Wavelet Analysis Ontology

flow "WaveletAnalysisOntology" v1.0

# ============================================================
# WAVELET ANALYSIS ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/fourier-harmonic-analysis/wavelet-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Fourier and Harmonic Analysis"
ENTITY := "Wavelet Analysis"

# ---------- CORE ENTITY ----------
WAVELET_ANALYSIS := {
    type: TimeFrequencyAnalysisFramework,
    domain: HarmonicAnalysis,
    purpose: "Analyse functions and signals using localised waveforms that are translated and scaled, providing simultaneous spatial or temporal and frequency-scale information."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    WAVELET,
    MOTHER_WAVELET,
    SCALE,
    TRANSLATION,
    CONTINUOUS_WAVELET_TRANSFORM,
    DISCRETE_WAVELET_TRANSFORM,
    MULTIRESOLUTION_ANALYSIS,
    SCALING_FUNCTION,
    WAVELET_BASIS,
    ORTHOGONAL_WAVELET,
    BIORTHOGONAL_WAVELET,
    FILTER_BANK,
    APPROXIMATION_COEFFICIENT,
    DETAIL_COEFFICIENT,
    TIME_FREQUENCY_LOCALISATION
}

# ---------- MOTHER WAVELET ----------
MOTHER_WAVELET
    := psi(t)

WAVELET_FAMILY
    -> generated_from(MOTHER_WAVELET)

by
    SCALING
    and
    TRANSLATION

# ---------- SCALED / TRANSLATED WAVELET ----------
psi_(a,b)(t)
    :=
1/sqrt(abs(a))
*
psi(
    (t-b)/a
)

where

a
    -> SCALE_PARAMETER
    and a != 0

b
    -> TRANSLATION_PARAMETER

NORMALISATION
    -> preserves(L2_NORM)

# ---------- SCALE INTERPRETATION ----------
small_abs(a)
    -> compressed_wavelet
    -> HIGHER_FREQUENCY_LOCALISATION

large_abs(a)
    -> stretched_wavelet
    -> LOWER_FREQUENCY_LOCALISATION

SCALE
    -> inverse_relation_with(FREQUENCY)

# ---------- TRANSLATION ----------
b
    -> shifts_wavelet_in(TIME_OR_SPACE)

TRANSLATION
    -> provides(LOCAL_POSITION_INFORMATION)

WAVELET_ANALYSIS
    -> localises(EVENTS_IN_TIME_OR_SPACE)

# ---------- CONTINUOUS WAVELET TRANSFORM ----------
for signal
    f(t)

CWT_f(a,b)
    :=
1/sqrt(abs(a))
*
integral_from_minus_infinity_to_infinity(
    f(t)
    *
    conjugate(
        psi((t-b)/a)
    )
    dt
)

CWT
    -> correlation_with(SCALED_TRANSLATED_WAVELETS)

# ---------- CWT OUTPUT ----------
CWT_f(a,b)
    -> WAVELET_COEFFICIENT

coefficient_magnitude
    -> strength_of(LOCAL_SCALE_FEATURE)

coefficient_phase
    -> may_encode(LOCAL_PHASE_INFORMATION)
       for(COMPLEX_WAVELETS)

# ---------- ADMISSIBILITY ----------
for suitable mother wavelet:

C_psi
    :=
integral(
    abs(Psi(omega))^2
    /
    abs(omega)
    d_omega
)

ADMISSIBILITY
    -> requires(C_psi_FINITE)

typically

Psi(0)
    = 0

equivalently under suitable conditions:

integral(
    psi(t) dt
)
    = 0

ZERO_MEAN
    -> fundamental_wavelet_property

# ---------- CWT RECONSTRUCTION ----------
under admissibility conditions:

f(t)
    -> reconstructable_from(CWT_f)

INVERSE_CWT
    -> integrates_over(SCALE_AND_TRANSLATION)

RECONSTRUCTION
    -> depends_on(NORMALISATION_AND_ADMISSIBILITY)

