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Layerwise Tunable Lifting Scheme for the Convolutional Neural Network

Abdumannon Yovkochov, An Le, Sungbal Seo, You-Suk Bae, Truong Nguyen

arXiv:2609.09827v1cs.CV

TL;DR

CNN downsampling can lose fine-grained information and alias high-frequency content. This work introduces layer-wise tunable lattice lifting schemes for low-pass, high-pass, and sequential wavelet tuning, achieving consistent improvements in classification and anomaly detection.

  • Problem

    CNN downsampling operations are deterministically low-pass, causing fine-grained information loss and aliasing, while many alternatives retain only low-pass information.

  • Method

    The paper introduces layer-wise tunable biorthogonal wavelet lifting schemes with low-pass, high-pass, and sequential strategies integrated into ResNet-18.

  • Results

    The proposed schemes consistently improve image classification and anomaly detection, including 45.32% for LS-LayLatt-HP and 99.75% AUROC for LS-LayLatt-Sequential on MVTec-AD hazelnut.

  • Takeaways & Limitations

    Wavelet-domain tuning improves task-specific capture of high- and low-frequency features while preserving perfect reconstruction constraints.

Abstract

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This work introduces a family of tunable lifting schemes for biorthogonal wavelet filter banks. We propose three lifting strategies: low-pass tuning (LS-LayLatt-LP), high-pass tuning (LS-LayLatt-HP), and a sequential lifting scheme that jointly adapts low- and high-frequency branches (LS-LayLatt-Sequential). All proposed designs are formulated using a lattice-based lifting structure, which guarantees invertibility and stability for arbitrary parameter values within the lifting functions. We evaluated the proposed methods by integrating them into a ResNet-18 backbone for image classification on the Describable Textures Dataset (DTD), as well as for anomaly detection on hazelnut images from the MVTec-AD dataset and private KRC102S dataset. Experimental results demonstrate consistent performance improvements across all evaluated tasks.

I. INTRODUCTION

CNN downsampling can lose fine-grained, high-frequency information, while existing wavelet alternatives often retain only low-pass content. This work introduces layer-wise tunable wavelet lifting schemes and evaluates them across classification and anomaly detection tasks.

  • I. INTRODUCTION: Tunable lifting schemes are proposed to address information loss and limited flexibility in CNN downsampling.The approach targets both high-frequency detail preservation and low-frequency feature learning.
  • I. INTRODUCTION: High-frequency components preserve discriminative visual information, whereas retaining only low-frequency content can remove crack structures.The DTD-cracked example illustrates why suppressing high-frequency details can harm classification and anomaly detection.
  • I. INTRODUCTION: The method studies high-pass tuning, low-pass tuning, and sequential tuning of both frequency branches.The sequential scheme updates low-pass and then high-pass representations while preserving perfect reconstruction.
  • I. INTRODUCTION: The proposed methods are integrated into ResNet-18 and evaluated on DTD classification plus MVTec-AD and KRC102S anomaly detection.The experiments use CFLOW-AD for anomaly detection.
  • II. RELATED WORKS: Layer-wise tuning lets each downsampling layer learn its own wavelet parameters and supports longer low-pass or high-pass filters.The parameters are learned through cross-entropy–based optimization, allowing frequency responses to adapt across the network hierarchy.

III. PROPOSED METHODS

The proposed filter banks use a lattice-based lifting structure with separate functions for tuning high- and low-pass filters. Its determinant-based construction preserves invertibility, while the resulting filters retain FIR and linear-phase properties.

  • III. PROPOSED METHODS: The lifting scheme uses P_k(z) and U_k(z) to tune the high-pass and low-pass filters, respectively.A K-step scheme is constructed by composing these lifting operations.
  • III. PROPOSED METHODS: Figure 2 depicts the analysis filter-bank structure containing the two lifting functions U_k(z^2) and P_k(z^2).The figure provides the structural view corresponding to the general formulation.
  • III. PROPOSED METHODS: Table I reports filter coefficients after one-step high-pass tuning.The supplied passage identifies the table scope but does not provide coefficient values.
  • III. PROPOSED METHODS: Unit diagonal entries make each lifting matrix determinant 1, so the complete structure is invertible for arbitrary lifting coefficients.The synthesis filters can be exactly reconstructed by reversing the analysis operations.
  • III. PROPOSED METHODS: The lifting construction preserves finite impulse response and linear-phase properties through delays and finite-length filtering operations.Delayed linear combinations with real coefficients introduce only constant phase shifts, while alignment delays maintain a consistent phase center.

B. High-Pass Tuning (LS-LayLatt-HP)

LS-LayLatt-HP increases high-pass filter adaptability by using the low-pass filter within the lifting structure. It sets U_k to zero and learns distinct coefficients for P_k across lifting steps.

  • B. High-Pass Tuning (LS-LayLatt-HP): LS-LayLatt-HP increases high-pass filter length to improve adaptability to high-frequency details such as textures and edges.The method uses the low-pass filter to construct the tunable high-pass branch.

C. Low-Pass Tuning (LS-LayLatt-LP)

LS-LayLatt-LP adapts the low-pass branch to improve coarse feature representation within CNNs. It uses a lifting construction analogous to high-pass tuning but learns parameters in the low-pass update function.

