Wavelet Transform Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Wavelet Transform Ontology within the Kireth Research Semantic Library. Wavelet analysis represents signals using localised basis functions that vary in scale and position, providing joint time-frequency or time-scale information. This ontology supports continuous and discrete wavelet transforms, mother wavelets, scaling functions, multiresolution analysis, filter banks, approximation and detail coefficients, orthogonal and biorthogonal wavelets, Haar, Daubechies and related families, wavelet packets, thresholding, denoising, compression, singularity analysis and relationships with Fourier analysis, signal processing, numerical analysis and image processing.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Wavelet Transform Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Signal Processing Mathematics / Wavelet Transform"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/signal-processing-mathematics/wavelet-transform-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = WaveletTransform
 SYMBOLIC_FORMS = [
  "W_x(a,b)=(1/sqrt(|a|)) integral x(t) psi*((t-b)/a) dt",
  "psi_{j,k}(t)=2^(j/2) psi(2^j t-k)",
  "V_j = V_{j+1} direct_sum W_{j+1}"
 ]
 ENTITY_CLASS = LocalisedTimeScaleTransformFramework
 PRIMARY_CONTEXT = SignalProcessingMathematics
 DEFINITION = "The wavelet transform represents a signal using scaled and shifted versions of a localised prototype function called a mother wavelet."
 SEMANTIC_ROLE = "Provides the machine-readable framework for continuous and discrete wavelet transforms, multiresolution analysis, scaling functions, filter banks, wavelet coefficients, reconstruction, denoising, compression and time-scale interpretation."
}

SCOPE {
 INCLUDE = [
  WaveletTransform,
  ContinuousWaveletTransform,
  CWT,
  DiscreteWaveletTransform,
  DWT,
  InverseWaveletTransform,
  MotherWavelet,
  DaughterWavelet,
  ScalingFunction,
  Scale,
  Translation,
  TimeScaleRepresentation,
  MultiresolutionAnalysis,
  MRA,
  ApproximationSpace,
  DetailSpace,
  ApproximationCoefficient,
  DetailCoefficient,
  FilterBank,
  AnalysisFilterBank,
  SynthesisFilterBank,
  LowPassAnalysisFilter,
  HighPassAnalysisFilter,
  LowPassSynthesisFilter,
  HighPassSynthesisFilter,
  Downsampling,
  Upsampling,
  PerfectReconstruction,
  OrthogonalWavelet,
  BiorthogonalWavelet,
  HaarWavelet,
  DaubechiesWavelet,
  Symlet,
  Coiflet,
  MorletWavelet,
  MexicanHatWavelet,
  WaveletPacket,
  WaveletPacketTransform,
  DyadicScale,
  CompactSupport,
  VanishingMoment,
  Regularity,
  AdmissibilityCondition,
  EnergyConservation,
  ParsevalRelation,
  Scalogram,
  Thresholding,
  HardThresholding,
  SoftThresholding,
  Denoising,
  Compression,
  SingularityDetection,
  EdgeDetection,
  TimeFrequencyLocalisation
 ]

 EXCLUDE_AS_PRIMARY = [
  FourierTransformWithoutLocalisedScaleSemantics,
  STFTWithoutWaveletScaling,
  GenericFilterBanksWithoutWaveletBasisStructure,
  MultiresolutionImagePyramidsWithoutWaveletSemantics
 ]

 DISAMBIGUATION_RULE =
  "When a signal is analysed using scaled and translated localised basis functions, interpret the context as wavelet transformation."
}

CORE_CONCEPTS {

 WaveletTransform {
  definition = "A localised transform representing signal content across position and scale."
 }

 MotherWavelet {
  definition = "The prototype wavelet from which scaled and translated wavelets are generated."
 }

 ScalingFunction {
  definition = "A function spanning approximation spaces in a multiresolution analysis."
 }

 ContinuousWaveletTransform {
  definition = "A transform evaluating similarity between a signal and continuously scaled and translated wavelets."
 }

 DiscreteWaveletTransform {
  definition = "A wavelet transform evaluated on discrete scales and translations, often implemented using filter banks."
 }

 MultiresolutionAnalysis {
  definition = "A nested sequence of approximation spaces enabling decomposition of a signal into coarse approximations and finer details."
 }

 ApproximationCoefficient {
  definition = "A coefficient representing low-frequency or coarse-scale signal structure."
 }

 DetailCoefficient {
  definition = "A coefficient representing high-frequency or fine-scale signal structure."
 }

