[1]Aicha Baya Goumeidane

1]DTSI, Research Center in industrial Technologies, CRTI , P.O.BOX 64,
Algiers, 16000, Algeria

2]Département informatique, Université de Sherbrooke , Sherbrooke, Québec,
Canada

# Guided Scale Space Radon Transform for linear structures detection

<EMAIL_ADDRESS><EMAIL_ADDRESS>Djemel Ziou  Nafaa Nacereddine  [
[

###### Abstract

Using integral transforms to the end of lines detection in images with complex
background, makes the detection a hard task needing additional processing to
manage the detection. As an integral transform, the Scale Space Radon
Transform (SSRT) suffers from such drawbacks, even with its great abilities
for thick lines detection. In this work, we propose a method to address this
issue for automatic detection of thick linear structures in gray scale and
binary images using the SSRT, whatever the image background content. This
method involves the calculated Hessian orientations of the investigated image
while computing its SSRT, in such a way that linear structures are emphasized
in the SSRT space. As a consequence, the subsequent maxima detection in the
SSRT space is done on a modified transform space freed from unwanted parts
and, consequently, from irrelevant peaks that usually drown the peaks
representing lines. Besides, highlighting the linear structure in the SSRT
space permitting, thus, to efficiently detect lines of different thickness in
synthetic and real images, the experiments show also the method robustness
against noise and complex background.

###### keywords:

Scale Space Radon Transform, Linear Structures Detection, Hessian, Object
Orientation

##  1 Introduction

Automatic linear structures detection in digital images is a classical topic
addressed by many researchers for years. Each research has tackled the issue
from a different point of view, exploiting different techniques leading to
application-based approaches. Indeed, images containing linear structures can
be produced with various image formation process, and therefore, can be very
different from each other?s, having, consequently, different characteristics,
so the proposed approach must, therefore, manage the image content. As
previously noticed, automatic detection of lines has been employed in a wide
range of applications, including robot guidance [1, 2], Autonomous vehicles
and Advanced Driver Assistance Systems [3, 4, 5], high-voltage lines detection
[6], Barcode detection [7], road network extraction [8, 9]. The exploited
techniques vary from deep Learning to integral transforms-driven methods or
their association [10], as well as, derivatives-based proposals [11]. However,
the most robust approaches to achieve the task of straight line detection are
the ones involving integral transforms as Radon Transform (RT)[12], Hough
Transform (HT) [13], and the transforms derived therefrom associated to basic
image processing [14]. In fact, RT and HT and their extensions have been
widely applied to the above-mentioned purpose. They consist in converting the
global line detection problem in the image domain into a peak detection
problem in the transform one, even in presence of noise [15]. These peaks
positions correspond to the lines parameters. Unfortunately, most of the
integral transform-based proposed methods include pre-processing steps,
additional tricks, supplementary procedures and predefined thresholds to
manage the detection, as often ambiguities regarding the relevant and
irrelevant peaks in the transforms spaces rise. In fact, applied to images,
these transforms correspond to projections and pixels intensity accumulation
over the image in all directions. Consequently, sinograms or accumulators can
present peaks where there are no linear features. To overcome such drawbacks,
some proposals suggest to limit the computation to dynamically adjusted Region
of Interests (ROI) in the image obtained with deep learning, with the aim of
avoiding irrelevant image parts [16]. Others propose to exclude a large number
of useless peaks in the transform space by keeping only the ones related to
lines passing through the vanishing point in lane detection application [17],
whereas voting schemes is exploited to select the real peaks among the others
by using statistics and entropy in [18] and [19], respectively. Concerning RT
and HT, it is worth to note that their respective spaces are constructed by
pixels values accumulation (in a line-wise manner) done line by line which
make them adequate only for filiform lines, unless involving additional
processing and artifices to do the job accuratly for thick structures as
proposed by the authors in [20]. Fortunately, the work proposed in [21] have
introduced a novel integral transform called Scale Space Radon transform
(SSRT), which can be viewed as a significant generalized form of the Radon
Transform by replacing the Dirac function in RT by a Gaussian kernel tuned by
a scale space parameter. In the early paper about the SSRT, the authors have
shown that this transform can be used to detect accurately and in an elegant
manner thick lines when the scale parameter is tuned correctly. Nevertheless,
the SSRT has inherited some RT drawbacks, especially its sensitivity to
complex backgrounds, particularly when the feature is too short comparing to
the image dimensions. This fact produces peaks in the SSRT space that do not
correspond to lines. To overcome this shortcoming, authors in [22] have
proposed to construct the SSRT space only around pre-computed SSRT parameters
yielded with multiscales Hessian of an image, with the aim of discarding
orientations and positions bringing useless information. However, the pixels
values integration is done inside a band for the SSRT in all directions in the
entire image, which increases the interference of the irrelevant parts of the
image when constructing the SSRT space, leading, as a result, to the
introduction of a bias in the detected linear structures. This situation
occurs even if this space is constructed around precomputed SSRT parameters
and excludes the remaining of the space, because still the transform space
computation incorporates image elements parts that lay in the precomputed
orientation and position of the researched linear structures. These image
elements parts will interfere while computing the SSRT space and may deviate
the detected line from its correct position, especially when the processed
images are gray valued and/or the linear structure is a short one. In this
paper, we propose a solution to this issue to, simultaneously, emphasis the
linear features in the SSRT space and exclude major or all parts of the other
features. Furthermore, in each SSRT projection computation, the proposed
approach removes the contribution of all image elements that do not have the
orientation of the actual projection. These procedures operate by the means of
the SSRT and the Hessian of the processed image.

