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Unsupervised Image Regression for Heterogeneous Change Detection

Luigi T. Luppino, Filippo M. Bianchi, Gabriele Moser, Stian N. Anfinsen

arXiv:1909.05948v1cs.CVcs.LGeess.IVstat.ML

TL;DR

Unsupervised heterogeneous change detection needs a way to identify unchanged samples across images from different domains without labeled data. This paper compares affinity matrices to select pseudo-training data, then applies bidirectional image regression; experiments on two datasets show improved change detection, with random forest offering similar accuracy at lower computational cost and easier tuning.

  • Problem

    Unsupervised heterogeneous change detection must identify unchanged areas for cross-domain mapping without the labeled samples required by supervised approaches.

    Source: §I · p. 1

  • Method

    The framework compares affinity matrices to select likely unchanged pixels as pseudo-training data for bidirectional image regression between the two domains.

    Source: §I · pp. 1–2

  • Results

    Experiments on two datasets found the framework effective for training-set selection and change detection, with training-set quality more crucial than the particular regression method.

    Source: §V · p. 13

  • Takeaways & Limitations

    Random forest regression is recommended for its reliability, robustness, speed, and ease of hyperparameter tuning.

    Source: §V · p. 15

    The affinity comparison depends on patch size: small windows may contain too little structure, especially for higher-resolution imagery.

    Source: §V · p. 14

Abstract

from arXiv · show

Change detection in heterogeneous multitemporal satellite images is an emerging and challenging topic in remote sensing. In particular, one of the main challenges is to tackle the problem in an unsupervised manner. In this paper we propose an unsupervised framework for bitemporal heterogeneous change detection based on the comparison of affinity matrices and image regression. First, our method quantifies the similarity of affinity matrices computed from co-located image patches in the two images. This is done to automatically identify pixels that are likely to be unchanged. With the identified pixels as pseudo-training data, we learn a transformation to map the first image to the domain of the other image, and vice versa. Four regression methods are selected to carry out the transformation: Gaussian process regression, support vector regression, random forest regression, and a recently proposed kernel regression method called homogeneous pixel transformation. To evaluate the potentials and limitations of our framework, and also the benefits and disadvantages of each regression method, we perform experiments on two real data sets. The results indicate that the comparison of the affinity matrices can already be considered a change detection method by itself. However, image regression is shown to improve the results obtained by the previous step alone and produces accurate change detection maps despite of the heterogeneity of the multitemporal input data. Notably, the random forest regression approach excels by achieving similar accuracy as the other methods, but with a significantly lower computational cost and with fast and robust tuning of hyperparameters.

I. INTRODUCTION

Heterogeneous change detection must compare images from different domains without relying on labeled unchanged areas. The paper reviews prior approaches and proposes automatic pseudo-training data selection followed by image regression.

I. INTRODUCTION

  • Unsupervised heterogeneous change detection must identify unchanged areas to learn mappings between domains, while avoiding the labeled data required by supervised methods.The images may differ in measured quantities, class signatures, and statistical behavior, making homogeneous methods unsuitable without additional processing.Source: §I · p. 1
  • The framework selects likely unchanged pixel pairs automatically, then uses them to learn bidirectional image-domain mappings without prior knowledge about the data.The paper compares Gaussian process, support vector, random forest, and homogeneous pixel transformation regression.Source: §I · pp. 1–2

II. RELATED WORK

  • The review groups heterogeneous bitemporal methods into supervised and unsupervised approaches, with unsupervised methods further divided into traditional and deep learning methods.The supervised-method review includes mappings between image domains, transformations to joint spaces, post-classification, manifold approaches, and kernel alignment.Source: §II · p. 2, A. Supervised methods · pp. 2–3

A. Supervised methods

  • Supervised examples learn cross-domain relationships from unchanged samples, but their training can depend on manually selected or labeled data.Examples include quantile regression and copulas, HPT, and kernel canonical correlation analysis.Source: A. Supervised methods · pp. 2–3

B. Unsupervised methods

Unsupervised heterogeneous change detection depends on finding unchanged areas suitable for learning cross-domain mappings. Prior methods use iterative patch, pixel-similarity, or learned-feature strategies to identify changes.

