Valero Laparra

José A. Padrón-Hidalgo, Valero Laparra, Gustau Camps-Valls

Anomaly detection is a field of intense research. Identifying low probability events in data/images is a challenging problem given the high-dimensionality of the data, especially when no (or little) information about the anomaly is available a priori. While plenty of methods are available, the vast majority of them do not scale well to large datasets and require the choice of some (very often critical) hyperparameters. Therefore, unsupervised and computationally efficient detection methods become strictly necessary. We propose an unsupervised method for detecting anomalies and changes in remote sensing images by means of a multivariate Gaussianization methodology that allows to estimate multivariate densities accurately, a long-standing problem in statistics and machine learning. The methodology transforms arbitrarily complex multivariate data into a multivariate Gaussian distribution. Since the transformation is differentiable, by applying the change of variables formula one can estimate the probability at any point of the original domain. The assumption is straightforward: pixels with low estimated probability are considered anomalies. Our method can describe any multivariate distribution, makes an efficient use of memory and computational resources, and is parameter-free. We show the efficiency of the method in experiments involving both anomaly detection and change detection in different remote sensing image sets. Results show that our approach outperforms other linear and nonlinear methods in terms of detection power in both anomaly and change detection scenarios, showing robustness and scalability to dimensionality and sample sizes.

Daniel Heestermans Svendsen, Daniel Hernández-Lobato, Luca Martino et al.

Earth observation from satellites offers the possibility to monitor our planet with unprecedented accuracy. Radiative transfer models (RTMs) encode the energy transfer through the atmosphere, and are used to model and understand the Earth system, as well as to estimate the parameters that describe the status of the Earth from satellite observations by inverse modeling. However, performing inference over such simulators is a challenging problem. RTMs are nonlinear, non-differentiable and computationally costly codes, which adds a high level of difficulty in inference. In this paper, we introduce two computational techniques to infer not only point estimates of biophysical parameters but also their joint distribution. One of them is based on a variational autoencoder approach and the second one is based on a Monte Carlo Expectation Maximization (MCEM) scheme. We compare and discuss benefits and drawbacks of each approach. We also provide numerical comparisons in synthetic simulations and the real PROSAIL model, a popular RTM that combines land vegetation leaf and canopy modeling. We analyze the performance of the two approaches for modeling and inferring the distribution of three key biophysical parameters for quantifying the terrestrial biosphere.

Nuria Alabau-Bosque, Paula Daudén-Oliver, Jorge Vila-Tomás et al.

Subjective image quality metrics are usually evaluated according to the correlation with human opinion in databases with distortions that may appear in digital media. However, these oversee affine transformations which may represent better the changes in the images actually happening in natural conditions. Humans can be particularly invariant to these natural transformations, as opposed to the digital ones. In this work, we propose a methodology to evaluate any image quality metric by assessing their invariance to affine transformations, specifically: rotation, translation, scaling, and changes in spectral illumination. Here, invariance refers to the fact that certain distances should be neglected if their values are below a threshold. This is what we call invisibility threshold of a metric. Our methodology consists of two elements: (1) the determination of a visibility threshold in a subjective representation common to every metric, and (2) a transduction from the distance values of the metric and this common representation. This common representation is based on subjective ratings of readily available image quality databases. We determine the threshold in such common representation (the first element) using accurate psychophysics. Then, the transduction (the second element) can be trivially fitted for any metric: with the provided threshold extension of the method to any metric is straightforward. We test our methodology with some well-established metrics and find that none of them show human-like invisibility thresholds. This means that tuning the models exclusively to predict the visibility of generic distortions may disregard other properties of human vision as for instance invariances or invisibility thresholds. The data and code are publicly available to test other metrics.

Valero Laparra, Alexander Hepburn, J. Emmanuel Johnson et al.

In this paper we elaborate an extension of rotation-based iterative Gaussianization, RBIG, which makes image Gaussianization possible. Although RBIG has been successfully applied to many tasks, it is limited to medium dimensionality data (on the order of a thousand dimensions). In images its application has been restricted to small image patches or isolated pixels, because rotation in RBIG is based on principal or independent component analysis and these transformations are difficult to learn and scale. Here we present the \emph{Convolutional RBIG}: an extension that alleviates this issue by imposing that the rotation in RBIG is a convolution. We propose to learn convolutional rotations (i.e. orthonormal convolutions) by optimising for the reconstruction loss between the input and an approximate inverse of the transformation using the transposed convolution operation. Additionally, we suggest different regularizers in learning these orthonormal convolutions. For example, imposing sparsity in the activations leads to a transformation that extends convolutional independent component analysis to multilayer architectures. We also highlight how statistical properties of the data, such as multivariate mutual information, can be obtained from \emph{Convolutional RBIG}. We illustrate the behavior of the transform with a simple example of texture synthesis, and analyze its properties by visualizing the stimuli that maximize the response in certain feature and layer.

Pablo Hernández-Cámara, Jorge Vila-Tomás, Valero Laparra et al.

Subjective image quality measures based on deep neural networks are very related to models of visual neuroscience. This connection benefits engineering but, more interestingly, the freedom to optimize deep networks in different ways, make them an excellent tool to explore the principles behind visual perception (both human and artificial). Recently, a myriad of networks have been successfully optimized for many interesting visual tasks. Although these nets were not specifically designed to predict image quality or other psychophysics, they have shown surprising human-like behavior. The reasons for this remain unclear. In this work, we perform a thorough analysis of the perceptual properties of pre-trained nets (particularly their ability to predict image quality) by isolating different factors: the goal (the function), the data (learning environment), the architecture, and the readout: selected layer(s), fine-tuning of channel relevance, and use of statistical descriptors as opposed to plain readout of responses. Several conclusions can be drawn. All the models correlate better with human opinion than SSIM. More importantly, some of the nets are in pair of state-of-the-art with no extra refinement or perceptual information. Nets trained for supervised tasks such as classification correlate substantially better with humans than LPIPS (a net specifically tuned for image quality). Interestingly, self-supervised tasks such as jigsaw also perform better than LPIPS. Simpler architectures are better than very deep nets. In simpler nets, correlation with humans increases with depth as if deeper layers were closer to human judgement. This is not true in very deep nets. Consistently with reports on illusions and contrast sensitivity, small changes in the image environment does not make a big difference. Finally, the explored statistical descriptors and concatenations had no major impact.