The remainder of this paper is presented as follows. In Sect. 2 we introduce
the SSRT transform and the image Hessian. Section 3 is consecrated to the
proposed method for linear structure exatraction. Section 4 is dedicated to
experiments and results. Finally, conclusion is drawn in Sect. 5.

##  2 Methods and material

###  2.1 Scale Space Radon Transform

The Scale Space Radon Transform (SSRT), $\check{S_{f}}$, of an image $f$, is a
matching of a kernel and an embedded parametric shape in this image. If the
parametric shape is a line parametrized by the location parameter $\rho$ and
the angle $\theta$ and the kernel is a Gaussian one, then $\check{S_{f}}$ is
given by [21]

$\check{S_{f}}(\rho,\theta,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}\int_{\cal
X}\int_{\cal
Y}f(x,y)e^{-\frac{(x\cos\theta+y\sin\theta-\rho)^{2}}{2\sigma^{2}}}dxdy$ (1)

Here, $\sigma$ is the scale space parameter. It returns that the RT
$\check{R_{f}}$ is a special case of $\check{S_{f}}$ (when $\sigma\rightarrow\
0$) [21]. The replacement of the Dirac function $\delta$ in RT by a Gaussian
kernel in the SSRT allows $\check{S_{f}}$ to handle correctly embedded shapes
even if they are not filiform, unlike $\check{R_{f}}$. The role of the chosen
kernel is to control the parametric shape position inside the embedded object
via the scale parameter and then, the detection is reduced to maxima detection
in the SSRT space.

Figure 1: Directional Gaussian $DG_{\theta,\rho,\sigma}$ used in SSRT
computation, the unidimensional Gaussian $g_{\sigma}(\rho)$ and direction of
SSRT projection defined with red arrows.

The directional Gaussian in (1),
$DG_{\theta,\rho,\sigma}(x,y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x\cos\theta+y\sin\theta-\rho)^{2}}{2\sigma^{2}}}$,
can be viewed in Fig.1. It is an infinity of parallel 1d Gaussians,
$g_{\sigma}(\rho)$ having their mean values belonging to the line $\Delta$
with equation $x\cos\theta+y\sin\theta-\rho=0$.

Detection of linear structures performed with SSRT employed alone, gives
accurate results when the image is not complex and the SSRT space present
maxima related to the available linear structures as shown in [21]
Nevertheless, as a detection method, it can be subjected to problems related
to the detection issues, i.e: over-detection, under-detection and accuracy.
The over-detection occurs when the number of linear structures increases or
when the background is a complex one. In fact, in such cases, the SSRT
detection becomes a complicated task and we can notice, therefore, the
apparition of maxima that do not represent structures. As a result, spurious
linear structures are detected, as depicted in Fig.2. Furthermore, when
applying a "naive" thresholding to remedy to the over detection and improve
the detection, not all spurious lines are deleted, while suppressing relevant
ones, as shown in Fig.3, this is the under detection.