B. Unsupervised methods

  • Unchanged areas are essential for learning mappings between heterogeneous domains, because changed samples can impair the learned mappings.This is the central challenge for unsupervised heterogeneous change detection.Source: B. Unsupervised methods · p. 3

1) Traditional methods:

  • Traditional methods detect change through coupled sparse patch codes, nonlocal pixel-interaction similarities, or neighborhood statistics followed by dimensionality reduction and thresholding.The coupled-dictionary method iteratively discards patch pairs with the largest reconstruction errors before updating its dictionaries.Source: 1) Traditional methods · p. 3

2) Deep learning methods:

  • Deep learning methods learn features or mappings with convolutional networks and stacked denoising autoencoders, often alternating network training with updates to change probabilities.The reviewed approaches include binary change maps, change-type clusters, and mapping between modality-specific representations.Source: 2) Deep learning methods · pp. 3–4

III. METHODOLOGY

The methodology compares local affinity structures to estimate change and select low-change pixels for regression training. It also discusses representativeness and operating assumptions that constrain this selection.

A. Comparison of Affinity Matrices

  • For each co-located patch pair, the method computes within-patch pixel distances, converts them to affinities, and compares the two affinity matrices.Euclidean distances are selected as a computationally efficient option under near-Gaussian data assumptions; SAR intensity can be log-transformed to approach that condition.Source: A. Comparison of Affinity Matrices · p. 4
  • The Frobenius norm of the affinity-matrix difference is averaged over all overlapping patches covering each pixel to produce its change possibility.The procedure treats the local affinity matrices as graph structures whose relations diverge when patches contain changes.Source: A. Comparison of Affinity Matrices · p. 4
  • The method selects the M pixels with the lowest change possibilities as training samples, but this set may not represent all surface types and can lead to false positives.A Hellinger-distance diagnostic checks whether selected samples represent the full image distributions, but does not correct a poor selection.Source: A. Comparison of Affinity Matrices · p. 5
  • The affinity comparison is a unit-patch special case of patch-based local and global self-affinity approaches, with a Frobenius norm summarizing patch-level differences.The paper assumes perfect co-registration, while suggesting reasonable robustness to small registration errors for medium-resolution images and changes extending beyond pixel level.Source: A. Comparison of Affinity Matrices · p. 5

B. Image regression

Image regression learns mappings in both directions between heterogeneous image domains using selected training pairs. Predicted images are compared with observations to produce change maps.

B. Image regression

  • Bidirectional regression predicts each image as it would appear in the other sensor domain, enabling conventional change metrics on predictions and observations.The two mappings are X → predicted Y and Y → predicted X.Source: B. Image regression · p. 5
  • The method averages normalized distances between each original image and its corresponding prediction, so consistently high discrepancies indicate change while one-sided high values are suppressed.Distances may be clipped before normalization, and the resulting difference image can be spatially filtered.Source: B. Image regression · pp. 5–6
  • A final change map is obtained by thresholding the filtered difference image, with Otsu’s method used in this work.Otsu’s method seeks a threshold that minimizes intraclass variance and maximizes interclass variance.Source: B. Image regression · p. 6

1) Gaussian Process Regression:

Gaussian process regression maps training inputs to target vectors and predicts outputs for new observations. Its prediction quality depends on the kernel and its hyperparameters.