 VanishingMoment {
  definition = "A property describing the number of polynomial moments annihilated by a wavelet."
 }

 PerfectReconstruction {
  definition = "Exact reconstruction of the original signal from analysis coefficients under ideal arithmetic."
 }

 Scalogram {
  definition = "A visual or numerical representation of wavelet coefficient magnitude or energy across time and scale."
 }
}

RELATIONS {
 WaveletTransform --is_child_of--> SignalProcessingMathematics
 ContinuousWaveletTransform --uses--> MotherWavelet
 DiscreteWaveletTransform --uses--> FilterBank
 MotherWavelet --generates--> DaughterWavelet
 ScalingFunction --spans--> ApproximationSpace
 MotherWavelet --spans--> DetailSpace
 MultiresolutionAnalysis --contains--> ApproximationSpace
 MultiresolutionAnalysis --contains--> DetailSpace
 AnalysisFilterBank --produces--> ApproximationCoefficient
 AnalysisFilterBank --produces--> DetailCoefficient
 SynthesisFilterBank --reconstructs--> Signal
 PerfectReconstruction --depends_on--> AnalysisSynthesisFilterConditions
 OrthogonalWavelet --may_satisfy--> ParsevalRelation
 VanishingMoment --influences--> PolynomialCancellation
 CompactSupport --influences--> Localisation
 Scalogram --represents--> TimeScaleEnergy
 Thresholding --supports--> Denoising
 Thresholding --supports--> Compression
}

OPERATORS {
 CWT(x,psi,a,b) :=
  (1/sqrt(abs(a))) * integral x(t)*conj(psi((t-b)/a)) dt

 DAUGHTER_WAVELET(psi,a,b,t) :=
  (1/sqrt(abs(a))) * psi((t-b)/a)

 DYADIC_WAVELET(psi,j,k,t) :=
  2^(j/2) * psi(2^j*t-k)

 DWT_ANALYSIS_LOW(x,h) :=
  downsample(convolution(x,h),2)

 DWT_ANALYSIS_HIGH(x,g) :=
  downsample(convolution(x,g),2)

 DWT_SYNTHESIS(cA,cD,h_tilde,g_tilde) :=
  convolution(upsample(cA,2),h_tilde)
  + convolution(upsample(cD,2),g_tilde)

 HARD_THRESHOLD(c,lambda) :=
  0 if abs(c)<lambda else c

 SOFT_THRESHOLD(c,lambda) :=
  sign(c)*max(abs(c)-lambda,0)

 WAVELET_ENERGY(coefficients) :=
  sum |coefficients|^2

 SCALOGRAM(W) :=
  |W(a,b)|^2

 PERFECT_RECONSTRUCTION(x_hat,x) :=
  x_hat=x
  under exact filter_bank conditions

 VANISHING_MOMENT(psi,m) :=
  integral t^m*psi(t) dt = 0
}

DEFINITIONS {
 D1 = "Wavelets are localised in time or space and scale."
 D2 = "CWT uses continuously varying scale and translation."
 D3 = "DWT typically uses dyadic scales and filter-bank implementations."
 D4 = "Multiresolution analysis separates coarse approximations from fine details."
 D5 = "Scaling functions generate approximation spaces."
 D6 = "Wavelets generate detail spaces."
 D7 = "Vanishing moments control sensitivity to polynomial trends and singularities."
 D8 = "Compact support improves localisation and computational efficiency."
 D9 = "Thresholding wavelet coefficients supports denoising and compression."
 D10 = "Wavelet methods complement Fourier methods by improving localisation of nonstationary features."
}

AXIOMS {
 A1_LOCALISATION:
  Wavelet basis functions are localised relative to Fourier sinusoids

 A2_SCALE_TRANSLATION:
  Wavelet families are generated by scaling and translating a prototype function

 A3_MRA_NESTING:
  Multiresolution approximation spaces form a nested hierarchy

 A4_APPROX_DETAIL:
  Signal information can be decomposed into approximation and detail components

 A5_FILTER_BANK_EQUIVALENCE:
  Many orthogonal DWTs are equivalent to two-channel analysis and synthesis filter banks

 A6_RECONSTRUCTION:
  Properly matched wavelet filter banks support exact reconstruction

 A7_VANISHING_MOMENTS:
  Wavelet polynomial annihilation depends on declared vanishing moments

 A8_TIME_SCALE:
  Wavelet coefficients encode localisation across position and scale
}