Figure 2: From left to right: SSRT space with highlighted maxima and the
corresponding line.  Figure 3: From left to right: Thresolded SSRT space wit
the remaining maxima and the corresponding lines  Figure 4: (a): Detection
with one structure. (b) Detection when adding in a certain way another
structure.

Moreover, some configurations, as shown by the image in Fig.2 and in Fig.4.b.
lead to the deviation of the detected lines. For example in Fig.4.b, the
longest linear structure, being in the continuation of the short one with
slightly different angle, has displaced the maximum of the latter in the SSRT
space and therefore its position in the image domain. We can see in Fig.4.a,
that the detection is accurate when the short thick line is alone. Regrading
the SSRT spaces of the two images, they are superposed and depicted in Fig.5
where we can see that the highlighted maximum in the blue SSRT space
(corresponding to Fig.4.a) has been subjected a modification (the transparent
space corresponds to Fig.4.b.) when adding the other structure in Fig.4.b.
This deformation has displaced the short structure maximum, and create a
problem of accuracy.

Figure 5: Superposition of the blue SSRT space of the image in Fig.4.a and the
transparent one of the image in Fig.4.b, and highlighted maxima.

The same thing can be said about the structures in Fig.2. In fact when each
structure is processed alone, the SSRT centerline detection result is the
adequate one, as seen in Fig.6.

Figure 6: Structures centerlines detections when each structue is alone in the
image

Consequently, some additional processing must be added to the SSRT detection
to deal with such situations and improve the detection. To this end, we joint
the image Hessian computation to the SSRT as we will see later.

###  2.2 The Hessian

The Hessian of an image has been used for identifying particular structures
centers when the scale $\sigma_{h}$ of the Hessian matches the size of the
local structures in images [23]. The Hessian matrix $H_{\sigma_{h}}$ of the
intensity image $f$ at a scale $\sigma_{h}$ is given by

$\small H_{\sigma_{h}}=\begin{pmatrix}H_{11}(\sigma_{h})&H_{12}(\sigma_{h})\\\
H_{21}(\sigma_{h})&H_{22}(\sigma_{h})\\\
\end{pmatrix}=\begin{pmatrix}\sigma_{h}^{2}f\circledast\frac{\partial^{2}}{\partial
x^{2}}G(\sigma_{h})&\sigma_{h}^{2}f\circledast\frac{\partial^{2}}{\partial
x\partial y}G(\sigma_{h})\\\
\sigma_{h}^{2}f\circledast\frac{\partial^{2}}{\partial y\partial
x}G(\sigma_{h})&\sigma_{h}^{2}f\circledast\frac{\partial^{2}}{\partial
y^{2}}G(\sigma_{h})\\\ \end{pmatrix}\normalsize$ (2)

$G(\sigma_{h})$ is the bivariate Gaussian kernel of standard deviation
$\sigma_{h}$ and $\circledast$ is the convolution operator. Under continuity
assumptions the Hessian matrix is symmetric and non-negative definite.

Figure 7: 3D representation of Hessian components; (a) $-\lambda_{1}$, (b)
$-\lambda_{2}$, (c) $-\lambda_{max}$, of the linear structure in (d). (e)
Superposition of the linear structure and the vectors field
$\textbf{n}^{\perp}$ with a zoomed part of it

Let $\lambda_{1}$ and $\lambda_{2}$ be the eigenvalues of $H_{\sigma_{h}}$
obtained by eigenvalue decomposition as given in [24]. Let, furthermore,
$\lambda_{max}$ be the eigenvalue with maximum absolute value at a point
$(x,y)$ and $\textbf{n}(n_{x},n_{y})$ the unitary vectors field, designating
the orientations ${\phi}$ associated with $\lambda_{max}$. The components
$n_{x}$ and $n_{y}$ are the components of n along the $x$-axis and the
$y$-axis respectively. They are computed as
$\small\left[n_{x}(x,y),n_{y}(x,y)\right]=\left[\frac{H_{11}(x,y),\
\lambda_{max}(x,y)-H_{11}(x,y)}{\sqrt{H_{21}(x,y)^{2}+(\lambda_{max}(x,y)-H_{21}(x,y))^{2}}}\right]\normalsize$.
Furthermore, let $\textbf{n}^{\perp}$ be the vectors field made of vectors
normal to those composing n. In case of linear structures, a part of the field
$\textbf{n}^{\perp}$ matching the inside portion of the structure, points
towards to the orientation of the latter. Before going further, it is worth to
mention that here, and for notation, we have chosen that
$|\lambda_{2}|\geq|\lambda_{1}|$. Consequently, for a bright linear structure
on a dark background, we can see a 3D representations of $\lambda_{1}$,
$\lambda_{2}$ and $\lambda_{max}$ in Fig.7.a, Fig.7.b and Fig.7.c, of the
linear structure in Fig.7.d, as well as the field $\textbf{n}^{\perp}$ and a
zoom on it, inside the structure in Fig.7.e .

