TGHop: An Explainable, Efficient and Lightweight Method for Texture Generation

1 Introduction

• Earlier pixel- and patch-based methods can be slow or produce textures with limited diversity.Texture research also matters for understanding texture regularity and randomness and has applications in computer graphics and vision.Source: §1 · p. 1
• Deep-learning texture models can look visually pleasing, but are often large, difficult to explain theoretically, and computationally expensive to train.The authors seek a method that combines small size, transparency, efficient training and inference, and high-quality textures.Source: §1 · p. 2
• TGHop crops patches from an example, analyzes them with PixelHop++ into fine-to-coarse subspaces, generates from the coarsest core, then stitches patches into larger images.The authors report superior texture quality with a small model size and fast speed.Source: §1 · p. 2
• TGHop extends NITES [27] with spatial PCA, which improves generated texture quality and reduces model size, alongside expanded efficiency experiments.The paper also adds results supporting faster computation and smaller model size.Source: §1 · p. 2

2.1 Early Work on Texture Generation

• Pixel- and patch-based methods grow textures from small units, but patch search can be slow and stitched patches may lack diversity; feature-space methods instead exploit texture statistics.Earlier feature transforms included Gabor filters and steerable pyramids, whose design relied heavily on human expertise and lacked adaptivity.Source: §2.1 · p. 2

2.2 Deep-Learning-based (DL-based) Texture Generation

• Deep-learning methods use feature statistics to generate visually pleasing textures, but iterative optimization is costly and pretrained-network features are difficult to justify for texture.Attempts to improve inference speed can increase training time, while other optimization terms add computational burden.Source: §2.2 · pp. 2–3

2.3 Successive Subspace Learning (SSL)

• Successive subspace learning uses data-driven Saab transforms, with PixelHop++ replacing Saab by channel-wise Saab to exploit weak spectral-channel correlations.PixelHop++ supplies hierarchical representations and dimension reduction for TGHop.Source: §2.3 · p. 3

3 Successive Subspace Analysis and Generation

• Affine transforms are represented as linear transforms by augmenting each vector with a constant 1, and the resulting dimensions decrease from D0 through Dn.The dimension ordering is Dn < Dn−1 < · · · < D1 < D0.Source: §3 · p. 3
• For 32 × 32 RGB patches, the augmented source space has dimension 3073, too high for an effective generation model to be found directly.The input dimension is 3P^2, with one added dimension for the augmented space.Source: §3 · p. 3
• The method builds decreasing-dimensional subspaces from source S0 to core Sn, using fine-to-coarse analysis that selects one decomposed subspace and records their relationship for reverse generation.Each next subspace is formed by an affine combination of elements in the preceding one.Source: §3 · pp. 3–4
• Generation starts by modeling statistics in core Sn, then proceeds coarse-to-fine by generating samples in Si conditioned on samples in Si+1.The paper describes this as a non-parametric method with flexibly chosen intermediate subspaces.Source: §3 · p. 4

4.1 System Overview

• TGHop crops 32×32 RGB patches into source space S0, analyzes them fine-to-coarse, then quilts generated patches into larger images.The system can generate as many same-size texture patches as desired before image quilting.Source: §4.1 · p. 4

4.2 Fine-to-Coarse Analysis

• Two-stage c/w Saab analysis retains selected DC/AC channels and allows flexible Hop channel counts, reducing source-to-core dimension by 56.3%.The Saab filters are determined by feedforward unsupervised PCAs rather than end-to-end backpropagation.Source: §4.2 · pp. 4–5

4.3 Core Sample Generation

• Core generation combines spatial dimension reduction with ICHM: K-means clusters are sampled by frequency, then Gaussian draws are histogram-matched and inverse-transformed.Cluster-conditional distributions are used instead of modeling the full distribution directly.Source: §4.3 · pp. 5–7
• With SDR, generated grass texture quality significantly improves over generation without SDR.Figure 3 compares generated grass texture with and without spatial dimension reduction.Source: Figure 3 · p. 6, §4.3 · p. 6

4.4 Coarse-to-Fine Generation

• Inverse c/w Saab transforms map generated S2 channel groups into S1 blocks, then repeat from S1 to S0, progressively adding local detail.Core DC channels provide grayscale low-resolution patterns.Source: §4.4 · p. 7

5.1 Experimental Setup

• Analysis and generation windows match and do not overlap; texture-dependent K1 and K2 are selected at the PCA-eigenvalue energy knee.The experiments also report input filter window sizes of 2 for both Hop-1 and Hop-2.Source: §5.1 · pp. 7–8

5.2 An Example: Brick Wall Generation

• Generated brick-wall patches retained brick shape and texture diversity, including unseen patterns, while 64 × 64 patches were sometimes blurry due to higher source dimension.The patches were generated in 100 rounds; randomness primarily came from cluster selection and seed-vector generation.Source: §5.2 · p. 8
• Larger 64 × 64 patches better preserved global brick-wall structure after quilting, but sometimes had blurrier local detail than 32 × 32 patches.The stitched 256 × 256 images used 5,000 generated patches and different initial patches for quilting.Source: §5.2 · p. 9

5.3.2 Comparison of Generation Time

• 291.25 seconds for TGHop versus 513.98 seconds for Gatys and 949.64 seconds for Ustyuzhaninov to generate one 256 × 256 image.All methods were tested on the same machine; the two deep-learning methods used a GPU, whereas TGHop did not.Source: Table 2 · p. 9, §5.3.2 · p. 9

5.4 Comparison of Model Sizes

• 2.4M parameters for TGHop, versus 0.852M for [19] and 2.055M for, under the reported setting.The setting used γ = 0.01, N = 50, K1 = 9, K2 = 22, Dr = 909, F = 2,518, and W = 200.Source: Table 4 · p. 11, §5.4 · pp. 11, 13

5.3.1 Visual Quality Comparison

• TGHop generated high-quality, visually pleasant textures and avoided the brown fog artifact seen with [19] on meshed and cloud textures.The comparison used 10 input textures and results from the two cited methods and TGHop, with and without spatial dimension reduction.Source: §5.3.1 · p. 9, Figure 9 · p. 12

5.4 Comparison of Model Sizes

• With 50 clusters, generated patches showed over-saturation; increasing N to 80 made it less apparent, and quality improved at N = 100 with little further gain at N = 200.The passage also lists experiments at N = 50, 80, 110, and 200.Source: Figure 11 · p. 13, §5.4 · p. 14
• Reducing spatial dimension from 1408 to 553 caused little quality degradation, while further reduction produced blur as PCA components discarded details.A larger SDR threshold γ reduces Dr and N and therefore model size; the same threshold was used across channels.Source: §5.4 · pp. 13, 15, Figure 12 · p. 14

• Experiments demonstrate that TGHop generates texture images of superior quality with a small model size and at fast speed.Source: §6 · p. 15
• TGHop analyzes texture with multi-stage c/w Saab transforms and joint spatial-spectral representations, generates samples in the core, then reverses the path to expand spatial dimensions before stitching patches.Source: §6 · p. 15
• Future work includes controlling intermediate-subspace dimension growth, testing other image types and inpainting, and comparing the model with GANs.Source: §6 · p. 15