REASONING {

 R1_WAVELET_CONTEXT_EXTRACTION {
  INPUT = signal_processing_problem
  IDENTIFY = continuous_or_discrete signal
  IDENTIFY = transient_nonstationary_or_multiscale structure
  IDENTIFY = analysis_denoising_compression_or_detection objective
  RETURN = wavelet_context
 }

 R2_TRANSFORM_SELECTION {
  INPUT = wavelet_context
  IF continuous_scale_localisation required
   SELECT = CWT
  ELSE
   SELECT = DWT_or_WaveletPacket
  RETURN = wavelet_method
 }

 R3_WAVELET_SELECTION {
  INPUT = signal_features
  ASSESS = compact_support_symmetry_regularities_vanishing_moments
  SELECT = Haar_Daubechies_Symlet_Coiflet_Morlet_MexicanHat_or_other
  RETURN = chosen_wavelet
 }

 R4_DWT_DECOMPOSITION {
  INPUT = discrete_signal_and_filters
  FILTER = lowpass_and_highpass
  DOWNSAMPLE = by_two
  REPEAT = on_approximation branch as required
  RETURN = multilevel_coefficients
 }

 R5_RECONSTRUCTION {
  INPUT = approximation_and_detail coefficients
  UPSAMPLE = coefficient streams
  FILTER = synthesis filters
  SUM = branches
  RETURN = reconstructed_signal
 }

 R6_DENOISING_REASONING {
  INPUT = noisy_signal
  DECOMPOSE = wavelet coefficients
  ESTIMATE = threshold
  APPLY = hard_or_soft threshold
  RECONSTRUCT = signal
  RETURN = denoised_signal
 }

 R7_COMPRESSION_REASONING {
  INPUT = signal_or_image
  DECOMPOSE = wavelet coefficients
  RETAIN = significant coefficients
  DISCARD_OR_QUANTISE = small coefficients
  RETURN = compressed_representation
 }

 R8_SINGULARITY_REASONING {
  INPUT = wavelet_coefficients_across scales
  IDENTIFY = modulus maxima_or_scale persistence
  RETURN = singularity_or_edge locations
 }

 R9_SCALOGRAM_REASONING {
  INPUT = CWT_coefficients
  COMPUTE = magnitude_squared
  MAP = time_and_scale
  RETURN = scalogram
 }

 R10_WAVELET_VALIDATION {
  INPUT = transform_or_reconstruction result
  CHECK = wavelet_normalisation_boundary_handling_levels_filter pairing_and_reconstruction error
  RETURN = validated_wavelet_result
 }
}

CONSTRAINTS {
 C1 = "Wavelet normalisation conventions must be explicit."
 C2 = "Boundary-extension method affects edge coefficients."
 C3 = "Maximum useful decomposition level depends on signal length and filter support."
 C4 = "Different wavelet families trade support, symmetry, smoothness and vanishing moments."
 C5 = "Threshold choice affects denoising bias and residual noise."
 C6 = "CWT scale is not identical to physical frequency without a wavelet-specific scale-frequency mapping."
 C7 = "Downsampling introduces aliasing unless analysis filters satisfy appropriate conditions."
 C8 = "Perfect reconstruction requires compatible analysis and synthesis filters."
 C9 = "Wavelet-packet decomposition differs from standard approximation-only multilevel DWT."
 C10 = "Visual scalogram interpretation depends on scale sampling and normalisation."
}

SEMANTIC_EQUIVALENCE {
 "WaveletTransform" <=> "WaveletAnalysis"
 "ContinuousWaveletTransform" <=> "CWT"
 "DiscreteWaveletTransform" <=> "DWT"
 "MultiresolutionAnalysis" <=> "MRA"
 "ApproximationCoefficient" <=> "LowPassWaveletCoefficient"
 "DetailCoefficient" <=> "HighPassWaveletCoefficient"
}