If one wants to consider only vectors $n^{\perp}(x,y)$ inside the structures
then, the ones matching the outside of the structure can be discarded by
keeping only those corresponding to $-\lambda_{max}(x,y)>0$, i.e ; if
$-\lambda_{max}(x,y)<0$, then $\textbf{n}^{\perp}(x,y)\leftarrow 0$ as shown
in Fig.8. Let us call the resulting field, illustrated in 8.c,
$\textbf{n}^{\perp}_{-\lambda_{max}>0}$.

Figure 8: (a):$-\lambda_{max}$, (b):$-\lambda_{max}>0$, (c): the corresponding
vectors field $\textbf{n}^{\perp}_{-\lambda_{max}>0}$ for the linear structure
in Fig.7.d.

##  3 Guided SSRT

For the sake of simplicity, let us rename, in the following,
$\textbf{n}^{\perp}_{-\lambda_{max}>0}$ as $\textbf{n}^{\perp}$. To take
advantage of the SSRT benefits without being subjected to the drawbacks
mentioned before, SSRT computation will be guided by the orientation of the
linear structures. This orientation is provided via the vector field
$\textbf{n}^{\perp}$ computed with the Hessian of the image under
investigation $f$. Let us call this SSRT guided by the Hessian of the image
$f$, $\check{S_{f_{H}}}$. We define $\textbf{p}_{\theta}$, the unitary vector
field applied on all image points. The directions of vectors composing
$\textbf{p}_{\theta}$ is the one of the actual SSRT projection as seen in
Fig.1. If
$\Gamma(x,y)=|(\textbf{n}^{\perp}(x,y)\odot\textbf{p}_{\theta}(x,y))|$, where
$\odot$ is the dot product operator, let us, moreover, define $f_{H}$, the
image yielded by the operation $f_{H}(x,y)=f(x,y)\Gamma(x,y)$. The function
$\Gamma(x,y)$ can be written as
$\Gamma(x,y)=|{p_{\theta}}_{x}(x,y)n^{\perp}_{x}(x,y)+{p_{\theta}}_{y}(x,y)n^{\perp}_{y}(x,y)|$,
where ${p_{\theta}}_{x}$, $n^{\perp}_{x}$, ${p_{\theta}}_{y}$ and
$n^{\perp}_{y}$ are vectors $\textbf{n}^{\perp}$, $\textbf{p}_{\theta}$
components along the $x$-axis and the $y$-axis, which makes this function
taking its values in $\left[0\ 1\right]$. If fact, when the directions of the
vectors $\textbf{n}^{\perp}(x,y)$ and $\textbf{p}_{\theta}(x,y)$ are
approximately of the same order in absolute value, then $\Gamma(x,y)\approx
1$, but when their difference approximates in absolute value $\pi/2$, then,
$\Gamma(x,y)\approx 0$. We can see that $f_{H}$ changes for each projection
orientation , as $\textbf{p}_{\theta}$ depends on the latter. Consequently, in
each SSRT projection, all image features in $f$ that have not as orientation
the actual projection direction, are forced down in $f_{H}(x,y)$. Furthermore,
to lessen further the influence of previously mentioned image parts,
$\Gamma(x,y)$ is raised to a power $M$, with $M>>1$, dragging values of all
pixels in $f$ of location (x,y) to zero in $f_{H}$ when
${p_{\theta}}_{x}(x,y)\neq n^{\perp}_{x}(x,y)$ and ${p_{\theta}}_{y}(x,y)\neq
n^{\perp}_{y}(x,y)$, enhancing meanwhile, image features having SSRT
projection direction as orientation. It worth mentioning here, that vectors in
$\textbf{n}^{\perp}(x,y)$ are normalized so that their respective magnitude is
equal to 1, making them unitary vectors.