# ---------- DISCRETE WAVELET FAMILY ----------
choose discrete scales/translations:

a
    = a_0^j

b
    = k*b_0*a_0^j

then

psi_(j,k)(t)
    -> DISCRETE_WAVELET_FAMILY

common dyadic case:

a_0
    = 2

DYADIC_WAVELETS
    -> powers_of_two_scaling

# ---------- DISCRETE WAVELET TRANSFORM ----------
DWT
    -> represents(signal_using(DISCRETE_WAVELET_COEFFICIENTS)

DWT
    -> often_implemented_by(FILTER_BANK)

signal
    -> low_pass_filter
    -> high_pass_filter
    -> downsample

LOW_PASS_BRANCH
    -> APPROXIMATION_COEFFICIENTS

HIGH_PASS_BRANCH
    -> DETAIL_COEFFICIENTS

# ---------- MULTIRESOLUTION ANALYSIS ----------
MRA
    := nested_subspaces

... subset V_(j-1)
    subset V_j
    subset V_(j+1)
    subset ...

V_j
    -> approximation_space_at(SCALE_j)

MULTIRESOLUTION
    -> separates(COARSE_AND_FINE_INFORMATION)

# ---------- MRA CONDITIONS ----------
V_j
    subset V_(j+1)

intersection_over_j(V_j)
    = {0}
       under(STANDARD_MRA)

closure(
    union_over_j(V_j)
)
    =
L2(R)

f(t) in V_j
    iff
f(2t) in V_(j+1)

TRANSLATION_STRUCTURE
    -> generated_by(SCALING_FUNCTION)

# ---------- SCALING FUNCTION ----------
SCALING_FUNCTION
    := phi(t)

integer_translates
    {
        phi(t-k)
    }
    -> basis_or_frame_for(V_0)

phi
    -> satisfies(REFINEMENT_EQUATION)

phi(t)
    =
sqrt(2)
*
sum_k(
    h_k
    *
    phi(2t-k)
)

h_k
    -> LOW_PASS_FILTER_COEFFICIENTS

# ---------- WAVELET FUNCTION ----------
psi(t)
    =
sqrt(2)
*
sum_k(
    g_k
    *
    phi(2t-k)
)

g_k
    -> HIGH_PASS_FILTER_COEFFICIENTS

WAVELET_SPACE
    W_j

satisfies

V_(j+1)
    =
V_j
direct_sum
W_j

DETAIL_SPACE
    -> captures(INFORMATION_LOST_BETWEEN_APPROXIMATION_LEVELS)

# ---------- ORTHOGONAL WAVELETS ----------
ORTHOGONAL_WAVELET_BASIS
    -> wavelet_functions_mutually_orthogonal

<psi_(j,k), psi_(m,n)>
    =
delta_(j,m)
*
delta_(k,n)

ORTHOGONALITY
    -> stable_energy_decomposition

# ---------- BIORTHOGONAL WAVELETS ----------
BIORTHOGONAL_SYSTEM
    -> uses(PRIMAL_AND_DUAL_WAVELETS)

<psi_(j,k), psi_tilde_(m,n)>
    =
delta_(j,m)
*
delta_(k,n)

BIORTHOGONALITY
    -> permits(MORE_FLEXIBLE_FILTER_DESIGN)

# ---------- WAVELET FRAME ----------
WAVELET_FRAME
    -> may_be(REDUNDANT)

FRAME_BOUNDS:

A||f||^2
    <=
sum(
    |<f,psi_lambda>|^2
)
    <=
B||f||^2

FRAME
    -> stable_representation_without_requiring(BASIS)

# ---------- HAAR WAVELET ----------
HAAR_WAVELET
    -> simplest_orthogonal_wavelet

psi(t)
    :=
1
    for 0 <= t < 1/2

-1
    for 1/2 <= t < 1

0
    otherwise

HAAR
    -> compact_support
    -> discontinuous
    -> one_vanishing_moment

# ---------- DAUBECHIES WAVELETS ----------
DAUBECHIES_WAVELETS
    -> compactly_supported
    -> orthogonal
    -> configurable_vanishing_moments