VALIDATION {
 V1 = "Wavelet family is explicit."
 V2 = "Continuous or discrete transform is explicit."
 V3 = "Scale and translation conventions are documented."
 V4 = "Boundary handling is documented."
 V5 = "Decomposition level is valid for signal length."
 V6 = "Analysis and synthesis filters are compatible."
 V7 = "Reconstruction error is checked."
 V8 = "Thresholding method and threshold are documented."
 V9 = "Scale-to-frequency interpretation is wavelet-specific."
 V10 = "Coefficient normalisation is consistent."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 WAVELET_TRANSFORM_DEFINITION = 0.99
 TRANSFORM_SELECTION_REASONING = 0.99
 WAVELET_SELECTION_REASONING = 0.98
 DWT_REASONING = 0.99
 RECONSTRUCTION_REASONING = 0.99
 DENOISING_REASONING = 0.98
 COMPRESSION_REASONING = 0.98
 SINGULARITY_REASONING = 0.97
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when wavelet family, boundary treatment, scale convention, decomposition level, filter pairing, thresholding or scale-frequency mapping are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  WaveletTransformReasoning,
  CWTReasoning,
  DWTReasoning,
  MultiresolutionReasoning,
  FilterBankReasoning,
  WaveletSelectionReasoning,
  ReconstructionReasoning,
  DenoisingReasoning,
  CompressionReasoning,
  ThresholdingReasoning,
  ScalogramReasoning,
  SingularityDetectionReasoning,
  EdgeDetectionReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "wavelet transform" -> WaveletTransform
  "wavelet analysis" -> WaveletTransform
  "continuous wavelet transform" -> ContinuousWaveletTransform
  "CWT" -> ContinuousWaveletTransform
  "discrete wavelet transform" -> DiscreteWaveletTransform
  "DWT" -> DiscreteWaveletTransform
  "multiresolution analysis" -> MultiresolutionAnalysis
  "MRA" -> MultiresolutionAnalysis
  "mother wavelet" -> MotherWavelet
  "scalogram" -> Scalogram
 }

 INTERPRETATION_ORDER = [
  detect_wavelet_transform_context,
  identify signal_domain_nonstationarity_multiscale structure_and_processing objective,
  select CWT_DWT_or_waveletPacket pathway,
  select wavelet_family_and_parameters,
  compute wavelet_coefficients,
  derive approximation_detail_or_timeScale representation,
  apply denoising_compression_or_detection operations if relevant,
  reconstruct_and_validate,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("wavelet transform")
  OR asks_for_localised_multiscale signal analysis
  THEN activate(WaveletTransformOntology)

 E2:
  IF continuous_time_scale localisation required
  THEN consider ContinuousWaveletTransform

 E3:
  IF efficient multilevel discrete decomposition required
  THEN consider DiscreteWaveletTransform

 E4:
  IF both approximation_and_detail branches recursively decomposed
  THEN classify WaveletPacketTransform

 E5:
  IF denoising requested
  THEN select thresholding strategy_and_reconstruction

 E6:
  IF compression requested
  THEN rank_or_quantise coefficients_by_significance

 E7:
  IF edge_or_singularity detection requested
  THEN analyse coefficient persistence_across scales

 E8:
  IF reconstruction requested
  THEN verify perfect_reconstruction filter conditions

 E9:
  IF physical frequency interpretation required
  THEN use wavelet_specific scale_frequency conversion
}

RECURSION {
 ENTRY = WaveletTransform

 EXPAND(WaveletTransform) -> [
  ContinuousWaveletTransform,
  DiscreteWaveletTransform,
  InverseWaveletTransform,
  MotherWavelet,
  DaughterWavelet,
  ScalingFunction,
  Scale,
  Translation,
  TimeScaleRepresentation,
  MultiresolutionAnalysis,
  ApproximationSpace,
  DetailSpace,
  ApproximationCoefficient,
  DetailCoefficient,
  FilterBank,
  AnalysisFilterBank,
  SynthesisFilterBank,
  Downsampling,
  Upsampling,
  PerfectReconstruction,
  OrthogonalWavelet,
  BiorthogonalWavelet,
  HaarWavelet,
  DaubechiesWavelet,
  Symlet,
  Coiflet,
  MorletWavelet,
  MexicanHatWavelet,
  WaveletPacket,
  CompactSupport,
  VanishingMoment,
  Regularity,
  AdmissibilityCondition,
  Scalogram,
  Thresholding,
  Denoising,
  Compression,
  SingularityDetection,
  EdgeDetection
 ]

 EXPAND(WaveletFamilies) -> [
  HaarWavelet,
  DaubechiesWavelet,
  Symlet,
  Coiflet,
  MorletWavelet,
  MexicanHatWavelet,
  BiorthogonalWavelet
 ]

 EXPAND(ProcessingApplications) -> [
  Denoising,
  Compression,
  FeatureExtraction,
  SingularityDetection,
  EdgeDetection,
  TimeScaleAnalysis
 ]