Figure 9: (a) Superposition of the fields $\textbf{n}^{\perp}$ (in blue) and
field $\textbf{p}_{\theta}$ (in red) for $\theta=20$ of the linear structure
depicted in Fig.7.d, and (b) the corresponding image $f_{H}$

Fig.9 shows both $\textbf{n}^{\perp}$ and $\textbf{p}_{\theta}$ for
$\theta=20$ degrees as well as the corresponding $f_{H}$ for $M$ set equals to
100. We can see in this case that the linear structure has almost disappeared
in $f_{H}$, i.e. almost pixels are set to zero (0) , which means that the
projection becomes almost blind regarding this image content for this
orientation. The SSRT $\check{S_{f_{H}}}$ exploiting $f_{H}$ is computed as

$\check{S_{f_{H}}}(\rho,\theta,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}\int_{\cal
X}\int_{\cal
Y}f_{H}(x,y)e^{-\frac{(x\cos\theta+y\sin\theta-\rho)^{2}}{2\sigma^{2}}}dxdy$
(3)

###  3.1 Hessian scale

The question that arises in case of exploiting the Hessian of an image is what
Hessian scale should be used when no information about the linear structures
width is available. It should be recalled that, the only Hessian features to
be exploited here are the vectors composing $\textbf{n}^{\perp}$, as the SSRT
computation is guided by their orientations. Let us see in the following, the
effects of changing the Hessian scale, on the directions of the mentioned
vectors and the consequences on the Hessian guided SSRT spaces. The linear
structure shown in Fig.7.d has as width $w=23$ pixels, so, as demonstrated in
[23], the scale $\sigma_{h}$ should satisfy
$\sigma_{h}\geq\frac{w}{\sqrt{3}}\geq 13.28$, for linear structures
enhancement and detection. Let us see what happens if we take scales greater
and lower than this value for our guided SSRT based-detection, i.e
$\sigma_{h}=13.28\pm 10$. So we use $\sigma_{h}\in\left\\{3.28,\
23.28\right\\}$ and we observe the results on the Hessian guided SSRT spaces
and the lines representing the detected maxima. Let us begin by
$\sigma_{h}=3.28$. We can see in Fig.10 the vectors $\textbf{n}^{\perp}$ for
this scale. We remark that inside the structure they are pointing in the
latter direction.

Figure 10:  (a) the field $\textbf{n}^{\perp}$ generated with
$\sigma_{h}=3.28$ and a zoom on it

Concerning the maxima detection in the SSRT space, it is worth to recall that
as the RT, the SSRT has aperiodicity of $\pi$, i.e.
$\check{S_{f}}(\rho,\theta)=\check{S_{f}}(\rho,\theta+\pi)$, which makes us do
not highlight in Fig.11 the redundant maxima and do not take them into
consideration. Consequently, we can notice in Fig.11 that the guided SSRT
space has one maximum for the structure and the detected line is located in
the center of the structure as shown in the same figure.

Figure 11:  Hessian guided SSRT space with $\sigma_{h}=3.28$ and a the
detected centerline of the structure with the Hessian guided SSRT.

The result of using $\sigma_{h}=23.28$ is illustrated in Fig.12 with the field
$\textbf{n}^{\perp}$ in Fig.12.a and the Hessian guided SSRT in Fig.12.b. Note
that, the detected centerline is the same as the one illustrated in Fig.11.
This is to say that, it is not necessary to have an exact information about
the width of the linear structure to be detected, and to choose accordingly
the Hessian scale. In fact, a coarse knowledge about the structure derived
from the involved application, and consequently an approximate Hessian scale
is sufficient to achieve detection with the proposed method, conversely to the
other integral transforms when they are similarly associated to the Hessian,
like the Radon transform, for example. Indeed, if we consider the Radon
transform guided by the Hessian and, analogously, named it $\check{R_{f_{H}}}$
and computed it as $\small\check{R_{f_{H}}}(\rho,\theta)=\int_{\cal
X}\int_{\cal Y}f_{H}(x,y)\delta(x\cos\theta+y\sin\theta-\rho)dxdy\normalsize$,
we can illustrate the corresponding Hessian guided RT space for
$\sigma_{h}=3.28$ in Fig.13. We can notice that the structure is represented
by three maxima instead of one which confirms that RT, even associated with
Hessian as done in this work, is not adapted for thick structures, as the
element which allows the flexibility regarding the structure width permitting,
thus, a good detection is the SSRT scale parameter $\sigma$. Additionally, the
detection robustness can be enhanced by averaging the function $\Gamma(x,y)$
for two very different Hessian scales.

