Frequency-Decomposed Similarity with Haar Wavelet

• Frequency-decomposed similarity is a framework that measures similarities in signals or images by decomposing them into frequency-specific components using the Haar wavelet.
• It applies efficient multi-scale Haar transforms with dyadic separation to capture both coarse and fine patterns, enhancing interpretability and computational efficiency.
• Key implementations—cosine-based aggregation, WaveSim, and HaarPSI—demonstrate robust performance in image quality assessment, climate field evaluation, and statistical signal analysis.

Frequency-decomposed similarity with Haar wavelet refers to a methodological framework for quantifying similarity between signals or images by decomposing them into frequency components using the Haar wavelet transform, then comparing the resulting multi-scale representations. This approach captures differences and agreements across spatial or temporal scales, leveraging the advantages of the Haar basis: orthogonality, dyadic band separation, and computational efficiency. Frequency-decomposed similarity metrics have been applied in domains such as image dataset analysis, climate field comparison, perceptual quality assessment, and statistical signal analysis, frequently outperforming global similarity scores by providing interpretable, scale-resolved insight (Yousefzadeh, 2020, &&&1&&&, Kissell et al., 4 Nov 2025, Reisenhofer et al., 2016).

1. Haar Wavelet Transform Fundamentals

The Haar wavelet is the simplest orthogonal wavelet, with scaling (father) and wavelet (mother) functions:

• $\varphi(x) = 1$ for $0 \leq x < 1$; 0 otherwise
• $\psi(x) = 1$ for $0 \leq x < \frac12$, $-1$ for $\frac12 \leq x < 1$; 0 otherwise

The associated filter coefficients (unit $\ell_2$ norm) are $h = [1/\sqrt{2}, 1/\sqrt{2}]$ (low-pass) and $g = [1/\sqrt{2}, -1/\sqrt{2}]$ (high-pass). Applying the discrete wavelet transform (DWT) via iterated filtering and downsampling yields multi-level, separable decompositions for images and signals. In 2D, each level decomposes the signal into four subsampled subbands: LL (approximation), LH (horizontal detail), HL (vertical detail), and HH (diagonal detail) (Yousefzadeh, 2020, Accarino et al., 16 Dec 2025).

The decomposition is performed recursively, typically to $J = \lfloor \log_2 \min(M,N) \rfloor$ levels for an $M \times N$ image. Fast algorithms such as Mallat’s pyramid exploit filter separability for $O(P)$ operations with $P$ pixels (Yousefzadeh, 2020, Reisenhofer et al., 2016).

2. Frequency-Decomposed Similarity: General Methodology

Frequency-decomposed similarity quantifies agreement between two signals (or images) not in the spatial/temporal domain, but in their wavelet (i.e., frequency–scale–space) representations. At a conceptual level:

• Signals/images $A$ and $B$ are transformed to produce, at each level $j$ and subband $S$, coefficient matrices $W_S^{(j)}(A)$, $W_S^{(j)}(B)$.
• Similarity at each scale–subband pair is evaluated, e.g., using cross-correlation, normed Euclidean distance, or more elaborate functionals tailored to perceptual invariance or application-specific patterns.
• The similarities across bands and scales are then aggregated, often with scale-dependent weights to control the relative importance of fine or coarse patterns (Yousefzadeh, 2020, Kissell et al., 4 Nov 2025).

This approach yields a multi-level “similarity spectrum” capturing which frequency bands and spatial/temporal patterns are aligned or distinct between the objects under comparison.

3. Principal Implementations and Variants

a) Cosine Similarity-Based Aggregation

A central implementation (Yousefzadeh, 2020) uses normalized cross-correlation (cosine similarity) per subband:

$$S_S^{(j)}(A,B) = \frac{\sum_{m,n} W_S^{(j)}(A)[m,n]\, W_S^{(j)}(B)[m,n]} {\sqrt{\sum W_S^{(j)}(A)[m,n]^2} \sqrt{\sum W_S^{(j)}(B)[m,n]^2}}$$

The aggregate similarity is

$$S(A,B) = \sum_{j=1}^J \sum_{S \in \{LL,LH,HL,HH\}} w_{j,S} \cdot S_S^{(j)}(A,B)$$

Weights $w_{j,LL} \propto \alpha^j$, $w_{j,subband \neq LL} \propto \beta^j$ ($\alpha > \beta$) emphasize coarse versus fine scales. Renormalization ensures $\sum_{j,S} w_{j,S} = 1$.

This method, validated on CIFAR-10/100 and landmark datasets, rapidly identifies redundancy, near-duplicates, and outliers. Pivoted-QR can be used to focus on the most informative coefficients, reducing computation time and improving specificity (Yousefzadeh, 2020).

b) Multi-component WaveSim Metric

The WaveSim framework (Accarino et al., 16 Dec 2025) extends Haar-based similarity metrics with three orthogonal components computed per scale:

• Magnitude: Quantifies similarity in mean wavelet energy at each scale, using

$$\mathcal{M}^s = 1 - \delta^s;\quad \delta^s = \frac{|\bar E_X^s-\bar E_Y^s|}{\bar E_X^s + \bar E_Y^s + \varepsilon}$$

• Displacement: Measures alignment of energy centers-of-mass along rows/cols, with

$$\mathcal{D}^s = (1 - \Delta_{\mathrm{lat}}^s)(1 - \Delta_{\mathrm{lon}}^s)$$

• Structure: Assesses similarity of sorted, centralized and normalized coefficient patterns, with a cosine similarity and an amplitude penalty.