1) Gaussian Process Regression:

  • A zero-mean Gaussian process is specified by its mean and covariance functions, with training inputs X and targets Y used to predict outputs for new inputs X∗.The passage frames the regression data as M training vectors with P features and Q target features, and N new observed vectors.Source: 1) Gaussian Process Regression · p. 6
  • The posterior conditional mean prediction is obtained from the covariance matrices linking training inputs, new inputs, and training targets.The prediction is given as KX∗,X · KX,X^-1 · Y.Source: 1) Gaussian Process Regression · p. 6
  • Kernel choice and hyperparameters affect regression quality, while marginal-likelihood optimization can settle at a local maximum.The authors use an RBF kernel and recommend repeating optimization from different random starting points.Source: 1) Gaussian Process Regression · p. 6

2) Multiple-output Support Vector Regression:

The section describes multiple-output support vector regression as an alternative to fitting a separate support vector regressor for each target feature. Its regularization and kernel parameters govern fit and generalization.

2) Multiple-output Support Vector Regression:

  • Standard SVR implementations predict one output feature at a time, whereas the multiple-input multiple-output method models target features jointly.The passage notes that separate regressors ignore potentially nonlinear relations across target features.Source: 2) Multiple-output Support Vector… · p. 7
  • The penalty factor λ trades off regularization against error, with excessive values risking overfitting and small values risking underfitting.The method uses coefficients and biases to combine data points transformed by a kernel function.Source: 2) Multiple-output Support Vector… · p. 7
  • The insensitivity width ϵ and RBF kernel width σ also affect fit, and parameters can be selected by cross-validation error or an error bound.Too-small ϵ can increase support vectors and variance, while too-large ϵ can cause underfitting.Source: 2) Multiple-output Support Vector… · p. 7

3) Random Forest Regression:

Random forest regression averages predictions from decision trees whose splits use randomly selected features and learned thresholds. Bootstrap sampling and feature subsampling support generalization, while forest size trades accuracy against computational load.

3) Random Forest Regression:

  • Each tree routes an input through threshold-based feature splits to a leaf, and the forest averages the leaf outputs across its trees.A leaf is assigned the average target vector of its training data points.Source: 3) Random Forest Regression · p. 7
  • Trees use bootstrap samples and randomly selected feature subsets, with out-of-bag estimation used for validation.Forest size, features considered per node, and tree depth are the three parameters identified for tuning.Source: 3) Random Forest Regression · p. 7
  • Increasing the number of trees can improve outcomes only until accuracy saturates, so forest size balances accuracy gains against computational load.The section also gives empirical rules of thumb for the number of features considered at each node.Source: 3) Random Forest Regression · p. 7

4) Homogeneous Pixel Transformation:

Homogeneous pixel transformation uses kernel regression over the K nearest training vectors to map pixels between image domains. Its distance weighting depends on the kernel width and the normalization used.

4) Homogeneous Pixel Transformation:

  • For each pixel in the first image, homogeneous pixel transformation finds K nearest training vectors and forms a weighted-sum regression.The method is a kernel regression algorithm based on nearest neighbours.Source: 4) Homogeneous Pixel Transformation · p. 7
  • The kernel width γ controls how strongly farther neighbours are penalized, ranging from near-equal weighting to contributions dominated by a few neighbours.The passage describes the small-γ and large-γ behaviors without specifying an optimal value.Source: 4) Homogeneous Pixel Transformation · p. 7
  • The authors report using absolute rather than relative distance normalization in their implementation because testing favored the absolute form.Relative normalization uses the maximum distance from a pixel to its neighbours; absolute normalization uses the maximum among all computed distances.Source: 4) Homogeneous Pixel Transformation · p. 8

C. Methodological comparison among the considered regression methods

The regression methods differ in overfitting behavior, prediction cost, smoothness, and hyperparameter tuning. Experiments assess accuracy and computational speed across two datasets.

