Cross-Channel Vector Wavelet Front End

1. Integration point: use this module as the first two downsampling stages of a small CNN or U-Net, replacing a stride-2 convolution. The input is a tensor f of shape [batch, 2, H, W]; for RGB, first use a learned or fixed 3-to-2 projection, or generalize the two-channel index patterns to m=3. Construct short FIR filters sampled from compactly supported φ_1, φ_2, ψ_1, ψ_2. 2. Pseudocode: for scale j: for each listed subband r: y[r]=conv2d(f[0], h[r,0]) + conv2d(f[1], h[r,1]); y[r]=downsample2(y[r]); z[r]=pointwise_group_mix_r(y[r]); f_next=concat(z[r]); set h[r,i] from the two components of Ψ^r, and use the corresponding synthesis filters for reconstruction. Include the omitted scaling-function branch as the low-pass output. 3. Mathematical quantities computed exactly are the cross-paired filter coefficients and the synthesis bank. Estimate only normalization constants empirically by checking the discrete frame Gram matrix; rescale filters so its diagonal is one and its off-diagonal Frobenius norm is small. Add an optional diagnostic energy_error=abs(||f||²-Σ_r||y[r]||²)/||f||². Keep the wavelet filters frozen for the first experiment, then compare frozen filters with jointly learned filters initialized from them. 4. First experiment: train a two-channel image denoising CNN on synthetic correlated Gaussian fields or stereo patches, comparing a stride-2 convolution, a channelwise Haar/DWT front end, and this cross-channel bank at matched parameter count and FLOPs. Measure validation PSNR, reconstruction error after analysis/synthesis, and energy error. Success means higher PSNR at equal FLOPs, lower aliasing under 2x downsampling, and reconstruction error near numerical precision before any learned layers.