17 research outputs found
Sequential Dimensionality Reduction for Extracting Localized Features
Linear dimensionality reduction techniques are powerful tools for image
analysis as they allow the identification of important features in a data set.
In particular, nonnegative matrix factorization (NMF) has become very popular
as it is able to extract sparse, localized and easily interpretable features by
imposing an additive combination of nonnegative basis elements. Nonnegative
matrix underapproximation (NMU) is a closely related technique that has the
advantage to identify features sequentially. In this paper, we propose a
variant of NMU that is particularly well suited for image analysis as it
incorporates the spatial information, that is, it takes into account the fact
that neighboring pixels are more likely to be contained in the same features,
and favors the extraction of localized features by looking for sparse basis
elements. We show that our new approach competes favorably with comparable
state-of-the-art techniques on synthetic, facial and hyperspectral image data
sets.Comment: 24 pages, 12 figures. New numerical experiments on synthetic data
sets, discussion about the convergenc
Using Underapproximations for Sparse Nonnegative Matrix Factorization
Nonnegative Matrix Factorization consists in (approximately) factorizing a
nonnegative data matrix by the product of two low-rank nonnegative matrices. It
has been successfully applied as a data analysis technique in numerous domains,
e.g., text mining, image processing, microarray data analysis, collaborative
filtering, etc.
We introduce a novel approach to solve NMF problems, based on the use of an
underapproximation technique, and show its effectiveness to obtain sparse
solutions. This approach, based on Lagrangian relaxation, allows the resolution
of NMF problems in a recursive fashion. We also prove that the
underapproximation problem is NP-hard for any fixed factorization rank, using a
reduction of the maximum edge biclique problem in bipartite graphs.
We test two variants of our underapproximation approach on several standard
image datasets and show that they provide sparse part-based representations
with low reconstruction error. Our results are comparable and sometimes
superior to those obtained by two standard Sparse Nonnegative Matrix
Factorization techniques.Comment: Version 2 removed the section about convex reformulations, which was
not central to the development of our main results; added material to the
introduction; added a review of previous related work (section 2.3);
completely rewritten the last part (section 4) to provide extensive numerical
results supporting our claims. Accepted in J. of Pattern Recognitio
Two Algorithms for Orthogonal Nonnegative Matrix Factorization with Application to Clustering
Approximate matrix factorization techniques with both nonnegativity and
orthogonality constraints, referred to as orthogonal nonnegative matrix
factorization (ONMF), have been recently introduced and shown to work
remarkably well for clustering tasks such as document classification. In this
paper, we introduce two new methods to solve ONMF. First, we show athematical
equivalence between ONMF and a weighted variant of spherical k-means, from
which we derive our first method, a simple EM-like algorithm. This also allows
us to determine when ONMF should be preferred to k-means and spherical k-means.
Our second method is based on an augmented Lagrangian approach. Standard ONMF
algorithms typically enforce nonnegativity for their iterates while trying to
achieve orthogonality at the limit (e.g., using a proper penalization term or a
suitably chosen search direction). Our method works the opposite way:
orthogonality is strictly imposed at each step while nonnegativity is
asymptotically obtained, using a quadratic penalty. Finally, we show that the
two proposed approaches compare favorably with standard ONMF algorithms on
synthetic, text and image data sets.Comment: 17 pages, 8 figures. New numerical experiments (document and
synthetic data sets
Using machine learning on the sources of retinal images for diagnosis by proxy of diabetes mellitus and diabetic retinopathy
In current research in ophthalmology, images of the vascular system in the human retina are used as exploratory proxies for pathologies affecting different organs. This thesis addresses the analysis, using machine learning and computer vision techniques, of retinal images acquired with different techniques (Fundus retinographies, optical coherence tomography and optical coherence tomography angiography), with the objective of using them to assist diagnostic decision making in diabetes mellitus and diabetic retinopathy. This thesis explores the use of matrix factorization-based source extraction techniques, as the basis to transform the retinal images for classification. The proposed approach consists on preprocessing the images to enable the learning of an unsupervised parts-based representation prior to the classification. As a result of the use of interpretable models, with this approach we unveiled an important bias in the data. After correcting for the bias, promising results were still obtained which merit for further exploration
On Restricted Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is the problem of decomposing a given
nonnegative matrix into a product of a nonnegative matrix and a nonnegative matrix . Restricted NMF
requires in addition that the column spaces of and coincide. Finding
the minimal inner dimension is known to be NP-hard, both for NMF and
restricted NMF. We show that restricted NMF is closely related to a question
about the nature of minimal probabilistic automata, posed by Paz in his seminal
1971 textbook. We use this connection to answer Paz's question negatively, thus
falsifying a positive answer claimed in 1974. Furthermore, we investigate
whether a rational matrix always has a restricted NMF of minimal inner
dimension whose factors and are also rational. We show that this holds
for matrices of rank at most and we exhibit a rank- matrix for which
and require irrational entries.Comment: Full version of an ICALP'16 pape
A Block Coordinate Descent-based Projected Gradient Algorithm for Orthogonal Non-negative Matrix Factorization
This article utilizes the projected gradient method (PG) for a non-negative
matrix factorization problem (NMF), where one or both matrix factors must have
orthonormal columns or rows. We penalise the orthonormality constraints and
apply the PG method via a block coordinate descent approach. This means that at
a certain time one matrix factor is fixed and the other is updated by moving
along the steepest descent direction computed from the penalised objective
function and projecting onto the space of non-negative matrices.
Our method is tested on two sets of synthetic data for various values of
penalty parameters. The performance is compared to the well-known
multiplicative update (MU) method from Ding (2006), and with a modified global
convergent variant of the MU algorithm recently proposed by Mirzal (2014). We
provide extensive numerical results coupled with appropriate visualizations,
which demonstrate that our method is very competitive and usually outperforms
the other two methods
A hybrid approach for stain normalisation in digital histopathological images
Stain in-homogeneity adversely affects segmentation and quantifi-cation of tissues in histology images. Stain normalisation techniques have been used to standardise the appearance of images. However, most the available stain normalisation techniques only work on a particular kind of stain images. In addition, some of these techniques fail to utilise both the spatial and tex-tural information in histology images, leading to image tissue distortion. In this paper, a hybrid approach has been developed, based on an octree colour quantisation algorithm combined with the Beer-Lambert law, a modified blind source separation algorithm, and a modified colour transfer approach. The hybrid method consists of two stages the stain separation stage and colour transfer stage. An octree colour quantisation algorithm combined with Beer-Lambert law, and a modified blind source separation algorithm are used during the stain separation stage to computationally estimate the amount of stain in an histology image based on its chromatic and luminous response. A modified colour transfer algorithm is used during the colour transfer stage to minimise the effect of varying staining and illumination. The hybrid method addresses the colour variation problem in both H&DAB (Haemotoxylin and Diaminoben-zidine) and H&E (Haemotoxylin and Eosin) stain images. The stain normali-sation method is validated against ground truth data. It is widely known that the Beer-Lambert law applies to only stains (such as haematoxylin, eosin) that absorb light. We demonstrate that the Beer-Lambert law applies is applicable to images containing a DAB stain. Better stain normalisation results are obtained in both H&E and H&DAB images