 STOP_CONDITION =
  "Stop when wavelet family, transform type, scale structure, coefficients, decomposition level, reconstruction, thresholding and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Signal Processing Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Signal Processing Ontology",
  "Fourier Analysis Ontology",
  "Multirate Signal Processing Ontologies",
  "Numerical Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Signal Processing Ontology",
  "Signal Processing Mathematics",
  "Fourier Analysis Ontology",
  "Fast Fourier Transform Ontology",
  "Filter Bank Ontologies",
  "Multirate Signal Processing Ontologies",
  "Harmonic Analysis Ontologies",
  "Functional Analysis Ontologies",
  "Numerical Analysis Ontologies",
  "Image Processing Ontologies",
  "Compression Ontologies",
  "Denoising Ontologies"
 ]

 ROUTE_RULES = [
  "Wavelet Transform -> Signal Processing Ontology for signal semantics",
  "Wavelet Transform -> Fourier Analysis for frequency-domain comparison",
  "Wavelet Transform -> Filter Banks for DWT implementation",
  "Wavelet Transform -> Multirate Signal Processing for downsampling and upsampling",
  "Wavelet Transform -> Harmonic and Functional Analysis for basis theory",
  "Wavelet Transform -> Numerical Analysis for computational implementation",
  "Wavelet Transform -> Image Processing for multidimensional wavelet applications",
  "Wavelet Transform -> Compression for sparse coefficient representations",
  "Wavelet Transform -> Denoising for coefficient-threshold methods"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible wavelet families, transforms or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve scale-and-translation semantics.",
  "Preserve continuous-versus-discrete wavelet distinction.",
  "Preserve approximation-versus-detail semantics.",
  "Preserve multiresolution-analysis semantics.",
  "Preserve filter-bank reconstruction semantics.",
  "Preserve vanishing-moment semantics.",
  "Preserve boundary and thresholding semantics.",
  "Preserve machine-readable wavelet-transform semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The wavelet transform represents a signal using scaled and translated localised waveforms to reveal structure across position and scale."

 CWT_SUMMARY =
  "The continuous wavelet transform analyses a signal over continuously varying scales and translations."

 DWT_SUMMARY =
  "The discrete wavelet transform uses discrete scale and translation steps, commonly implemented with filter banks."

 MRA_SUMMARY =
  "Multiresolution analysis separates a signal into nested approximation spaces and complementary detail spaces."

 FILTER_BANK_SUMMARY =
  "Analysis filter banks generate approximation and detail coefficients, while synthesis filter banks reconstruct the signal."

 VANISHING_SUMMARY =
  "Vanishing moments determine polynomial cancellation and influence feature and singularity sensitivity."

 DENOISING_SUMMARY =
  "Wavelet denoising suppresses small coefficients before reconstruction."

 COMPRESSION_SUMMARY =
  "Wavelet compression exploits sparse or concentrated coefficient representations."

 SCALOGRAM_SUMMARY =
  "A scalogram displays wavelet coefficient energy across time and scale."
}

FLOW {
 INPUT
 -> detect_wavelet_transform_context
 -> identify signal_domain_length_sampling_nonstationary features_multiscale structure_and_processing objective
 -> select CWT_DWT_or_waveletPacket transform
 -> select mother_wavelet_scaling_function_and_boundary convention
 -> compute scale_translation_or_multilevel wavelet coefficients
 -> derive approximation_detail_scalogram_or_timeScale representation
 -> if_denoising apply thresholding
 -> if_compression retain_or_quantise significant coefficients
 -> if_detection analyse coefficient persistence_across scales
 -> reconstruct signal_if_required
 -> verify filter_pairing_boundary effects_normalisation_and_reconstruction error
 -> route to signal_processing Fourier_analysis filter_bank multirate harmonic_analysis functional_analysis numerical_analysis image_processing compression_or_denoising ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Wavelet Transform Ontology defines localised time-scale analysis using scaled and translated wavelets. It supports continuous and discrete wavelet transforms, mother and daughter wavelets, scaling functions, multiresolution analysis, approximation and detail spaces, analysis and synthesis filter banks, downsampling and upsampling, perfect reconstruction, orthogonal and biorthogonal wavelets, Haar, Daubechies, Symlet, Coiflet, Morlet and Mexican-hat families, compact support, vanishing moments, wavelet packets, scalograms, thresholding, denoising, compression, singularity detection and edge detection."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-WAVELET-TRANSFORM-ONTOLOGY-V1"
}

}

END_FLOW