Pablo Hernández-Cámara, Jorge Vila-Tomás, Paula Dauden-Oliver et al.

Autonomous driving is a challenging scenario for image segmentation due to the presence of uncontrolled environmental conditions and the eventually catastrophic consequences of failures. Previous work suggested that a biologically motivated computation, the so-called Divisive Normalization, could be useful to deal with image variability, but its effects have not been systematically studied over different data sources and environmental factors. Here we put segmentation U-nets augmented with Divisive Normalization to work far from training conditions to find where this adaptation is more critical. We categorize the scenes according to their radiance level and dynamic range (day/night), and according to their achromatic/chromatic contrasts. We also consider video game (synthetic) images to broaden the range of environments. We check the performance in the extreme percentiles of such categorization. Then, we push the limits further by artificially modifying the images in perceptually/environmentally relevant dimensions: luminance, contrasts and spectral radiance. Results show that neural networks with Divisive Normalization get better results in all the scenarios and their performance remains more stable with regard to the considered environmental factors and nature of the source. Finally, we explain the improvements in segmentation performance in two ways: (1) by quantifying the invariance of the responses that incorporate Divisive Normalization, and (2) by illustrating the adaptive nonlinearity of the different layers that depends on the local activity.

Pablo Hernández-Cámara, Valero Laparra, Jesús Malo

One of the key problems in computer vision is adaptation: models are too rigid to follow the variability of the inputs. The canonical computation that explains adaptation in sensory neuroscience is divisive normalization, and it has appealing effects on image manifolds. In this work we show that including divisive normalization in current deep networks makes them more invariant to non-informative changes in the images. In particular, we focus on U-Net architectures for image segmentation. Experiments show that the inclusion of divisive normalization in the U-Net architecture leads to better segmentation results with respect to conventional U-Net. The gain increases steadily when dealing with images acquired in bad weather conditions. In addition to the results on the Cityscapes and Foggy Cityscapes datasets, we explain these advantages through visualization of the responses: the equalization induced by the divisive normalization leads to more invariant features to local changes in contrast and illumination.

Alexander Hepburn, Valero Laparra, Raúl Santos-Rodriguez et al.

Previously, Barlow and Attneave hypothesised a link between biological vision and information maximisation. Following Shannon, information was defined using the probability of natural images. Several physiological and psychophysical phenomena have been derived from principles like info-max, efficient coding, or optimal denoising. However, it remains unclear how this link is expressed in mathematical terms from image probability. Classical derivations were subjected to strong assumptions on the probability models and on the behaviour of the sensors. Moreover, the direct evaluation of the hypothesis was limited by the inability of classical image models to deliver accurate estimates of the probability. Here, we directly evaluate image probabilities using a generative model for natural images, and analyse how probability-related factors can be combined to predict the sensitivity of state-of-the-art subjective image quality metrics, a proxy for human perception. We use information theory and regression analysis to find a simple model that when combining just two probability-related factors achieves 0.77 correlation with subjective metrics. This probability-based model is validated in two ways: through direct comparison with the opinion of real observers in a subjective quality experiment, and by reproducing basic trends of classical psychophysical facts such as the Contrast Sensitivity Function, the Weber-law, and contrast masking.

Tashi Namgyal, Alexander Hepburn, Raul Santos-Rodriguez et al.

The subjective quality of natural signals can be approximated with objective perceptual metrics. Designed to approximate the perceptual behaviour of human observers, perceptual metrics often reflect structures found in natural signals and neurological pathways. Models trained with perceptual metrics as loss functions can capture perceptually meaningful features from the structures held within these metrics. We demonstrate that using features extracted from autoencoders trained with perceptual losses can improve performance on music understanding tasks, i.e. genre classification, over using these metrics directly as distances when learning a classifier. This result suggests improved generalisation to novel signals when using perceptual metrics as loss functions for representation learning.

Nuria Alabau-Bosque, Jorge Vila-Tomas, Paula Dauden-Oliver et al.

Convolutional Neural Networks (CNNs) are widely assumed to be translation-invariant, yet standard architectures exhibit a startling fragility: even a single-pixel shift can drastically degrade performance due to their reliance on spatially dependent fully connected layers. In this work, we resolve this vulnerability by proposing a lightweight 'Online Architecture' strategy. By strategically inserting Global Average Pooling (GAP) layers at various network depths, we effectively decouple feature recognition from spatial location. Using VGG-16 as a primary case study, we demonstrate that this architectural modification achieves a massive 98% reduction in trainable parameters (from 5.2M to just 82K) and a 90% reduction in total network size (138M to 14M). Despite this drastic pruning, our variants maintain competitive Top-1 accuracy on ImageNet (66.4%) while doubling translational robustness, reducing average relative loss from 0.09 to 0.05. Furthermore, our analysis identifies a fundamental limit to invariance: while GAP resolves macroscopic sensitivity, discrete pooling operations introduce a residual periodic aliasing that prevents perfect pixel-level stability. Finally, we extend these findings to Perceptual Image Quality Assessment (IQA) by integrating our invariant backbones into the LPIPS framework. The resulting metric significantly outperforms the retrained baseline in generalization across the KADID-10k dataset (Spearman 0.89 vs. 0.75) and achieves a near-perfect alignment with human psychophysical response curves on the RAID dataset (Spearman 0.95). These results confirm that enforcing architectural invariance is a far more efficient and biologically plausible path to robustness than traditional data augmentation. Data and code are publicly available. The data and code are publicly available to facilitate validation and further research.

Alexandra Gomez-Villa, Pablo Hernández-Cámara, Muhammad Atif Butt et al.

Color serves as a fundamental dimension of human visual perception and a primary means of communicating about objects and scenes. As vision-language models (VLMs) become increasingly prevalent, understanding whether they name colors like humans is crucial for effective human-AI interaction. We present the first systematic evaluation of color naming capabilities across VLMs, replicating classic color naming methodologies using 957 color samples across five representative models. Our results show that while VLMs achieve high accuracy on prototypical colors from classical studies, performance drops significantly on expanded, non-prototypical color sets. We identify 21 common color terms that consistently emerge across all models, revealing two distinct approaches: constrained models using predominantly basic terms versus expansive models employing systematic lightness modifiers. Cross-linguistic analysis across nine languages demonstrates severe training imbalances favoring English and Chinese, with hue serving as the primary driver of color naming decisions. Finally, ablation studies reveal that language model architecture significantly influences color naming independent of visual processing capabilities.