C. Methodological comparison among the considered regression methods

  • RF and SVM have properties associated with overfitting robustness, whereas dense GPR expansions and HPT with too-small K may be more sensitive.RF combines bagging with random feature selection; SVM uses a generally sparse expansion over selected support vectors, while GPR terms are centered on all training samples.Source: C. Methodological comparison among the… · p. 8
  • Prediction cost scales with support-vector count for SVM and training-sample count for GPR, while RF cost grows linearly with forest tree count.HPT can limit computational burden using K-d tree formulations.Source: C. Methodological comparison among the… · p. 8
  • RF prediction is piecewise constant and may be undesirable when a smooth predictor is sought, though the paper says this is not a limitation for its change-detection method.SVM and GPR are smooth when their kernels are smooth, and HPT is generally regular.Source: C. Methodological comparison among the… · p. 8
  • GPR offers likelihood-based autocovariance-hyperparameter optimization, RF tuning is usually straightforward, and automatic HPT optimization methods had not been developed.SVM hyperparameter approaches are case-specific; cross-validation is a general option but increases computation time.Source: C. Methodological comparison among the… · p. 8

IV. EXPERIMENTAL RESULTS

  • Experiments varied training-set size from 10^2 to 10^5 and affinity patch size from 5 to 20, and evaluated AUC, OA, Kappa, and regression runtime.The largest training set was 8.07% of N in the first case study and 1.43% in the second; runtime comparisons are indicators rather than exact algorithm comparisons because implementations use Python and MATLAB.Source: §IV · pp. 8–9

A. Forest fire in Texas

The Texas test pairs Landsat 5 imagery from before a forest fire with post-event EO-1 ALI imagery. Affinity-based training-set selection yields change maps and identifies samples that largely avoid changed pixels while covering the image domain.

A. Forest fire in Texas

  • The Texas dataset pairs 1534 × 808 Landsat 5 TM imagery acquired before the fire with post-event EO-1 ALI imagery; their 7 and 10 channels partly share spectral bands.The paper characterizes this as one of the easier heterogeneous change-detection scenarios.Source: A. Forest fire in Texas · p. 9, Figure 2 · p. 9

1) Training set selection:

  • 84.9%, 91.3%, and 93.1% AUC were achieved for k = 5, 10, and 20, respectively, by thresholding the affinity heat-map.The full procedure took about 1, 5, and 20 minutes, respectively.Source: 1) Training set selection · p. 9
  • The average matrix norm increases monotonically with the percentage of changed pixels, a trend more evident for k = 20 than k = 5.This supports using low-norm patches to identify likely unchanged areas.Source: 1) Training set selection · p. 9, Figure 4 · p. 9
  • Across the tested k and M combinations, the method largely avoids changed pixels in T, with false negatives remaining negligible and going to zero as k increases.Larger M slightly raises the chance of including changed pixels; Hellinger distances tend toward zero as more training points are considered.Source: 1) Training set selection · pp. 9–10, Table I · p. 10
  • For k = 20, selected samples cover Y’s domain even at M = 10^2, while avoiding outliers and pixels in changed areas.As M increases, the selected-sample histograms tend to match the full-data histogram; for M = 10^5, few or no selected points fall inside the ground-truth change area.Source: 1) Training set selection · p. 10

2) Image regression:

On the Texas dataset, the regression methods generally achieved high change-detection accuracy, and the proposed unsupervised method outperformed the referenced alternatives in OA and KC. Automatic training-set selection provides substantially more samples, though kernel methods face computational limits at the largest sample size.

2) Image regression:

