|
A multiresolution analysis of the [[Lp space|Lebesgue space]] <math>L^2(\mathbb{R})</math> consists of a [[sequence]] of nested [[linear subspace|subspaces]]
::<math>\{0\}\subset \dots\subset V_1\subset V_0\subset V_{-1}\subset\dots\subset V_{-n}\subset V_{-(n+1)}\subset\dots\subset L^2(\R)</math>
that satisfies certain [[self-similarity]] relations in time-space and scale-frequency, as well as [[Complete metric space|completeness]] and regularity relations.
* ''Regularity'' demands that the model [[linear subspace|subspace]] ''V<sub>0</sub>'' be generated as the [[linear hull]] ([[algebraic closure|algebraically]] or even [[topologically closed]]) of the integer shifts of one or a finite number of generating functions <math>\phi</math> or <math>\phi_1,\dots,\phi_r</math>. Those integer shifts should at least form a frame for the subspace <math> V_0\subset L^2(\R) </math>, which imposes certain conditions on the decay at [[infinity]]. The generating functions are also known as '''[[Wavelet#Scaling function|scaling functions]]''' or '''[[father wavelets]]'''. In most cases one demands of those functions to be [[piecewise continuous]] with [[compact support]].
* ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be [[dense set|dense]] in <math> L^2(\R) </math>, and that they are not too redundant, i.e., their [[intersection]] should only contain the [[zero element]].
== Algorithms ==
This section explores the core algorithms that form the foundation of multiresolution analysis, enabling its wide range of applications.
=== Subdivision Schemes ===
'''Subdivision schemes''' are [[Iterative method|iterative algorithms]] used to generate smooth curves and surfaces from an initial set of control points. These schemes progressively refine the control polygon or mesh to produce increasingly detailed representations.<ref name="cohen2003">{{Cite book | author=Albert Cohen | title=Chapter 2 - Multiresolution approximation | editor=Albert Cohen | publisher=Elsevier | volume=32 | year=2003 | pages=43–153 | url=https://www.sciencedirect.com/science/article/pii/S016820240380005X }}</ref><ref name="bruce1998">{{Cite book | author=Bruce W. Suter | title=Chapter 5 - Wavelet Signal Processing | editor=Bruce W. Suter | series=Wavelet Analysis and Its Applications | publisher=Academic Press | volume=8 | year=1998 | pages=167–190 | url=https://www.sciencedirect.com/science/article/pii/S1874608X98800502 }}</ref>
Key characteristics of subdivision schemes include:
* '''Masks''': Define the rules for generating new points at each refinement step.
* '''Flexibility''': Enable local modifications at varying resolution levels, making them ideal for multiresolution editing.
A notable example is the '''Lane-Riesenfeld algorithm''', which constructs smooth [[B-spline]] curves by iteratively averaging control points. Subdivision schemes are widely applied in geometric modeling, particularly for creating and editing shapes with varying levels of detail.
=== Discrete Wavelet Transform (DWT) ===
The '''[[Discrete wavelet transform|Discrete Wavelet Transform]] (DWT)''' is a pivotal algorithm in multiresolution analysis, offering a multiscale representation of signals through decomposition into different frequency sub-bands.
Key features of DWT:
* '''Decomposition''': The signal is passed through high-pass and low-pass filters, yielding detail coefficients (high frequencies) and approximation coefficients (low frequencies).
* '''Reconstruction''': The original signal is reconstructed using inverse filters.
* '''Efficiency''': With a computational complexity of <math> O(N) </math>, DWT is well-suited for large-scale data processing tasks like image compression and feature extraction.
=== Pyramidal Algorithms ===
'''Pyramidal algorithms''' leverage a [[hierarchical structure]], akin to a pyramid, where each level represents the signal at a progressively coarser resolution.<ref name="cohen2003" />
Core steps include:
* '''Decomposition''': Downsampling and smoothing the signal at each level to create a hierarchy of representations.
* '''Reconstruction''': Upsampling and combining information from different levels to restore the original signal.
These algorithms are computationally efficient and extensively used in image processing, computer vision, and pattern recognition.
=== Fast Decomposition and Reconstruction Algorithms ===
The '''Mallat algorithm''' is a fast, hierarchical method for wavelet decomposition and reconstruction. It processes data at multiple scales, enabling efficient computation of wavelet coefficients and their reconstruction.<ref name="bruce1998" />
== Applications ==
=== Image Fusion in Remote Sensing ===
MRA is instrumental in merging images from sensors with varying resolutions and spectral bands <ref>{{Cite book | author=Bruno Aiazzi, Stefano Baronti, Massimo Selva | title=2 - Image fusion through multiresolution oversampled decompositions | editor=Tania Stathaki | publisher=Academic Press | year=2008 | pages=27–66 | isbn=978-0-12-372529-5 | url=https://www.sciencedirect.com/science/article/pii/B9780123725295000020 | ___location=Oxford }}</ref>
. For instance, a high-resolution [[panchromatic image]] can be fused with a low-resolution [[multispectral image]], producing a single output with enhanced spatial and spectral resolution. Techniques like the "[[À trous algorithm|à trous]]" wavelet algorithm and [[Laplacian pyramid|Laplacian pyramids]] preserve spatial connectivity and minimize artifacts.
=== Multiresolution Editing in Geometric Modeling ===
MRA enhances geometric modeling by enabling efficient representation and manipulation of complex shapes<ref name="Georges2008">{{Cite book | author=Georges-Pierre Bonneau, Gershon Elber, Stefanie Hahmann, Basile Sauvage | editor=Leila De Floriani, Michela Spagnuolo | title=Multiresolution Analysis | publisher=Springer Berlin Heidelberg | year=2008 | pages=83–114 | isbn=978-3-540-33265-7 | url=https://doi.org/10.1007/978-3-540-33265-7_3 | ___location=Berlin, Heidelberg }}</ref>:
* '''Hierarchical B-splines''': Allow local and global modifications, simplifying both coarse adjustments and detailed refinements.
* '''Flexible design''': Provides a multiresolution framework for iterative editing, streamlining the creative process in [[computer-aided design]] (CAD).
=== Shape Compression Using Semi-Regular Remeshing ===
MRA contributes to efficient 3D model compression through '''semi-regular remeshing'''<ref name="Georges2008" />:
* '''Simplification''': Reduces unnecessary connectivity and parameterization data.
* '''Parameterization''': Maps the input mesh onto base triangular domains, resulting in a compact representation.
This approach facilitates the efficient storage, transmission, and rendering of 3D models in applications like gaming, virtual reality, and scientific visualization.
=== Emerging Fields ===
* '''Machine Learning''': MRA aids in multiscale feature extraction for tasks like image recognition and natural language processing.
* '''Quantum Wavelet Transforms''': Leveraging quantum computing principles, MRA is being explored for high-dimensional datasets.
* '''[[Seismic analysis|Seismic Analysis]]''': MRA enhances the interpretation of seismic data, identifying subsurface structures with high precision.
== Practical Examples ==
=== Case Study: Image Compression ===
[[JPEG 2000]], a widely used image compression standard, relies on MRA through the DWT. By retaining critical wavelet coefficients, it achieves high compression ratios with minimal loss of image quality.
=== Additional Case Studies ===
* '''Climate Data Analysis''': Detects patterns in multiscale climate datasets.
* '''Financial Market Trends''': Analyzes stock market data for trend detection and anomaly identification.
* '''Medical Imaging''': Enhances feature detection and clarity in MRI and CT scans.
== Important conclusions ==
| 