Nuria Alabau-Bosque, Jorge Vila-Tomás, Paula Daudén-Oliver et al.

Playing games is inherently human, and a lot of games are created to challenge different human characteristics. However, these tasks are often left out when evaluating the human-like nature of artificial models. The objective of this work is proposing a new approach to evaluate artificial models via board games. To this effect, we test the color perception and color naming capabilities of CLIP by playing the board game Hues & Cues and assess its alignment with humans. Our experiments show that CLIP is generally well aligned with human observers, but our approach brings to light certain cultural biases and inconsistencies when dealing with different abstraction levels that are hard to identify with other testing strategies. Our findings indicate that assessing models with different tasks like board games can make certain deficiencies in the models stand out in ways that are difficult to test with the commonly used benchmarks.

Jorge Vila-Tomás, Pablo Hernández-Cámara, Valero Laparra et al.

Human vision models are at the core of image processing. For instance, classical approaches to the problem of image quality are based on models that include knowledge about human vision. However, nowadays, deep learning approaches have obtained competitive results by simply approaching this problem as regression of human decisions, and training an standard network on human-rated datasets. These approaches have the advantages of being easily adaptable to a particular problem and they fit very efficiently when data is available. However, mainly due to the excess of parameters, they have the problems of lack of interpretability, and over-fitting. Here we propose a vision model that combines the best of both worlds by using a parametric neural network architecture. We parameterize the layers to have bioplausible functionality, and provide a set of bioplausible parameters. We analyzed different versions of the model and compared it with the non-parametric version. The parametric models achieve a three orders of magnitude reduction in the number of parameters without suffering in regression performance. Furthermore, we show that the parametric models behave better during training and are easier to interpret as vision models. Interestingly, we find that, even initialized with bioplausible trained for regression using human rated datasets, which we call the feature-spreading problem. This suggests that the deep learning approach is inherently flawed, and emphasizes the need to evaluate and train models beyond regression.

Tashi Namgyal, Alexander Hepburn, Raul Santos-Rodriguez et al.

Perceptual metrics are traditionally used to evaluate the quality of natural signals, such as images and audio. They are designed to mimic the perceptual behaviour of human observers and usually reflect structures found in natural signals. This motivates their use as loss functions for training generative models such that models will learn to capture the structure held in the metric. We take this idea to the extreme in the audio domain by training a compressive autoencoder to reconstruct uniform noise, in lieu of natural data. We show that training with perceptual losses improves the reconstruction of spectrograms and re-synthesized audio at test time over models trained with a standard Euclidean loss. This demonstrates better generalisation to unseen natural signals when using perceptual metrics.

Pablo Hernández-Cámara, Jesus Malo, Valero Laparra

A number of scientists suggested that human visual perception may emerge from image statistics, shaping efficient neural representations in early vision. In this work, a bio-inspired architecture that can accommodate several known facts in the retina-V1 cortex, the PerceptNet, has been end-to-end optimized for different tasks related to image reconstruction: autoencoding, denoising, deblurring, and sparsity regularization. Our results show that the encoder stage (V1-like layer) consistently exhibits the highest correlation with human perceptual judgments on image distortion despite not using perceptual information in the initialization or training. This alignment exhibits an optimum for moderate noise, blur and sparsity. These findings suggest that the visual system may be tuned to remove those particular levels of distortion with that level of sparsity and that biologically inspired models can learn perceptual metrics without human supervision.

Pablo Hernández-Cámara, Alexandra Gomez-Villa, Jose Manuel Jaén-Lorites et al.

Understanding how Multimodal Large Language Models (MLLMs) process low-level visual features is critical for evaluating their perceptual abilities and has not been systematically characterized. Inspired by human psychophysics, we introduce a behavioural method for estimating the Contrast Sensitivity Function (CSF) in MLLMs by treating them as end-to-end observers. Models are queried with structured prompts while viewing noise-based stimuli filtered at specific spatial frequencies. Psychometric functions are derived from the binary verbal responses, and contrast thresholds (and CSFs) are obtained without relying on internal activations or classifier-based proxies. Our results reveal that some models resemble human CSFs in shape or scale, but none capture both. We also find that CSF estimates are highly sensitive to prompt phrasing, indicating limited linguistic robustness. Finally, we show that CSFs predict model performance under frequency-filtered and adversarial conditions. These findings highlight systematic differences in frequency tuning across MLLMs and establish CSF estimation as a scalable diagnostic tool for multimodal perception.

Pablo Hernández-Cámara, Jose Manuel Jaén-Lorites, Jorge Vila-Tomás et al.

Vision Transformers (ViTs) achieve remarkable performance in image recognition tasks, yet their alignment with human perception remains largely unexplored. This study systematically analyzes how model size, dataset size, data augmentation and regularization impact ViT perceptual alignment with human judgments on the TID2013 dataset. Our findings confirm that larger models exhibit lower perceptual alignment, consistent with previous works. Increasing dataset diversity has a minimal impact, but exposing models to the same images more times reduces alignment. Stronger data augmentation and regularization further decrease alignment, especially in models exposed to repeated training cycles. These results highlight a trade-off between model complexity, training strategies, and alignment with human perception, raising important considerations for applications requiring human-like visual understanding.

Pablo Hernández-Cámara, Jose Manuel Jaén-Lorites, Jorge Vila-Tomás et al.

During the training of multi-modal models like CLIP, we observed an intriguing phenomenon: the correlation with low-level human image quality assessments peaks in the early epochs before gradually declining. This study investigates this observation and seeks to understand its causes through two key factors: shape-texture bias alignment and classification accuracy drop under noise. Our findings suggest that CLIP initially learn low-level visual features, enhancing its alignment with low-level human perception but also increasing its sensitivity to noise and its texture bias. As training progresses, the model shifts toward more abstract shape-based representations, improving noise robustness but reducing alignment with low-level human perception. These results suggest that these factors shared an underlying learning mechanism and provide new insights into optimizing the trade-off between perceptual alignment and robustness in vision-language models.