  • Fig. 6 shows selected training sets for M = 10^5 at k = 5 and k = 20, while Fig. 5 compares full-image and selected-sample histograms across M values.The Fig. 5 comparison uses two channels of Y and k = 20.Source: Figure 6 · p. 10, Figure 5 · p. 10
  • The unsupervised method consistently outperforms the referenced methods in OA and KC on the Texas dataset.Table II reports elapsed time, filtered AUC, and thresholded OA and KC for the regression methods and parameter combinations.Source: 2) Image regression · p. 10, Table II · p. 11
  • Automatic selection supplies a training sample one or two orders of magnitude larger than the few hundred points practical to select by hand.The authors identify training-sample size as a likely reason for the performance gaps against supervised methods.Source: 2) Image regression · pp. 10–11
  • SVR and GPR results are unavailable at M = 10^5 because their kernel matrices exceed memory limits and make matrix multiplications unfeasible.The reported computational time grows exponentially as M increases.Source: 2) Image regression · p. 11
  • GPR produced low KC across examples, while at M = 10^2 the methods generated lake- and river-related false positives when those classes were underrepresented in T.The false positives are attributed to too few training samples from those classes; Fig. 7 depicts a poor GPR result and a successful SVR example.Source: 2) Image regression · p. 11, Figure 7 · p. 11

B. Flood in California

The California experiment tests the approach on a flood observed in heterogeneous optical and SAR imagery. Heat-map change detection improves in AUC with larger windows but takes longer, while training-set selection avoids changed pixels despite their spectral similarity to unchanged land covers.

B. Flood in California

  • The flood dataset pairs a nine-channel Landsat 8 image with Sentinel-1A VV and VH SAR data plus their ratio; its ground truth comes from two other single-polarisation SAR images.The acquisitions show California before and after the flood, with the ground truth displayed alongside them.Source: B. Flood in California · p. 12, Figure 8 · p. 12

1) Training set selection:

  • 64.9%, 71.6%, and 76.0% AUC were achieved by heat-map detection for k = 5, 10, and 20, respectively, in about 20 minutes, 2 hours, and 5 hours.The smaller windows provide a preliminary change feature of lower quality than in the Texas experiment, and the matrix-norm trend is less clear, especially for k = 5.Source: 1) Training set selection · p. 12, Figure 10 · p. 12
  • Hellinger distances are high at M = 10^2 and 10^3, then decrease as M and k grow; the training samples avoid changed pixels despite flood features resembling rice fields.The image contains N = 7 · 10^6 data points, and Table III reports AUC, FN, and both image-to-training-set Hellinger distances.Source: Table III · p. 12, 1) Training set selection · pp. 12–13
  • As M rises from 10^2 to 10^5, the selected-sample histogram first covers the image domain and then increasingly fits the full-image histogram.Fig. 11 illustrates this progression for Y and its first two channels.Source: 1) Training set selection · p. 13

2) Image regression:

The California results favor random forest regression for its accuracy, speed, and ease of tuning. The conclusions emphasize training-set selection and window size as important factors, while identifying further opportunities to improve affinity-based selection and reduce computation.

2) Image regression:

  • GPR had acceptable AUC and OA but struggled to converge to a reasonable KC; other regression methods had KC values of 0.3–0.46.The authors associate the lower KC values with many false positives and note that ground-truth reliability may be involved.Source: 2) Image regression · p. 13

V. CONCLUSIONS

  • The consistency of results across experiments indicates that selecting a good training set matters more than the specific regression method used afterward.Source: §V · p. 13
  • Larger window sizes improved self-supervision and final change maps, but increased computational load and memory requirements.Affinity comparison targets local rather than global structure; overly small patches may capture only noise, especially for higher-resolution imagery, and corrupt the output.Source: §V · pp. 13–14

2) Image regression:

  • Across California configurations, AUC and OA were good; RFR was slightly better than HPT and SVR while significantly faster.The comparison includes the results summarized in Table IV.Source: 2) Image regression · p. 13, Table IV · p. 14

V. CONCLUSIONS

  • Random forest regression achieved performance comparable to state-of-the-art methods with considerably shorter computation and robust, easy-to-tune hyperparameters.The authors recommend random forest for its reliability, robustness, speed, and ease of tuning.Source: §V · pp. 14–15
  • Future work could improve Algorithm 1 and reduce its computational cost, improve feature-space coverage in training-set sampling, and test multi-pixel patch shifts.Shifting patches could speed computation, but its effects on training-set selection remain to be investigated.Source: §V · p. 15
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