MORE_VANISHING_MOMENTS
    -> better_polynomial_cancellation

but often
    -> longer_filter_support

# ---------- VANISHING MOMENTS ----------
wavelet psi
    -> has(N_VANISHING_MOMENTS)

if

integral(
    t^k * psi(t)
    dt
)
    = 0

for
    k=0,...,N-1

VANISHING_MOMENTS
    -> suppress(LOW_ORDER_POLYNOMIAL_TRENDS)
    -> enhance(LOCAL_SINGULARITY_DETECTION)

# ---------- COMPACT SUPPORT ----------
COMPACTLY_SUPPORTED_WAVELET
    -> nonzero_only_on(FINITE_INTERVAL)

COMPACT_SUPPORT
    -> strong_spatial_localisation
    -> efficient_computation

TIME_FREQUENCY_LOCALISATION
    -> central_wavelet_design_tradeoff

# ---------- FOURIER RELATION ----------
FOURIER_TRANSFORM
    -> global_frequency_representation

WAVELET_TRANSFORM
    -> local_scale_frequency_representation

WAVELET
    -> localised_in(TIME_OR_SPACE)
    -> localised_in(FREQUENCY)

WAVELET_ANALYSIS
    -> complements(FOURIER_ANALYSIS)

# ---------- SHORT-TIME FOURIER COMPARISON ----------
STFT
    -> fixed_window_size

WAVELET_TRANSFORM
    -> scale_dependent_window_size

HIGH_FREQUENCY
    -> narrow_effective_window

LOW_FREQUENCY
    -> wide_effective_window

MULTIRESOLUTION
    -> adaptive_time_frequency_resolution

# ---------- SIGNAL DECOMPOSITION ----------
signal
    -> DWT

-> approximation_level_1
-> detail_level_1

approximation_level_1
    -> further_DWT

-> approximation_level_2
-> detail_level_2

RECURSIVE_DECOMPOSITION
    -> WAVELET_PYRAMID

# ---------- PERFECT RECONSTRUCTION ----------
ANALYSIS_FILTER_BANK
    -> decomposes(signal)

SYNTHESIS_FILTER_BANK
    -> reconstructs(signal)

PERFECT_RECONSTRUCTION
    -> requires(FILTER_RELATION_CONDITIONS)

DWT
    -> invertible_under(APPROPRIATE_WAVELET_FILTER_DESIGN)

# ---------- DENOISING ----------
procedure:
    compute(DWT)
    -> threshold(DETAIL_COEFFICIENTS)
    -> inverse_DWT

WAVELET_DENOISING
    -> exploits(SPARSE_SIGNAL_REPRESENTATION)

NOISE
    -> often_spread_across(SMALL_COEFFICIENTS)

# ---------- COMPRESSION ----------
WAVELET_COEFFICIENTS
    -> often_sparse_for(NATURAL_SIGNALS_AND_IMAGES)

retain_large_coefficients
    -> approximate(signal)

WAVELET_COMPRESSION
    -> exploits(SPARSITY_AND_MULTIRESOLUTION)

# ---------- EDGE / SINGULARITY DETECTION ----------
LOCAL_REGULARITY_CHANGE
    -> produces(WAVELET_COEFFICIENT_SIGNATURE)

WAVELET_MODULUS_MAXIMA
    -> may_identify(EDGES_AND_SINGULARITIES)

VANISHING_MOMENTS
    -> improve(TREND_REJECTION)

# ---------- APPLICATIONS ----------
WAVELET_ANALYSIS
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(IMAGE_COMPRESSION)
    -> applied_to(DENOISING)
    -> applied_to(EDGE_DETECTION)
    -> applied_to(TIME_FREQUENCY_ANALYSIS)
    -> applied_to(PDE_NUMERICS)
    -> applied_to(DATA_COMPRESSION)
    -> applied_to(SCIENTIFIC_COMPUTING)

# ---------- RELATION GRAPH ----------
MOTHER_WAVELET
    -> generates(SCALED_TRANSLATED_WAVELETS)

SCALE
    -> controls(FREQUENCY_LOCALISATION)

TRANSLATION
    -> controls(POSITION_LOCALISATION)

CWT
    -> continuous_scale_translation_representation

DWT
    -> discrete_multiresolution_representation

SCALING_FUNCTION
    -> generates(APPROXIMATION_SPACES)