Paula Daudén-Oliver, David Agost-Beltran, Emilio Sansano-Sansano et al.

Image quality databases are used to train models for predicting subjective human perception. However, most existing databases focus on distortions commonly found in digital media and not in natural conditions. Affine transformations are particularly relevant to study, as they are among the most commonly encountered by human observers in everyday life. This Data Descriptor presents a set of human responses to suprathreshold affine image transforms (rotation, translation, scaling) and Gaussian noise as convenient reference to compare with previously existing image quality databases. The responses were measured using well established psychophysics: the Maximum Likelihood Difference Scaling method. The set contains responses to 864 distorted images. The experiments involved 105 observers and more than 20000 comparisons of quadruples of images. The quality of the dataset is ensured because (a) it reproduces the classical Piéron's law, (b) it reproduces classical absolute detection thresholds, and (c) it is consistent with conventional image quality databases but improves them according to Group-MAD experiments.

Tashi Namgyal, Alexander Hepburn, Raul Santos-Rodriguez et al.

In this study, we investigate the feasibility of utilizing state-of-the-art image perceptual metrics for evaluating audio signals by representing them as spectrograms. The encouraging outcome of the proposed approach is based on the similarity between the neural mechanisms in the auditory and visual pathways. Furthermore, we customise one of the metrics which has a psychoacoustically plausible architecture to account for the peculiarities of sound signals. We evaluate the effectiveness of our proposed metric and several baseline metrics using a music dataset, with promising results in terms of the correlation between the metrics and the perceived quality of audio as rated by human evaluators.

Alexander Hepburn, Valero Laparra, Raul Santos-Rodriguez et al.

It has been demonstrated many times that the behavior of the human visual system is connected to the statistics of natural images. Since machine learning relies on the statistics of training data as well, the above connection has interesting implications when using perceptual distances (which mimic the behavior of the human visual system) as a loss function. In this paper, we aim to unravel the non-trivial relationships between the probability distribution of the data, perceptual distances, and unsupervised machine learning. To this end, we show that perceptual sensitivity is correlated with the probability of an image in its close neighborhood. We also explore the relation between distances induced by autoencoders and the probability distribution of the training data, as well as how these induced distances are correlated with human perception. Finally, we find perceptual distances do not always lead to noticeable gains in performance over Euclidean distance in common image processing tasks, except when data is scarce and the perceptual distance provides regularization. We propose this may be due to a \emph{double-counting} effect of the image statistics, once in the perceptual distance and once in the training procedure.

Fernando Mateo, Jordi Munoz-Mari, Valero Laparra et al.

Gaussian Processes (GPs) has experienced tremendous success in geoscience in general and for bio-geophysical parameter retrieval in the last years. GPs constitute a solid Bayesian framework to formulate many function approximation problems consistently. This paper reviews the main theoretical GP developments in the field. We review new algorithms that respect the signal and noise characteristics, that provide feature rankings automatically, and that allow applicability of associated uncertainty intervals to transport GP models in space and time. All these developments are illustrated in the field of geoscience and remote sensing at a local and global scales through a set of illustrative examples.

Juan Emmanuel Johnson, Valero Laparra, Gustau Camps-Valls

Gaussian Processes (GPs) are a class of kernel methods that have shown to be very useful in geoscience applications. They are widely used because they are simple, flexible and provide very accurate estimates for nonlinear problems, especially in parameter retrieval. An addition to a predictive mean function, GPs come equipped with a useful property: the predictive variance function which provides confidence intervals for the predictions. The GP formulation usually assumes that there is no input noise in the training and testing points, only in the observations. However, this is often not the case in Earth observation problems where an accurate assessment of the instrument error is usually available. In this paper, we showcase how the derivative of a GP model can be used to provide an analytical error propagation formulation and we analyze the predictive variance and the propagated error terms in a temperature prediction problem from infrared sounding data.

David Malmgren-Hansen, Allan Aasbjerg Nielsen, Valero Laparra et al.

The Infrared Atmospheric Sounding Interferometer (IASI) on board the MetOp satellite series provides important measurements for Numerical Weather Prediction (NWP). Retrieving accurate atmospheric parameters from the raw data provided by IASI is a large challenge, but necessary in order to use the data in NWP models. Statistical models performance is compromised because of the extremely high spectral dimensionality and the high number of variables to be predicted simultaneously across the atmospheric column. All this poses a challenge for selecting and studying optimal models and processing schemes. Earlier work has shown non-linear models such as kernel methods and neural networks perform well on this task, but both schemes are computationally heavy on large quantities of data. Kernel methods do not scale well with the number of training data, and neural networks require setting critical hyperparameters. In this work we follow an alternative pathway: we study transfer learning in convolutional neural nets (CNN s) to alleviate the retraining cost by departing from proxy solutions (either features or networks) obtained from previously trained models for related variables. We show how features extracted from the IASI data by a CNN trained to predict a physical variable can be used as inputs to another statistical method designed to predict a different physical variable at low altitude. In addition, the learned parameters can be transferred to another CNN model and obtain results equivalent to those obtained when using a CNN trained from scratch requiring only fine tuning.

Emiliano Díaz, Adrián Pérez-Suay, Valero Laparra et al.

This paper introduces a novel statistical regression framework that allows the incorporation of consistency constraints. A linear and nonlinear (kernel-based) formulation are introduced, and both imply closed-form analytical solutions. The models exploit all the information from a set of drivers while being maximally independent of a set of auxiliary, protected variables. We successfully illustrate the performance in the estimation of chlorophyll content.

José A. Padrón-Hidalgo, Valero Laparra, Nathan Longbotham et al.

Anomalous change detection (ACD) is an important problem in remote sensing image processing. Detecting not only pervasive but also anomalous or extreme changes has many applications for which methodologies are available. This paper introduces a nonlinear extension of a full family of anomalous change detectors. In particular, we focus on algorithms that utilize Gaussian and elliptically contoured (EC) distribution and extend them to their nonlinear counterparts based on the theory of reproducing kernels' Hilbert space. We illustrate the performance of the kernel methods introduced in both pervasive and ACD problems with real and simulated changes in multispectral and hyperspectral imagery with different resolutions (AVIRIS, Sentinel-2, WorldView-2, and Quickbird). A wide range of situations is studied in real examples, including droughts, wildfires, and urbanization. Excellent performance in terms of detection accuracy compared to linear formulations is achieved, resulting in improved detection accuracy and reduced false-alarm rates. Results also reveal that the EC assumption may be still valid in Hilbert spaces. We provide an implementation of the algorithms as well as a database of natural anomalous changes in real scenarios http://isp.uv.es/kacd.html.