Figure 12:  (a) The field $\textbf{n}^{\perp}$ generated with
$\sigma_{h}=23.28$ and a zoom on it. (b) Hessian guided SSRT space.  Figure
13:  Hessian guided RT space

####  3.1.1 Speeding the transform space computation up

The authors in [25] have shown that there is a relationship between RT and
SSRT, expressed as follows

$\check{S_{f}}(\rho)=\check{R_{f}}(\rho)\circledast g_{\sigma}(\rho)$ (4)

The formulae (4) constitutes a straightforward and a fast way to compute the
SSRT which is the convolution of the RT with the 1d kernel $g_{\sigma}(\rho)$
tuned by the SSRT scale space parameter $\sigma$, as seen in Sect2.
Analogously, $\check{S_{f}}_{H}(\rho)$, is computed via
$\check{R_{f}}_{H}(\rho)$ as

$\check{S_{f}}_{H}(\rho)=\check{R_{f}}_{H}(\rho)\circledast g_{\sigma}(\rho)$
(5)

####  3.1.2 Practical considerations

Figure 14:  Hessian guided RT space and false maxima Figure 15:  Constructed
3d lines in scale space for SSRT space of Fig.14

We can notice the presence of false maxima in Fig.11. This means that even
with the application of the Hessian-guided SSRT, some irrelevant maxima remain
as shown in Fig.14. These maxima can be removed as following. First maxima are
thresholded to reject low maxima (below a threshold $TH$). Afterward, behavior
in scale space of SSRT maxima above the threshold $TH$, are studied. The
method applied to carry out this study is proposed in [26] to discard non
meaningful modes in histograms. By extending this study to 3d
($\theta,\rho,\check{S_{f_{H}}}(\theta,\rho)$) and to maxima, SSRT space is
convoluted repetitively by a 3d Gaussian kernel which remove progressively
local maxima and produce as well a scale space representation of all computed
maxima.The number of local maxima in the scale space is decreasing with an
increase of the scale step $ss$. At the end the 3d coordinates of all maxima
construct 3d lines in scale space , as seen in Fig.15. On the basis of the
corresponding line length, a maximum can be retained or rejected. This is step
is considered as a SSRT space refinement stage.

##  4 Experimentation and results

To apply the proposed method, a set of synthetic and real images containing
lines of one or of several pixels widths, have been used. We begin by the
image of the example given in Fig.2 and a very noisy version of it generated
by adding white Gaussian noise (AWGN) with standard deviation $\zeta$ equal to
200. We have seen that applying SSRT alone even with a thresholding does not
provide the desired results. In Fig.16, we show the result of applying the
Hessian-guided SSRT method on this image and on its noisy version, in terms of
guided SSRT spaces, $\check{S_{f_{H}}}$, and in terms of centerlines
detection. To visually appreciate the modifications undergone by the SSRT
spaces $\check{S_{f}}$ when combining the Hessian and the SSRT, we depict in
Fig.16.a and Fig.16.a’ the SSRT spaces as well as the SSRT guided by Hessian
ones, $\check{S_{f_{H}}}$, for both cases in Fig.16.b. and Fig.16.b’.
Furthermore, the detected centerlines are provided in Fig.16.c and Fig.16.c’.
The investigated images are of size $501\times 501$, consisting of five linear
structures, $S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$, $S_{5}$ of widths equal to 15,
37,27,23 and 15 respectively. The centerlines of those structures whose
equations can be written in the form
$\rho_{i}=x\cos\theta_{i}+y\sin\theta_{i}$, are
$\left\\{\rho_{1}=362.6011,\theta_{1}=10^{\circ}\right\\}$,
$\left\\{\rho_{2}=181,\theta_{2}=45^{\circ}\right\\}$,
$\left\\{\rho_{3}=-428.8724,\theta_{1}=120^{\circ})\right\\}$,
$\left\\{\rho_{4}=-276.6168,\theta_{4}=140^{\circ}\right\\}$ and
$\left\\{\rho_{5}=24,\theta_{5}=0^{\circ}\right\\}$.

