adjoint operators and objective function gradients in PACT
Description 11: This section covers the computation of adjoint operators and the gradients of objective functions, essential for solving optimization-based reconstruction problems.

Section 12: Joint reconstruction of initial pressure distribution and acoustic medium parameters
Description 12: This section reviews methodologies for joint reconstruction of the initial pressure distribution and the acoustically heterogeneous parameters within the medium.

Section 13: Other challenges
Description 13: This section outlines additional challenges and opportunities for advanced research in practical applications of PACT, including 2D approximations, spatiotemporal reconstruction, and imaging in elastic media.

Section 14: Summary
Description 14: This section provides a summary and conclusion, recapping the survey of computational frameworks for solving the acoustic inverse problem in PACT and highlighting key findings.