David Malmgren-Hansen, Valero Laparra, Allan Aasbjerg Nielsen et al.

In this paper we present a combined strategy for the retrieval of atmospheric profiles from infrared sounders. The approach considers the spatial information and a noise-dependent dimensionality reduction approach. The extracted features are fed into a canonical linear regression. We compare Principal Component Analysis (PCA) and Minimum Noise Fraction (MNF) for dimensionality reduction, and study the compactness and information content of the extracted features. Assessment of the results is done on a big dataset covering many spatial and temporal situations. PCA is widely used for these purposes but our analysis shows that one can gain significant improvements of the error rates when using MNF instead. In our analysis we also investigate the relationship between error rate improvements when including more spectral and spatial components in the regression model, aiming to uncover the trade-off between model complexity and error rates.

Adrián Pérez-Suay, Julia Amorós-López, Luis Gómez-Chova et al.

Dealing with land cover classification of the new image sources has also turned to be a complex problem requiring large amount of memory and processing time. In order to cope with these problems, statistical learning has greatly helped in the last years to develop statistical retrieval and classification models that can ingest large amounts of Earth observation data. Kernel methods constitute a family of powerful machine learning algorithms, which have found wide use in remote sensing and geosciences. However, kernel methods are still not widely adopted because of the high computational cost when dealing with large scale problems, such as the inversion of radiative transfer models or the classification of high spatial-spectral-temporal resolution data. This paper introduces an efficient kernel method for fast statistical retrieval of bio-geo-physical parameters and image classification problems. The method allows to approximate a kernel matrix with a set of projections on random bases sampled from the Fourier domain. The method is simple, computationally very efficient in both memory and processing costs, and easily parallelizable. We show that kernel regression and classification is now possible for datasets with millions of examples and high dimensionality. Examples on atmospheric parameter retrieval from hyperspectral infrared sounders like IASI/Metop; large scale emulation and inversion of the familiar PROSAIL radiative transfer model on Sentinel-2 data; and the identification of clouds over landmarks in time series of MSG/Seviri images show the efficiency and effectiveness of the proposed technique.

J. Emmanuel Johnson, Valero Laparra, Gustau Camps-Valls et al.

Density destructors are differentiable and invertible transforms that map multivariate PDFs of arbitrary structure (low entropy) into non-structured PDFs (maximum entropy). Multivariate Gaussianization and multivariate equalization are specific examples of this family, which break down the complexity of the original PDF through a set of elementary transforms that progressively remove the structure of the data. We demonstrate how this property of density destructive flows is connected to classical information theory, and how density destructors can be used to get more accurate estimates of information theoretic quantities. Experiments with total correlation and mutual information inmultivariate sets illustrate the ability of density destructors compared to competing methods. These results suggest that information theoretic measures may be an alternative optimization criteria when learning density destructive flows.

J. Emmanuel Johnson, Valero Laparra, Maria Piles et al.

Information theory is an excellent framework for analyzing Earth system data because it allows us to characterize uncertainty and redundancy, and is universally interpretable. However, accurately estimating information content is challenging because spatio-temporal data is high-dimensional, heterogeneous and has non-linear characteristics. In this paper, we apply multivariate Gaussianization for probability density estimation which is robust to dimensionality, comes with statistical guarantees, and is easy to apply. In addition, this methodology allows us to estimate information-theoretic measures to characterize multivariate densities: information, entropy, total correlation, and mutual information. We demonstrate how information theory measures can be applied in various Earth system data analysis problems. First we show how the method can be used to jointly Gaussianize radar backscattering intensities, synthesize hyperspectral data, and quantify of information content in aerial optical images. We also quantify the information content of several variables describing the soil-vegetation status in agro-ecosystems, and investigate the temporal scales that maximize their shared information under extreme events such as droughts. Finally, we measure the relative information content of space and time dimensions in remote sensing products and model simulations involving long records of key variables such as precipitation, sensible heat and evaporation. Results confirm the validity of the method, for which we anticipate a wide use and adoption. Code and demos of the implemented algorithms and information-theory measures are provided.

Valero Laparra, J. Emmanuel Johnson, Gustau Camps-Valls et al.

Information theory is an outstanding framework to measure uncertainty, dependence and relevance in data and systems. It has several desirable properties for real world applications: it naturally deals with multivariate data, it can handle heterogeneous data types, and the measures can be interpreted in physical units. However, it has not been adopted by a wider audience because obtaining information from multidimensional data is a challenging problem due to the curse of dimensionality. Here we propose an indirect way of computing information based on a multivariate Gaussianization transform. Our proposal mitigates the difficulty of multivariate density estimation by reducing it to a composition of tractable (marginal) operations and simple linear transformations, which can be interpreted as a particular deep neural network. We introduce specific Gaussianization-based methodologies to estimate total correlation, entropy, mutual information and Kullback-Leibler divergence. We compare them to recent estimators showing the accuracy on synthetic data generated from different multivariate distributions. We made the tools and datasets publicly available to provide a test-bed to analyze future methodologies. Results show that our proposal is superior to previous estimators particularly in high-dimensional scenarios; and that it leads to interesting insights in neuroscience, geoscience, computer vision, and machine learning.

J. Emmanuel Johnson, Valero Laparra, Adrián Pérez-Suay et al.

Kernel methods are powerful machine learning techniques which implement generic non-linear functions to solve complex tasks in a simple way. They Have a solid mathematical background and exhibit excellent performance in practice. However, kernel machines are still considered black-box models as the feature mapping is not directly accessible and difficult to interpret.The aim of this work is to show that it is indeed possible to interpret the functions learned by various kernel methods is intuitive despite their complexity. Specifically, we show that derivatives of these functions have a simple mathematical formulation, are easy to compute, and can be applied to many different problems. We note that model function derivatives in kernel machines is proportional to the kernel function derivative. We provide the explicit analytic form of the first and second derivatives of the most common kernel functions with regard to the inputs as well as generic formulas to compute higher order derivatives. We use them to analyze the most used supervised and unsupervised kernel learning methods: Gaussian Processes for regression, Support Vector Machines for classification, Kernel Entropy Component Analysis for density estimation, and the Hilbert-Schmidt Independence Criterion for estimating the dependency between random variables. For all cases we expressed the derivative of the learned function as a linear combination of the kernel function derivative. Moreover we provide intuitive explanations through illustrative toy examples and show how to improve the interpretation of real applications in the context of spatiotemporal Earth system data cubes. This work reflects on the observation that function derivatives may play a crucial role in kernel methods analysis and understanding.