Figure 16: a: SSRT of the noise free image, b: Guided SSRT of the same image,
c: Centerlines detection, a’: SSRT of the noisy image, b’: Guided SSRT of the
same image, c’: Centerlines detection

To evaluate quantitatively the results, we summarize in Table.1 detection
accuracy in terms of absolute error values $(\Delta\rho,\Delta\theta)$ for the
noisefree image, where $\Delta\rho=|\rho-\hat{\rho}|$,
$\Delta\theta=|\theta-\hat{\theta}|$ and $(\Delta\rho_{N},\Delta\theta_{N})$
for the noisy case, where $\hat{x}$ is the computed value of $x$. In the case
of the noisy image, as the noise is generated randomly leading to slightly
different images when reproducing the noise generation, the detection is then
carried out forty (40) times on forty generated images and the detection error
results are averaged and reported on the same table. The used Hessian scales
here are $\sigma_{h}\in\left\\{6,16\right\\}$ and the SSRT scale $\sigma$ is
set equal to 10.

Table 1: Error values of lines detection Structure | $\Delta\mathbf{\rho}$ | $\Delta\mathbf{\theta}$ | $\Delta\mathbf{\rho}_{N}$ | $\Delta\mathbf{\theta}_{N}$
---|---|---|---|---
$S_{1}$ | 0.0001 | 0 | 0.0004 | 0
$S_{2}$ | 0 | 0 | 0 | 0
$S_{3}$ | 0 | 0 | 0.0002 | 0
$S_{4}$ | 0.0001 | 0 | 0.0004 | 0
$S_{5}$ | 0 | 0 | 0 | 0
|  |  |  |

In the light of the detected structures centerlines depicted in Fig.16 that
are equal to the structures number, we can notice that the centerlines seem to
be drawn exactly in the center of the structure. Moreover, the results given
in Table.1 confirming the previously visually-based assumptions, show that the
proposed method provides an accurate detection of all structures in the image
represented through their centerlines. Indeed, the computed detection errors
are very close to zero for all structures even for those composing the very
bad image quality. Moreover, we can see in Fig.16 a., .b, a’. and .b’ that the
method has permitted to highlight only the linear structure in the SSRT space
retaining, therefore, only the ones related to those structure and discarding
simultaneously all space parts and consequently maxima that do not represent
these structure, eliminating , in addition and when computing the space,
irrelevant image parts from computation which modifies considerably the SSRT
space in the appropriate way as seen in Fig.16.b and .b’ and as ascertained by
numerical results. Moreover, we can notice that the used Hessian scale are not
the optimal ones for the structures widths but still leads to the desired
results.

The second used synthetic image is a challenging one as it composed of 35
structures of three different sizes and very close to each-other, which makes
the detection a hard task when exploiting the Hessian and the SSRT. In fact,
when the structures are too close to each other as it is the case for this
example, each structure can interfere in the Hessian and in the SSRT
computation with the adjoining structures, especially when the used $\sigma$
and $\sigma_{h}$ far from the optimal ones. The image structures are arranged
in six different orientations and have as widths 7, 13 and 19 pixels. SSRT
guided detection is performed on this image and a noisy version of it, where
the noise is a white Gaussian noise of standard deviation $\zeta=100$. Results
of such application are shown in Fig.17 and in Table.2, in terms of root mean
square error (RMSE), noted $\sum_{\Delta}$ and computed as
$\sum_{\Delta\theta}=\sqrt{\sum_{i=1}^{i=N}(\theta_{i}-\hat{\theta}_{i})^{2}/N}$
and $\sum_{\Delta\rho}=\sqrt{\sum_{i=1}^{i=N}(\rho_{i}-\hat{\rho}_{i})^{2}/N}$
where $N$ is the number of the structures. In consideration of Fig.17 and
Table.2, the noise free image detection seems to be accurate despite the
structures promiscuity. Indeed, as illustrated in Fig.17.c, the detected
centerline seem to be well located. This fact is ascertained by the detection
errors that are very low and approximate the zero value for both $\theta$ and
$\rho$, as shown in the mentioned table. Moreover, we can see how the guided
SSRT looks in Fig.17.b comparatively to the SSRT in Fig.17.a which shows that
joining the structures orientation by means of the Hessian the way we proceed,
has operated a big modification to the SSRT space by emphasizing the linear
structures while removing all unwanted interference’s. Nevertheless, regarding
the noisy image, the detection errors have slightly increased especially for
the parameter $\rho$, which means that the detected centerlines have been
subjected to displacement. However, the detection can considered as acceptable
for this last case since the image is of a poor quality. In conclusion of the
previously tests we can conclude that having structures too close to each
other can alter the detection for very poor quality images, even if this
detection can be considered as acceptable. To end tests for synthetic images
we can note that one of the structures has not been detected. It is the
smallest one. This is due to the fact that the linear part of it is too small
to be assimilated to a linear structure as it has a very low maximum in the
guided SSRT space and has been, consequently, deleted in the refinement stage.
To finish for this experiments, the used Hessian scales here are
$\sigma_{h}\in\left\\{3,8\right\\}$ and the SSRT scale $\sigma$ is set equal
to 5.

