
Robust threshold estimation for images with unimodal histograms
Nicolas Coudray, Jean-Luc Buessler, Jean-Philippe Urban
*
Université de Haute-Alsace, Laboratoire MIPS, 4 Rue des Frères Lumière, 68093 Mulhouse, France
article info
Article history:
Received 13 January 2009
Received in revised form 1 October 2009
Available online 4 January 2010
Communicated by J.A. Robinson
Keywords:
Automatic thresholding
Image histogram
Unimodal distribution
Edge detection
abstract
This article introduces a method to determine in a robust manner the threshold in highly noisy gradient
images. To enhance the robustness, the proposed technique is based on a piecewise linear regression to fit
the whole descending slope of the histogram, rather than the search of some specific points. The algo-
rithm gives a reliable estimation of the threshold, and is practically insensitive to the noise distribution,
to the quantity of edge pixels to segment, and to random histogram fluctuations.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
The study of electron microscopy images of biological speci-
mens, which are especially noisy and low contrasted, has led us
to develop a new method to automatically threshold the edges in
a robust manner, compatible with a multiresolution approach. Glo-
bal edge thresholding methods have been widely studied, but the
robustness of the threshold determination remains a problem.
Our objective is to obtain a reliable threshold whatever the amount
of edges, the noise distribution or the histogram statistical fluctu-
ations. These parameters vary considerably between different
acquisitions and different levels of a multiresolution analysis.
We propose a thresholding method called the T-point algo-
rithm. The original minimization criterion to set the threshold is
studied in this article. The algorithm is simple, has low computa-
tional complexity, and requires little a priori knowledge concern-
ing the distribution and proportion of pixels to segment.
To achieve these objectives, a novel minimization criterion is
proposed. It consists of the minimization of the error between
the descending slope of the histogram and its piecewise linear
regression. It belongs to the histogram-based algorithms, but, un-
like the main available methods, its computation relies on a wide
portion of the histogram rather than a single bin or a few local bins.
The technique is presented for gradient images, but can be gen-
eralized to other types of images. It does not rely on a specific gray-
level unimodal distribution, but on the hypothesis of a less popu-
lated class belonging to the tail of the histogram.
The main automatic thresholding algorithms are reviewed in
Section 2. The proposed method is described in Section 3. Experi-
ments and results are discussed in Section 4. Examples of applica-
tions are given in Section 5.
2. Thresholding of unimodal histograms
Setting thresholds is a non-trivial problem. Ideally, each coher-
ent set of pixels of the image is characterized by its gray-
level and the histogram presents several corresponding maxima
(modes): the threshold can simply be placed at the local minima
between the peaks of the histograms, detected either directly or
indirectly, or it can be placed by analyzing the statistics of the
classes to minimize (Otsu, 1979; Hou et al., 2006; Nakib et al.,
2008). However, in many cases, when the sets are not clearly distin-
guished, the histogram becomes unimodal.
Fig. 1 illustrates an unimodal histogram of an image composed
of two sets of pixels to be separated: SET 1 and SET 2 . Two non-
exclusive reasons explain unimodality: SET 2 is very small com-
pared to SET 1, and/or the overall signal to noise ratio (SNR) is
low (both sets share many gray-levels). These types of histograms
can be viewed in low contrasted images (e.g., Baradez et al., 2004,
in microscopy images), but they are more frequently observed in
high-pass filtered images where SET 2 represents the edges to be
segmented, and SET 1 noisy homogeneous regions (non-edge pix-
els). The gray-level of the edges being higher than the mean
gray-level of the noise, the edges, which are less numerous, con-
tribute mainly to the upper elongated tail of the histogram, simply
referred to here as the tail. Edges have generally higher gradient
magnitudes than homogeneous regions, they are bound to be in
minority in the images, and their amplitude can vary over a wide
0167-8655/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.patrec.2009.12.025
* Corresponding author. Tel.: +33 3 89 33 64 32; fax: +33 3 89 33 60 84.
E-mail addresses: nicolas.coudray@uha.fr (N. Coudray), jean-luc.buessler@uha.fr
(J.-L. Buessler), jean-philippe.urban@uha.fr (J.-P. Urban).
Pattern Recognition Letters 31 (2010) 1010–1019
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Pattern Recognition Letters
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