Gonzalo Mateo-García, Valero Laparra, Dan López-Puigdollers et al.

The number of Earth observation satellites carrying optical sensors with similar characteristics is constantly growing. Despite their similarities and the potential synergies among them, derived satellite products are often developed for each sensor independently. Differences in retrieved radiances lead to significant drops in accuracy, which hampers knowledge and information sharing across sensors. This is particularly harmful for machine learning algorithms, since gathering new ground truth data to train models for each sensor is costly and requires experienced manpower. In this work, we propose a domain adaptation transformation to reduce the statistical differences between images of two satellite sensors in order to boost the performance of transfer learning models. The proposed methodology is based on the Cycle Consistent Generative Adversarial Domain Adaptation (CyCADA) framework that trains the transformation model in an unpaired manner. In particular, Landsat-8 and Proba-V satellites, which present different but compatible spatio-spectral characteristics, are used to illustrate the method. The obtained transformation significantly reduces differences between the image datasets while preserving the spatial and spectral information of adapted images, which is hence useful for any general purpose cross-sensor application. In addition, the training of the proposed adversarial domain adaptation model can be modified to improve the performance in a specific remote sensing application, such as cloud detection, by including a dedicated term in the cost function. Results show that, when the proposed transformation is applied, cloud detection models trained in Landsat-8 data increase cloud detection accuracy in Proba-V.

J. Emmanuel Johnson, Valero Laparra, Gustau Camps-Valls

Gaussian processes (GPs) are a class of Kernel methods that have shown to be very useful in geoscience and remote sensing applications for parameter retrieval, model inversion, and emulation. They are widely used because they are simple, flexible, and provide accurate estimates. GPs are based on a Bayesian statistical framework which provides a posterior probability function for each estimation. Therefore, besides the usual prediction (given in this case by the mean function), GPs come equipped with the possibility to obtain a predictive variance (i.e., error bars, confidence intervals) for each prediction. Unfortunately, the GP formulation usually assumes that there is no noise in the inputs, only in the observations. However, this is often not the case in earth observation problems where an accurate assessment of the measuring instrument error is typically available, and where there is huge interest in characterizing the error propagation through the processing pipeline. In this letter, we demonstrate how one can account for input noise estimates using a GP model formulation which propagates the error terms using the derivative of the predictive mean function. We analyze the resulting predictive variance term and show how they more accurately represent the model error in a temperature prediction problem from infrared sounding data.

Alexander Hepburn, Valero Laparra, Jesús Malo et al.

Traditionally, the vision community has devised algorithms to estimate the distance between an original image and images that have been subject to perturbations. Inspiration was usually taken from the human visual perceptual system and how the system processes different perturbations in order to replicate to what extent it determines our ability to judge image quality. While recent works have presented deep neural networks trained to predict human perceptual quality, very few borrow any intuitions from the human visual system. To address this, we present PerceptNet, a convolutional neural network where the architecture has been chosen to reflect the structure and various stages in the human visual system. We evaluate PerceptNet on various traditional perception datasets and note strong performance on a number of them as compared with traditional image quality metrics. We also show that including a nonlinearity inspired by the human visual system in classical deep neural networks architectures can increase their ability to judge perceptual similarity. Compared to similar deep learning methods, the performance is similar, although our network has a number of parameters that is several orders of magnitude less.

Alexander Hepburn, Valero Laparra, Ryan McConville et al.

In recent years there has been a growing interest in image generation through deep learning. While an important part of the evaluation of the generated images usually involves visual inspection, the inclusion of human perception as a factor in the training process is often overlooked. In this paper we propose an alternative perceptual regulariser for image-to-image translation using conditional generative adversarial networks (cGANs). To do so automatically (avoiding visual inspection), we use the Normalised Laplacian Pyramid Distance (NLPD) to measure the perceptual similarity between the generated image and the original image. The NLPD is based on the principle of normalising the value of coefficients with respect to a local estimate of mean energy at different scales and has already been successfully tested in different experiments involving human perception. We compare this regulariser with the originally proposed L1 distance and note that when using NLPD the generated images contain more realistic values for both local and global contrast. We found that using NLPD as a regulariser improves image segmentation accuracy on generated images as well as improving two no-reference image quality metrics.

Adrián Pérez-Suay, Valero Laparra, Gonzalo Mateo-García et al.

New social and economic activities massively exploit big data and machine learning algorithms to do inference on people's lives. Applications include automatic curricula evaluation, wage determination, and risk assessment for credits and loans. Recently, many governments and institutions have raised concerns about the lack of fairness, equity and ethics in machine learning to treat these problems. It has been shown that not including sensitive features that bias fairness, such as gender or race, is not enough to mitigate the discrimination when other related features are included. Instead, including fairness in the objective function has been shown to be more efficient. We present novel fair regression and dimensionality reduction methods built on a previously proposed fair classification framework. Both methods rely on using the Hilbert Schmidt independence criterion as the fairness term. Unlike previous approaches, this allows us to simplify the problem and to use multiple sensitive variables simultaneously. Replacing the linear formulation by kernel functions allows the methods to deal with nonlinear problems. For both linear and nonlinear formulations the solution reduces to solving simple matrix inversions or generalized eigenvalue problems. This simplifies the evaluation of the solutions for different trade-off values between the predictive error and fairness terms. We illustrate the usefulness of the proposed methods in toy examples, and evaluate their performance on real world datasets to predict income using gender and/or race discrimination as sensitive variables, and contraceptive method prediction under demographic and socio-economic sensitive descriptors.

Alexander Berardino, Johannes Ballé, Valero Laparra et al.