Figure 17: a: SSRT of the noise free image, b: Guided SSRT of the same image, c: Centerlines detection, a’: SSRT of the noisy image, b’: Guided SSRT of the same image, c’: Centerlines detection  Table 2: RMSE for line detection based on Guided SSRT Image | $\sum_{\Delta\mathbf{\rho}}$ | $\sum_{\Delta\mathbf{\theta}}$
---|---|---
Noise free image | 0.7678 | 0.7071
Noisy image | 2.8150 | 0.811
|  |
Figure 18: From left to right: Guided SSRT and SSRT spaces and the
corresponding paths detection in archaeological site and railroad detection

Next experiments have been done on several real images including bar-
code,paths in an archaeological site, railroad, crosswalks and lanes given
in18, Fig.19 .and .20, where we can see the guided SSRT in each case along the
structures centerlines detection. In 18 $\check{S_{f}}$ spaces are depicted
with $\check{S_{f_{H}}}$ ones to see how a complex background is reflected in
the SSRT space for real images and the amount of irrelevant maxima contained
in the SSRT space. We can notice, also, that for certain structures the
detected lines are not in the center of those structures. This is due to the
fact that the color is not uniform inside the structure as we can see for the
crosswalks image, for example, or for the railway one. Moreover, the
structures edges are irregular and not smooth, which influences the SSRT and
the Hessian computation. Concerning the detection parameters, the used Hessian
scales here are $\sigma_{h}\in\left\\{3,8\right\\}$ and the SSRT scale
$\sigma$ is set equal to 5 except for the image of crosswalks as the
structures are very thick, where $\sigma$ is set to 10. For Color images, all
computations are performed on gray scale version of them.

Figure 19: From left to right: Guided SSRT spaces and the corresponding bar-
code and neon detection  Figure 20: From left to right: Guided SSRT spaces and
the corresponding crosswalks and lane detections

##  5 Conclusion

In this paper, we have proposed to combine the SSRT and the Hessian of the
image to tackle, in an elegant manner, the issue of linear structures
detection in images in presence of noise and complex background. The method
modifies, via the Hessian, the SSRT computation so that its failing in the
detection when facing complex images is alleviated. Indeed, this is done by
making the SSRT projections effective only for features having the same
orientation as the projection direction. In fact, failing in SSRT detection
arise when linear structures are too close to each others. Moreover, when the
investigated structures are drown in a complex background, their corresponding
maxima in the SSRT space are also drown in a huge amount of other false
maxima. To overcome these drawbacks, all image features, including linear
structures that do not have the actual orientation of the SSRT projection are
considered irrelevant and therefore attenuated. The others are enhanced to
produce maxima that are easy to detect in the modified SSRT space.
Experimentation carried out on both synthetic and real images confirms the
effectiveness of the proposed method. In fact, besides correct location of the
structures centerlines, it has proven its robustness against noise and complex
background.

### Availability of data and materials

The datasets used and analyzed during the current study are available from the
corresponding author upon reasonable request

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