Layerwise Tunable Lifting Scheme for the Convolutional Neural Network

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.Source: §I · p. 1
• 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.Source: Figure 1 · p. 1, §I · p. 1
• 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.Source: §I · p. 1
• 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.Source: §I · p. 1

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.Source: §II · p. 2

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.Source: A. General Formulation · p. 2, Figure 2 · p. 2
• 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.Source: Figure 2 · p. 2
• Table I reports filter coefficients after one-step high-pass tuning.The supplied passage identifies the table scope but does not provide coefficient values.Source: Table I · p. 2
• 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.Source: A. General Formulation · p. 2
• 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.Source: A. General Formulation · p. 2

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.Source: B. High-Pass Tuning (LS-LayLatt-HP) · p. 2

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.Source: C. Low-Pass Tuning (LS-LayLatt-LP) · p. 2
• 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.Source: C. Low-Pass Tuning (LS-LayLatt-LP) · p. 3

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.Source: D. Sequential Tuning… · p. 3
• The recursive construction represents the sequential wavelet filter-bank update through matrix factors involving P_k(z^2) and U_k(z^2).Source: D. Sequential Tuning… · p. 3
• 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.Source: D. Sequential Tuning… · p. 3

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.Source: E. 2D Implementation · p. 3
• The low-pass and high-pass transform matrices are built from downsampling and Toeplitz convolution matrices derived from the learned filter coefficients.Source: E. 2D Implementation · p. 3
• The resulting 2D discrete wavelet transform operates on an input feature map represented by X.Source: E. 2D Implementation · p. 3

F. CNN Implementation

• Each proposed method replaces conventional downsampling operations in ResNet architectures with a tunable wavelet-based downsampling module.Source: F. CNN Implementation · p. 3
• The models were evaluated within ResNet-18 for DTD classification and within CFLOW-AD for anomaly detection on MVTec-AD hazelnut and KRC102S datasets.Source: §IV · p. 3
• Across all datasets and experimental settings, the proposed models consistently outperformed the baseline methods.Source: §IV · p. 3

A. Image Classification: DTD

• DTD contains 5,640 high-resolution images across 47 texture categories and emphasizes high-frequency texture features.Source: A. Image Classification: DTD · p. 3
• The comparison uses ResNet-18 with standard pooling, WaveCNet, LS-BiorUwU, OrthLatt-UwU, and BiorLatt-UwU baselines.Source: A. Image Classification: DTD · p. 3
• 45.32% accuracy for LS-LayLatt-HP improves 11.47 points over ResNet-18 and exceeds LS-BiorUwU-3Steps at 43.50%.Source: 1) LS-LayLatt Tuning Across different… · p. 3
• The LS-LayLatt-Sequential strategy achieves the best overall result in Table II, supporting joint adaptation of both frequency branches.Source: 1) LS-LayLatt Tuning Across different… · pp. 3–4, Table II · p. 4
• LS-LayLatt-LP consistently outperforms ResNet-18 but underperforms LS-LayLatt-HP, indicating stronger texture-classification value for high-frequency cues.Source: 1) LS-LayLatt Tuning Across different… · p. 4

• 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.Source: B. Anomaly Detection: MVTecAD and… · p. 4
• The MVTec-AD results include both detection and segmentation AUROC measurements averaged over five random seeds.Source: Table III · p. 4
• 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.Source: B. Anomaly Detection: MVTecAD and… · p. 4, Table IV · p. 4

• 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.Source: C. Lifting Parameters Analysis · p. 5
• 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.Source: C. Lifting Parameters Analysis · p. 5
• Each downsampling site adds only 2K learnable scalars for K lifting steps, while the main overhead comes from four-subband channel widening before convolution.Source: D. Computational Complexity and… · p. 5
• DTD training used 300 epochs with three 100-epoch stage-wise transfer-learning phases and checkpoint initialization between stages.Source: D. Computational Complexity and… · p. 5
• 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.Source: §V · p. 5
• Future work will extend the lifting strategy to different base wavelet filter banks with varying filter lengths.Source: §V · p. 5