  • C. Low-Pass Tuning (LS-LayLatt-LP): Low-pass tuning targets coarse structures, global context, and long-range dependencies needed for robust feature hierarchies.The method extends tunable lifting to low-frequency representation learning in CNN architectures.
  • C. Low-Pass Tuning (LS-LayLatt-LP): LS-LayLatt-LP sets P_k(z) = 0 and assigns a learnable parameter b_k to U_k(z) across k lifting steps.The complete low-pass lifting function is expressed recursively.

D. Sequential Tuning (LS-LayLatt-Sequential)

The sequential scheme updates the low-pass branch first, then uses it to guide high-pass lifting, coupling both frequency components while retaining flexible filter construction.

  • D. Sequential Tuning (LS-LayLatt-Sequential): Sequential lifting first updates the low-pass branch, then uses that representation to guide a subsequent high-pass update.This staged dependency couples low- and high-frequency components.
  • D. Sequential Tuning (LS-LayLatt-Sequential): The recursive construction represents the sequential wavelet filter-bank update through matrix factors involving P_k(z^2) and U_k(z^2).
  • D. Sequential Tuning (LS-LayLatt-Sequential): The lifting functions use tunable coefficients a_k and b_k defined through P_k(z) and U_k(z) for k from 1 to N.

E. 2D Implementation

The 2D implementation applies learned one-dimensional filters successively across horizontal and vertical dimensions, producing four frequency subbands from the input feature map.

  • E. 2D Implementation: The separable 2D transform applies learned low-pass and high-pass filters successively along horizontal and vertical dimensions.This produces XLL, XLH, XHL, and XHH subbands.
  • E. 2D Implementation: The low-pass and high-pass transform matrices are built from downsampling and Toeplitz convolution matrices derived from the learned filter coefficients.
  • E. 2D Implementation: The resulting 2D discrete wavelet transform operates on an input feature map represented by X.

F. CNN Implementation

The proposed tunable wavelet modules replace conventional CNN downsampling operations and learn layer-specific frequency responses jointly with the network.

  • F. CNN Implementation: Each proposed method replaces conventional downsampling operations in ResNet architectures with a tunable wavelet-based downsampling module.
  • F. CNN Implementation: The models were evaluated within ResNet-18 for DTD classification and within CFLOW-AD for anomaly detection on MVTec-AD hazelnut and KRC102S datasets.
  • F. CNN Implementation: Across all datasets and experimental settings, the proposed models consistently outperformed the baseline methods.

A. Image Classification: DTD

DTD evaluates tunable lifting schemes for texture classification, with high-pass tuning improving over key baselines and sequential tuning achieving the best overall result.

  • A. Image Classification: DTD: DTD contains 5,640 high-resolution images across 47 texture categories and emphasizes high-frequency texture features.
  • A. Image Classification: DTD: The comparison uses ResNet-18 with standard pooling, WaveCNet, LS-BiorUwU, OrthLatt-UwU, and BiorLatt-UwU baselines.
  • A. Image Classification: DTD: 45.32% accuracy for LS-LayLatt-HP improves 11.47 points over ResNet-18 and exceeds LS-BiorUwU-3Steps at 43.50%.
  • A. Image Classification: DTD: The LS-LayLatt-Sequential strategy achieves the best overall result in Table II, supporting joint adaptation of both frequency branches.
  • A. Image Classification: DTD: LS-LayLatt-LP consistently outperforms ResNet-18 but underperforms LS-LayLatt-HP, indicating stronger texture-classification value for high-frequency cues.

B. Anomaly Detection: MVTecAD and KRC102S

The proposed tunable lifting schemes were evaluated in CFLOW-AD on MVTec-AD hazelnut and KRC102S, consistently outperforming baseline methods across the reported settings.

  • CFLOW-AD used DTD-pretrained modified or baseline ResNet-18 feature extractors to evaluate anomaly detection on MVTec-AD hazelnut and KRC102S.The MVTec-AD evaluation included 391 defect-free training samples and 110 test images.
  • The MVTec-AD results include both detection and segmentation AUROC measurements averaged over five random seeds.
  • On KRC102S, low-pass tuning underperformed the ResNet-18 baseline, whereas sequential lifting achieved the best image-level detection performance.KRC102S evaluation was restricted to image-level AUROC because pixel-level ground-truth masks were unavailable.

C. Lifting Parameters Analysis

The study analyzes initialization and computational aspects of the lifting parameters, while reporting overall performance improvements from tunable lifting schemes in classification and anomaly detection.

  • All proposed methods initialize wavelet filters with the Haar/Bior1.1 wavelet, and lifting parameters are optimized with gradient-based training using cross-entropy loss.
  • LS-LayLatt-LP coefficients are initialized from LS-LayLatt-HP filters using perfect-reconstruction relations, while sequential branches reuse coefficients learned by the HP and LP models.
  • Each downsampling site adds only 2K learnable scalars for K lifting steps, while the main overhead comes from four-subband channel widening before convolution.
  • DTD training used 300 epochs with three 100-epoch stage-wise transfer-learning phases and checkpoint initialization between stages.
  • Tunable lifting schemes consistently improved image classification and anomaly detection performance across the evaluated tasks.The conclusion attributes the gains to improved capture of high- and low-frequency features through task-specific tuning.
  • Future work will extend the lifting strategy to different base wavelet filter banks with varying filter lengths.
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