Scores are multiplicatively blended and then aggregated across scales with exponentiation and weighting:

$$\text{WaveSim}(X,Y) = \sum_{s=1}^S w^s \left[ (\mathcal{M}^s)^\alpha (\mathcal{D}^s)^\beta (\mathcal{S}^s)^\gamma \right]$$

(with $\alpha=\beta=\gamma=1$ by default).

This enables interpretability along physical modes of dissimilarity (energy, spatial shift, pattern organization), critical for complex spatial field comparison (Accarino et al., 16 Dec 2025).

c) Wavelet-Based Statistical Correlations

The work in (Kissell et al., 4 Nov 2025) provides scale-resolved correlational analysis using Haar detail coefficients (for signals):

• At each scale $j$, compute correlation $R_{XY}(j)$ between corresponding detail coefficient vectors.
• Optionally, use partial correlation to control for other scales.
• Parseval’s identity gives that global Pearson correlation can be additively decomposed as a weighted sum of scale-wise correlations.

This enables signal dependence to be interpreted as a function of frequency band (octave), which is infeasible with global methods like Pearson correlation (Kissell et al., 4 Nov 2025).

d) HaarPSI: Perceptual Quality Index

HaarPSI (Reisenhofer et al., 2016) is an efficient similarity index for full-reference image quality assessment:

• Uses Haar detail coefficients at two high-frequency scales in two orientations (horizontal/vertical).
• Local similarity at each coefficient is measured by a stabilized ratio; nonlinearity is applied via a logistic function.
• A coarser-scale Haar response provides a saliency weight per pixel.
• The final similarity merges orientation- and scale-combined similarities with per-location weights, using an inverted logistic mapping and squaring to optimize agreement with human perception.

Empirical performance on large image quality datasets shows HaarPSI achieves state-of-the-art results with far lower computational cost than methods such as SSIM or FSIM (Reisenhofer et al., 2016).

4. Computational Properties and Scaling

All Haar-based similarity approaches leverage the minimal support and orthogonality of the Haar wavelet. Complexity for 2D Haar DWT is $O(MN J)$, with $J = O(\log_2 \min(M,N))$. Most similarity measures (inner products, cosine similarity, energy, etc.) reduce to $O(P)$ operations when precomputed coefficients are used. For large-scale datasets, dimensionality reduction by feature selection (e.g., QR pivoting) and parallel clustering enable rapid application to tens of thousands of images or signals (Yousefzadeh, 2020, Reisenhofer et al., 2016).

Edge effects are typically controlled by symmetric extension, and coefficient normalization (e.g., by $1/\sqrt{2}$ filter taps) maintains metric invariance (Accarino et al., 16 Dec 2025).

5. Empirical Performance and Applications

Applications of frequency-decomposed similarity with Haar wavelet include:

• Image dataset analysis: Identification of redundancy and outliers; clustering; pre-training analysis of classification datasets without requiring trained models. For CIFAR-10, with $J=4$ levels ($\sim$3,072 coeffs/image), informative feature selection and clustering recover reported redundancies and enable efficient cluster pruning (Yousefzadeh, 2020).
• Climate and spatial field evaluation: Scale-specific magnitude, displacement, and structure similarity scores provide interpretable diagnostics not possible with pointwise norms. For example, WaveSim quantifies physical scale errors, spatial displacements, and structural misalignments in Earth System Model outputs (Accarino et al., 16 Dec 2025).
• Perceptual image quality assessment: HaarPSI achieves superior correlation with subjective human scores compared to SSIM, FSIM, VSI on LIVE, TID2008/2013, CSIQ datasets at a fraction of the computational cost (Reisenhofer et al., 2016).
• Statistical signal dependence: Wavelet correlograms diagnose which frequency bands carry signal relationships, critical in applications such as turbulence analysis, AR modeling, and economic time series (Kissell et al., 4 Nov 2025).

6. Interpretability, Robustness, and Limitations

The scale-resolved nature of Haar-based similarity metrics provides interpretability—one can attribute matches/mismatches to specific frequency bands, spatial displacements, or structures. For high-dimensional data, this supports robust redundancy detection and model generalization insights (e.g., association between low similarity and test errors in CIFAR-100 classes) (Yousefzadeh, 2020).

Pearson and Kendall correlations can both be used at each scale, offering trade-offs between linear and monotonic dependence sensitivity (Kissell et al., 4 Nov 2025).

A limitation is that Haar, while fast and maximally localized, lacks frequency selectivity and directional richness compared to more complex wavelets (e.g., Daubechies, Symlets), which may be desirable in certain contexts. However, for speed and simplicity on large datasets or spatial fields, Haar-based decompositions remain preferred (Yousefzadeh, 2020, Reisenhofer et al., 2016).

7. Comparative Summary

Approach	Domain	Key Features	Reference
Cosine/Haar aggregation	Image, Dataset	Multi-level, cross-band similarity	(Yousefzadeh, 2020)
WaveSim	Climate/Fields	Magnitude, displacement, structure components	(Accarino et al., 16 Dec 2025)
Wavelet correlogram	Signals/Series	Scale-resolved Pearson/Kendall correlation	(Kissell et al., 4 Nov 2025)
HaarPSI	Image Quality	Local Haar similarity, saliency weighted	(Reisenhofer et al., 2016)

All of these methods rely on the same foundational operations—Haar wavelet transform, coefficient similarity at each scale, and judicious aggregation—demonstrating the versatility and interpretability of frequency-decomposed similarity for analyzing structured data across a range of applications.