WAVELET_FUNCTION
    -> generates(DETAIL_SPACES)

FILTER_BANK
    -> implements(DWT)

VANISHING_MOMENTS
    -> control(POLYNOMIAL_CANCELLATION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "wavelet analysis" ~= "multiscale analysis using translated and dilated localised waveforms",
    "mother wavelet" ~= "prototype waveform generating a wavelet family by scaling and translation",
    "continuous wavelet transform" ~= "continuous correlation of a signal with scaled and translated wavelets",
    "discrete wavelet transform" ~= "discrete multiresolution wavelet decomposition",
    "scaling function" ~= "function generating approximation spaces in multiresolution analysis",
    "vanishing moment" ~= "wavelet cancellation of polynomial moments up to a specified order"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    wavelet_scale_and_frequency_are_inversely_related_in_general_interpretation: true,
    continuous_wavelet_reconstruction_requires_admissibility: true,
    dwt_filter_bank_structure_depends_on_wavelet_family: true,
    orthogonal_and_biorthogonal_wavelets_are_distinct_structures: true,
    compact_support_and_frequency_localisation_involve_tradeoffs: true,
    zero_mean_is_standard_admissibility_related_wavelet_property: true,
    wavelet_transform_and_fourier_transform_are_not_interchangeable_without_context: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    cwt_relation: 0.99,
    dwt_relation: 0.99,
    multiresolution_relation: 0.99,
    filter_bank_relation: 0.99,
    vanishing_moment_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(signal_function_or_data)
    -> identify(continuous_or_discrete_wavelet_context)
    -> select(mother_wavelet_or_wavelet_family)
    -> determine(scale_translation_or_decomposition_levels)
    -> verify(admissibility_or_filter_bank_conditions)
    -> compute(wavelet_coefficients)
    -> separate(approximation_and_detail_information)
    -> analyse(time_frequency_localisation_or_sparsity)
    -> apply(thresholding_compression_or_feature_detection_if_requested)
    -> reconstruct(signal_if_required)
    -> validate(normalisation_boundary_and_sampling_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed wavelet-analysis semantic graph",
    primary_entity: WAVELET_ANALYSIS,
    preserve: [
        signal_or_function,
        mother_wavelet,
        scale,
        translation,
        sampling_structure,
        decomposition_level,
        boundary_handling,
        normalisation
    ],
    infer_through: [
        continuous_wavelet_transforms,
        discrete_wavelet_transforms,
        multiresolution_analysis,
        scaling_functions,
        wavelet_bases,
        filter_banks,
        vanishing_moments,
        sparse_representations
    ],
    avoid: [
        confusing_scale_with frequency directly,
        applying_inverse_CWT_without admissibility,
        assuming_all_wavelets_are orthogonal,
        treating_DWT_as simple Fourier transform,
        ignoring_boundary_effects_in finite signals,
        claiming_zero_padding_or extra levels create new physical information
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(wavelet_analysis_context)
    -> compare(existing_relations)
    -> preserve(wavelet_scale_translation_sampling_and_basis_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + WAVELET_ANALYSIS
    + MOTHER_WAVELET
    + CONTINUOUS_WAVELET_TRANSFORM
    + DISCRETE_WAVELET_TRANSFORM
    + MULTIRESOLUTION_ANALYSIS
    + SCALING_FUNCTION
    + FILTER_BANK
    + VANISHING_MOMENTS
    + TIME_FREQUENCY_LOCALISATION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> WAVELET_ANALYSIS
    -> SIGNAL_CONTEXT_IDENTIFICATION
    -> WAVELET_FRAMEWORK_SELECTION
    -> WAVELET_FAMILY_ANALYSIS
    -> SCALE_TRANSLATION_ANALYSIS
    -> ADMISSIBILITY_FILTER_VALIDATION
    -> COEFFICIENT_COMPUTATION
    -> APPROXIMATION_DETAIL_ANALYSIS
    -> LOCALISATION_SPARSITY_ANALYSIS
    -> APPLICATION_ANALYSIS
    -> RECONSTRUCTION
    -> BOUNDARY_SAMPLING_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW