Building interior layout reconstruction from throughthewall radar image using MSTbased method
 Bo Chen^{1},
 Tian Jin^{1}Email author,
 Biying Lu^{1} and
 Zhimin Zhou^{1}
https://doi.org/10.1186/16876180201431
© Chen et al.; licensee Springer. 2014
Received: 16 December 2013
Accepted: 27 February 2014
Published: 12 March 2014
Abstract
In order to reconstruct the building interior layout from the throughthewall radar image, this paper proposes a method based on minimum spanning tree (MST) which comes from the graph theory. The building layout graph (BLG) model is firstly established using the wallwallfloor trihedrals inside the building. The vertex set and the edge set including the weight between any two nodes of the graph are defined afterwards. Then, the MSTbased method is used to reconstruct the building interior layout. We have also improved the traditional method by applying the MST model with restricted conditions to solve the reconstruction problem when there exist interference targets in a building. The anechoic chamber measurements are carried out to demonstrate the suitability and efficacy of the MSTbased method.
Keywords
1. Introduction
Sensing through walls is an emerging technology in recent years. Throughthewall radar imaging (TWRI) is considered to be very effective to achieve the objectives of ‘seeing’ through walls using electromagnetic waves. TWRI is highly desirable for a range of organizations, including police, rescue personnel missions, surveillance, first responders, and defense forces[1, 2]. Most throughthewall radars in use often require a close position to the wall or they have to be pressed against the wall. Moreover, TWRI mainly focuses on behindthewall target. These systems only have the singlewall penetrating ability, providing range, crossrange, and motion information of behindthewall targets[3–5].
In recent years, some research institutes in America and Europe are focusing on throughthewall radar systems which have the ability of reconstructing the interior layout of a building. These systems will greatly improve security, response time, covertness, and reliability of the firefighting and law enforcement tasks. Typical systems include the SAPPHIRE system designed by the Netherlands Organization for Applied Scientific (TNO) and the Visibuilding program sponsored by the Defense Advanced Research Projects Agency (DARPA). The operating principle of SAPPHIRE is detecting and identifying the principal scatterers inside a building, and then deducing the interior layout[3, 4]. Visibuilding program uses modelbased signal processing method to reconstruct the building layout. In addition, the Army Research Laboratory (ARL) had conducted several field experiments to image an abandoned barrack. During the experiment, the radar system moves along two mutual vertical sides and the two radar images are synthesized coherently to reconstruct the interior layout[6, 7].
For specific building interior layout reconstruction methods in literatures, they are divided into two types: the first one is using multilook fusion approach to throughthewall radar images from multiview measurements, such as the experimental system of ARL mentioned above; the other one is analyzing throughthewall radar image and extracting the principal scatterers inside the building, namely, obtaining the position and orientation from region of interest (ROI) of the wallwallfloor trihedral, then using reasonable deducing method to infer the interior layout, such as SAPPHIRE system. Comparing with the first type, the advantage of the second type lies in using the electromagnetic scattering characteristics sufficiently, keeping away from the influence of clutter and multipath propagation and ensuring robustness[8].
In our previous work[9], a method of acquiring the attributes of a trihedral is presented. The orientation information of all the trihedrals in a throughthewall radar image can be obtained by a virtual aperture imaging model[10]. In addition, the estimation accuracy satisfies the requirements of reconstructing the building interior layout. Therefore, this paper mainly focuses on designing the reconstruction method. If the position and orientation information of all the trihedrals appeared in throughthewall radar image are known, the whole building interior layout can be obtained immediately by direct artificial judgment on onetime[3, 11]. However, they are based on simulation data only and lack high automation.
The problem of the onetime artificial judgment is that the reconstruction accuracy depends greatly on the signaltonoise ratio (SNR) of throughthewall radar image. Once there exist interference targets or undetected wallwallfloor trihedrals, the reconstructed layout will be different from the true layout. Therefore, some researchers propose an iterative reconstruction method based on electromagnetic scattering model[8, 12]. This method is formed by prediction stage and test stage. The prediction stage in the iterative procedure is the same as onetime reconstruction process. The predicted radar image can be obtained according to the predicted layout using electromagnetic modeling method. During the test stage, the measured radar image and the predicted radar image should be compared to modify the predicted building model, making a more accurate description of the building. Each iteration will add absent walls or cut out extra walls so as to be closer to the true layout of the building, comparing with the preiteration process. Then the positions of the wallwallfloor trihedrals will be more accurate after compensation of the wall parameters[13]. To improve the computation efficiency of the iterative reconstruction method, prevent manual intervention during the reconstruction process and realize online computation, an effective building interior layout autoreconstruction method need to be investigated.
Without loss of generality, this paper focuses on common building designs with regularity and rectilinearity. The interior walls are either perpendicular or parallel to the exterior walls. In addition, any closed structure is not allowed to be existed if the exterior walls of the building are eliminated. The graphbased reconstruction method proposed in this paper comes from recent papers on the subject of floor plan estimation[14, 15]. They also employ graph theory in its problem formulation. This paper proposes a graphtheoretical building interior layout reconstruction method using the prototype of their method. However, the way that graphs are employed in this paper is different: the edges of the graph are used to indicate the walls that are present or absent, and the nodes are used to represent the wallwallfloor trihedrals inside the building.
This paper is organized as follows: Section 2 gives a brief description of the experimental system in the anechoic chamber. Section 3 develops a graphtheoretical model for a rectangular floor plan which is called building layout graph (BLG). After the graph elements and the weight of each edge are defined, the minimum spanning tree (MST) method is used to reconstruct an experimental building using the radar system mentioned in Section 2. Section 4 probes into how to improve the MSTbased reconstruction method when there exist interference targets in the building and also presents the experimental results. Section 5 gives some remarkable conclusions.
2. Overview of the experimental system
3. MSTbased reconstruction method for building layout
The building interior layout reconstruction algorithm presented in this paper is based on MST which belongs to the graph theory. In this section, we will define the elements in BLG and establish the relationship between the building interior layout reconstruction problem and the MST.
3.1 Elements in BLG and MST
 (i)
Nodes (or Vertices): The wallwallfloor trihedrals inside the building are treated as nodes of the building layout graph.
 (ii)
Edges: Walls existing between two adjacent nodes, all the edges in BLG are undirected.
 (iii)
Degree: It is used to describe a node. It means the number of walls which relates to the current wallwallfloor trihedral node. For undirected graph, indegree and outdegree are both treated as uniform degree with no difference. The degree of node v is denoted as D(v).
 (iv)
Order: It is used to describe a graph. It means the number of trihedral nodes in the building graph and is denoted as V.
Positions and orientations of the wallwallfloor trihedrals are attributes of the node set. A weighted undirected graph can be developed if we have got all the trihedrals including their attributes through radar measurements. How to acquire these attributes has been researched. From[9], we have known that the pose angles of all the trihedrals in a radar image can be estimated by using a virtual aperture imaging model. It will not be discussed in this paper to avoid unnecessary repetition.
If a graph with any pair of nodes being connected by at least one edge, it is called connected graph. In a complete graph, any two nodes is connected by an edge. Therefore, a weighted complete undirected graph must be a connected graph. Given a weighted complete undirected graph G(V, E) with V = N, the number of trees (a subgraph of G without closed loops) that connects all the nodes of the graph is N^{N  2}. The MST is the tree with the minimum total weight, defined as the sum of the weight of each tree's edge.
As is mentioned afore, in a radar measurement data set consisting of reflected signals from principal scatterers, we consider the wallwallfloor trihedrals as the nodes of a graph, the horizontal or vertical wall lines being the edges joining the nodes. Any possible layout predicted according to present nodes corresponds a spanning tree of the weighted complete undirected graph BLG. If we have defined reasonable edge weight, we could make the total weight of the spanning tree corresponding to the correct interior layout to be minimum. By this way, we can translate the building interior layout reconstruction process into searching the MST of the complete weighted undirected graph. Therefore, how to construct the edge weight becomes the key of the reconstruction process.
3.2 Definition of edge weight in BLG
 (i)
Due to the rectangular shape of the building, if an edge connecting two nodes is closer to the horizontal or vertical direction than another edge, the weight of this edge should be smaller. In contrast, if an edge is farther from the node angle range which is defined above, the weight should be larger.
 (ii)
The edge weight should monotonically increase with its Euclid distance. Namely, one of two edges whose Euclid distance is smaller should also be smaller in weight if they have the same node angles.
 (iii)
If the edge weight is defined as an exponential form, the base number should not be equal to 1, and the power exponent should not be equal to 0. Otherwise, the weight cannot be distinguished from angle range and Euclid distance.
 (iv)
Considering that the positions of the nodes may be departed from their true positions and the position errors will influence the reconstructed interior layout, the edge weight should have a certain faulttolerant ability for this error. In other words, the edge weight function should be continuous at the boundary of the angle range. It is not allowed to have a sudden jump in value at the starting angle θ _{ s } and the ending angle θ _{ e }.
where$sgn\left(\mathit{y}\right)=\left\{\begin{array}{l}1,\phantom{\rule{1.75em}{0ex}}\mathit{y}\ge 0\\ 1\phantom{\rule{1.5em}{0ex}}\mathit{y}<0\end{array}\right.$.
In the edge weight definition, we will make some analysis on the exponential form a^{ x }(0 < a < 1). If 0 < a_{2} < a_{1} < 1 and power exponent x > 0, a_{2}^{ x } < a_{1}^{ x } when x is fixed. This feature matches the monotonicity between the Euclid distance and the edge weight. Moreover, a^{ x }(0 < a < 1) is a monotonically decreasing function with x, in order to keep equivalence between the MST and the correct layout, the power exponent γ(θ_{ i }, θ_{ j }) should get the maximum value when θ_{ i } or θ_{ j } equals 0, π/2, π, 3π/2, and 2π, making the edge weight between these two nodes get the minimum value. As a result, trigonometric function is the basic form of the power exponent γ(θ_{ i }, θ_{ j }) we considered, satisfying the first condition.
There is a variable β in the definition of power exponent γ(θ_{ i }, θ_{ j }). β is introduced in order to keep the exponential term above zero. Thus, d(i, j) will increase with the Euclid distance when θ_{ i } and θ_{ j } are within the angle range. This feature matches the second condition.
3.3 MST searching method
We have given the definition of edge weight of the BLG and made equivalence between the correct building interior layout and the MST. Now we should focus on how to find the MST of the building graph so as to obtain the interior layout of the building.