We develop a method for comparing hierarchical image representations in terms of their ability to explain perceptual sensitivity in humans. Specifically, we utilize Fisher information to establish a model-derived prediction of sensitivity to local perturbations of an image. For a given image, we compute the eigenvectors of the Fisher information matrix with largest and smallest eigenvalues, corresponding to the model-predicted most- and least-noticeable image distortions, respectively. For human subjects, we then measure the amount of each distortion that can be reliably detected when added to the image. We use this method to test the ability of a variety of representations to mimic human perceptual sensitivity. We find that the early layers of VGG16, a deep neural network optimized for object recognition, provide a better match to human perception than later layers, and a better match than a 4-stage convolutional neural network (CNN) trained on a database of human ratings of distorted image quality. On the other hand, we find that simple models of early visual processing, incorporating one or more stages of local gain control, trained on the same database of distortion ratings, provide substantially better predictions of human sensitivity than either the CNN, or any combination of layers of VGG16.

Valero Laparra, Alex Berardino, Johannes Ballé et al.

We develop a framework for rendering photographic images, taking into account display limitations, so as to optimize perceptual similarity between the rendered image and the original scene. We formulate this as a constrained optimization problem, in which we minimize a measure of perceptual dissimilarity, the Normalized Laplacian Pyramid Distance (NLPD), which mimics the early stage transformations of the human visual system. When rendering images acquired with higher dynamic range than that of the display, we find that the optimized solution boosts the contrast of low-contrast features without introducing significant artifacts, yielding results of comparable visual quality to current state-of-the art methods with no manual intervention or parameter settings. We also examine a variety of other display constraints, including limitations on minimum luminance (black point), mean luminance (as a proxy for energy consumption), and quantized luminance levels (halftoning). Finally, we show that the method may be used to enhance details and contrast of images degraded by optical scattering (e.g. fog).

Johannes Ballé, Valero Laparra, Eero P. Simoncelli

We describe an image compression method, consisting of a nonlinear analysis transformation, a uniform quantizer, and a nonlinear synthesis transformation. The transforms are constructed in three successive stages of convolutional linear filters and nonlinear activation functions. Unlike most convolutional neural networks, the joint nonlinearity is chosen to implement a form of local gain control, inspired by those used to model biological neurons. Using a variant of stochastic gradient descent, we jointly optimize the entire model for rate-distortion performance over a database of training images, introducing a continuous proxy for the discontinuous loss function arising from the quantizer. Under certain conditions, the relaxed loss function may be interpreted as the log likelihood of a generative model, as implemented by a variational autoencoder. Unlike these models, however, the compression model must operate at any given point along the rate-distortion curve, as specified by a trade-off parameter. Across an independent set of test images, we find that the optimized method generally exhibits better rate-distortion performance than the standard JPEG and JPEG 2000 compression methods. More importantly, we observe a dramatic improvement in visual quality for all images at all bit rates, which is supported by objective quality estimates using MS-SSIM.

Johannes Ballé, Valero Laparra, Eero P. Simoncelli

We introduce a general framework for end-to-end optimization of the rate--distortion performance of nonlinear transform codes assuming scalar quantization. The framework can be used to optimize any differentiable pair of analysis and synthesis transforms in combination with any differentiable perceptual metric. As an example, we consider a code built from a linear transform followed by a form of multi-dimensional local gain control. Distortion is measured with a state-of-the-art perceptual metric. When optimized over a large database of images, this representation offers substantial improvements in bitrate and perceptual appearance over fixed (DCT) codes, and over linear transform codes optimized for mean squared error.

Valero Laparra, Jesus Malo

This work includes all the technical details of the Sequential Principal Curves Analysis (SPCA) in a single document. SPCA is an unsupervised nonlinear and invertible feature extraction technique. The identified curvilinear features can be interpreted as a set of nonlinear sensors: the response of each sensor is the projection onto the corresponding feature. Moreover, it can be easily tuned for different optimization criteria; e.g. infomax, error minimization, decorrelation; by choosing the right way to measure distances along each curvilinear feature. Even though proposed in [Laparra et al. Neural Comp. 12] and shown to work in multiple modalities in [Laparra and Malo Frontiers Hum. Neuro. 15], the SPCA framework has its original roots in the nonlinear ICA algorithm in [Malo and Gutierrez Network 06]. Later on, the SPCA philosophy for nonlinear generalization of PCA originated substantially faster alternatives at the cost of introducing different constraints in the model. Namely, the Principal Polynomial Analysis (PPA) [Laparra et al. IJNS 14], and the Dimensionality Reduction via Regression (DRR) [Laparra et al. IEEE TGRS 15]. This report illustrates the reasons why we developed such family and is the appropriate technical companion for the missing details in [Laparra et al., NeCo 12, Laparra and Malo, Front.Hum.Neuro. 15]. See also the data, code and examples in the dedicated sites http://isp.uv.es/spca.html and http://isp.uv.es/after effects.html

Emma Izquierdo-Verdiguier, Valero Laparra, Robert Jenssen et al.

This work addresses two main issues of the standard Kernel Entropy Component Analysis (KECA) algorithm: the optimization of the kernel decomposition and the optimization of the Gaussian kernel parameter. KECA roughly reduces to a sorting of the importance of kernel eigenvectors by entropy instead of by variance as in Kernel Principal Components Analysis. In this work, we propose an extension of the KECA method, named Optimized KECA (OKECA), that directly extracts the optimal features retaining most of the data entropy by means of compacting the information in very few features (often in just one or two). The proposed method produces features which have higher expressive power. In particular, it is based on the Independent Component Analysis (ICA) framework, and introduces an extra rotation to the eigen-decomposition, which is optimized via gradient ascent search. This maximum entropy preservation suggests that OKECA features are more efficient than KECA features for density estimation. In addition, a critical issue in both methods is the selection of the kernel parameter since it critically affects the resulting performance. Here we analyze the most common kernel length-scale selection criteria. Results of both methods are illustrated in different synthetic and real problems. Results show that 1) OKECA returns projections with more expressive power than KECA, 2) the most successful rule for estimating the kernel parameter is based on maximum likelihood, and 3) OKECA is more robust to the selection of the length-scale parameter in kernel density estimation.

Valero Laparra, Sandra Jiménez, Gustavo Camps-Valls et al.

Mechanisms of human color vision are characterized by two phenomenological aspects: the system is nonlinear and adaptive to changing environments. Conventional attempts to derive these features from statistics use separate arguments for each aspect. The few statistical approaches that do consider both phenomena simultaneously follow parametric formulations based on empirical models. Therefore, it may be argued that the behavior does not come directly from the color statistics but from the convenient functional form adopted. In addition, many times the whole statistical analysis is based on simplified databases that disregard relevant physical effects in the input signal, as for instance by assuming flat Lambertian surfaces. Here we address the simultaneous statistical explanation of (i) the nonlinear behavior of achromatic and chromatic mechanisms in a fixed adaptation state, and (ii) the change of such behavior. Both phenomena emerge directly from the samples through a single data-driven method: the Sequential Principal Curves Analysis (SPCA) with local metric. SPCA is a new manifold learning technique to derive a set of sensors adapted to the manifold using different optimality criteria. A new database of colorimetrically calibrated images of natural objects under these illuminants was collected. The results obtained by applying SPCA show that the psychophysical behavior on color discrimination thresholds, discount of the illuminant and corresponding pairs in asymmetric color matching, emerge directly from realistic data regularities assuming no a priori functional form. These results provide stronger evidence for the hypothesis of a statistically driven organization of color sensors. Moreover, the obtained results suggest that color perception at this low abstraction level may be guided by an error minimization strategy rather than by the information maximization principle.

Valero Laparra, Gustavo Camps-Valls, Jesús Malo

Most signal processing problems involve the challenging task of multidimensional probability density function (PDF) estimation. In this work, we propose a solution to this problem by using a family of Rotation-based Iterative Gaussianization (RBIG) transforms. The general framework consists of the sequential application of a univariate marginal Gaussianization transform followed by an orthonormal transform. The proposed procedure looks for differentiable transforms to a known PDF so that the unknown PDF can be estimated at any point of the original domain. In particular, we aim at a zero mean unit covariance Gaussian for convenience. RBIG is formally similar to classical iterative Projection Pursuit (PP) algorithms. However, we show that, unlike in PP methods, the particular class of rotations used has no special qualitative relevance in this context, since looking for interestingness is not a critical issue for PDF estimation. The key difference is that our approach focuses on the univariate part (marginal Gaussianization) of the problem rather than on the multivariate part (rotation). This difference implies that one may select the most convenient rotation suited to each practical application. The differentiability, invertibility and convergence of RBIG are theoretically and experimentally analyzed. Relation to other methods, such as Radial Gaussianization (RG), one-class support vector domain description (SVDD), and deep neural networks (DNN) is also pointed out. The practical performance of RBIG is successfully illustrated in a number of multidimensional problems such as image synthesis, classification, denoising, and multi-information estimation.

Valero Laparra, Sandra Jiménez, Devis Tuia et al.

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

Valero Laparra, Juan Gutiérrez, Gustavo Camps-Valls et al.

A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. However, analytical estimates can be obtained only for particular combinations of analytical models of signal and noise, thus precluding its straightforward extension to deal with other arbitrary noise sources. In this paper, we propose an alternative non-explicit way to take into account the relations among natural image wavelet coefficients for denoising: we use support vector regression (SVR) in the wavelet domain to enforce these relations in the estimated signal. Since relations among the coefficients are specific to the signal, the regularization property of SVR is exploited to remove the noise, which does not share this feature. The specific signal relations are encoded in an anisotropic kernel obtained from mutual information measures computed on a representative image database. Training considers minimizing the Kullback-Leibler divergence (KLD) between the estimated and actual probability functions of signal and noise in order to enforce similarity. Due to its non-parametric nature, the method can eventually cope with different noise sources without the need of an explicit re-formulation, as it is strictly necessary under parametric Bayesian formalisms. Results under several noise levels and noise sources show that: (1) the proposed method outperforms conventional wavelet methods that assume coefficient independence, (2) it is similar to state-of-the-art methods that do explicitly include these relations when the noise source is Gaussian, and (3) it gives better numerical and visual performance when more complex, realistic noise sources are considered. Therefore, the proposed machine learning approach can be seen as a more flexible (model-free) alternative to the explicit description of wavelet coefficient relations for image denoising.

Valero Laparra, Jesus Malo, Gustau Camps-Valls

This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate regression to ensure the reduction in redundancy between he PCA coefficients, the reduction of the variance of the scores, and the reduction in the reconstruction error. More importantly, unlike other nonlinear dimensionality reduction methods, the invertibility, volume-preservation, and straightforward out-of-sample extension, makes DRR interpretable and easy to apply. The properties of DRR enable learning a more broader class of data manifolds than the recently proposed Non-linear Principal Components Analysis (NLPCA) and Principal Polynomial Analysis (PPA). We illustrate the performance of the representation in reducing the dimensionality of remote sensing data. In particular, we tackle two common problems: processing very high dimensional spectral information such as in hyperspectral image sounding data, and dealing with spatial-spectral image patches of multispectral images. Both settings pose collinearity and ill-determination problems. Evaluation of the expressive power of the features is assessed in terms of truncation error, estimating atmospheric variables, and surface land cover classification error. Results show that DRR outperforms linear PCA and recently proposed invertible extensions based on neural networks (NLPCA) and univariate regressions (PPA).

Johannes Ballé, Valero Laparra, Eero P. Simoncelli

We introduce a parametric nonlinear transformation that is well-suited for Gaussianizing data from natural images. The data are linearly transformed, and each component is then normalized by a pooled activity measure, computed by exponentiating a weighted sum of rectified and exponentiated components and a constant. We optimize the parameters of the full transformation (linear transform, exponents, weights, constant) over a database of natural images, directly minimizing the negentropy of the responses. The optimized transformation substantially Gaussianizes the data, achieving a significantly smaller mutual information between transformed components than alternative methods including ICA and radial Gaussianization. The transformation is differentiable and can be efficiently inverted, and thus induces a density model on images. We show that samples of this model are visually similar to samples of natural image patches. We demonstrate the use of the model as a prior probability density that can be used to remove additive noise. Finally, we show that the transformation can be cascaded, with each layer optimized using the same Gaussianization objective, thus offering an unsupervised method of optimizing a deep network architecture.