Step 1: Using definition in (1) ~ (5), sort the edge weight of the weighted undirected building graph in ascending order;

Step 2: Set i = 1 and let the initial edge be E_{0} = ∅;

Step 3: Select an edge e_{ i } of minimum weight value not in E_{i  1} such that T_{ i } = <E_{i  1}∪ {e_{ i }} > is acyclic and define E_{ i } = E_{i  1}∪e_{ i }. If no such edge exists, let T = <E_{ i } > and stop;

Step 4: Replace i + 1 by i. Return to step 3.
After N  1 iterations, where N is the order of the building layout graph, the complete MST is found. Thus, the building interior layout is obtained subsequently.
3.4 Experimental results
In the above sections, we have translated the building layout reconstruction problem into the MST searching problem. With the edge weight defined as (1) ~ (5) and the Kruskal algorithm, some simulations have been implemented and some valuable results are obtained. We have known that after the outmost trihedrals located on the outline which are called priori nodes are removed, the MST of the graph formed by the remaining nodes represents the interior layout of the building[19].
The result in Figure 6 represents the correct interior layout of the building in the anechoic chamber. Combining the priori outline information (shown by the dashed line), the whole layout of the building can be obtained.
4. Reconstruction of building layout with interferences
4.1 Drawback of the traditional MSTbased method
4.2 Improved MSTbased method

Step 1: Extract the trihedral nodes and dihedral edges information in the radar image;

Step 2: Search the MST with restricted conditions based on the nodes and edges information, denote the restricted MST as T_{1};

Step 3: Traverse every nodes of T_{1} and calculate the node angle;

Step 4: Pick out the nodes whose angle is not divisible by π/2;

Step 5: Apply Kruskal algorithm to the remaining nodes. The generated MST T_{2} represents the final structure of the building.
Since the method of acquiring the principal scatterers such as dihedral and trihedral has been given in[8], how to extract the nodes and edges information of the BLG will not be discussed in this paper. We will focus on how to apply the MSTbased method in building reconstruction when there exist some interference targets. From the procedure shown above, the key to this problem is how to solve the MST with restricted conditions[20]. The restricted conditions are the nodes and edges information obtained from the building radar image. An edge extracted from the radar image means that the two nodes which are related to this edge should be connected in the final generated MST definitely. This problem is described as follows using mathematical language.
Assume G(V, E) is a weighted undirected connected graph, G_{1}(V_{1}, E_{1}), …, G_{ r }(V_{ r }, E_{ r }) are the r subtrees which do not intersect with each other. If a spanning tree T = (V, E') of the graph G contains all the existed subtree, namely, E_{ i } ⊂ E' (i = 1, 2, …, r), moreover, the sum of all the edge weight of T is the minimum, then T is called the MST of G with restricted conditions G_{1}, …, G_{ r }.

Step 1: Assume U_{0} = V_{1}∪ … ∪V_{ r }, B_{0} = E_{1}∪ … ∪E_{ r }, T_{0} = (U_{0}, B_{0}) and W_{0} = V  U_{0}, take iteration in the following way:$\mathit{d}\left({\mathit{u}}_{0},{\mathit{w}}_{0}\right)=\underset{\mathit{w}\in {\mathit{W}}_{\mathit{k}}}{\underset{\mathit{u}\in {\mathit{U}}_{\mathit{k}}}{\text{min}}}\mathit{d}\left(\mathit{u},\mathit{w}\right)$(6)
Let U_{k + 1} = U_{ k }∪ {w_{0}}, W_{k + 1} = W_{ k }  {w_{0}}, B_{k + 1} = B_{ k }∪ {(u_{0}, w_{0})} and T_{k + 1} = (U_{k + 1}, B_{k + 1}), keep iteration until W_{ n } = 0. The forest characteristics will not be broken in the above iteration process due to the link of only one edge when adding nodes to U_{ k } each time. Moreover, we have not carried connection inside the forest, so the number of trees in the forest keeps unchanged. When the recursion stops, T_{ n } = (U_{ n }, B_{ n }) will be a forest covering all the nodes in G. The original G_{ i } will be expanded into${\mathit{G}}_{\mathit{i}}^{\text{'}}\left({\mathit{V}}_{\mathit{i}}^{\text{'}},{\mathit{E}}_{\mathit{i}}^{\text{'}}\right)\phantom{\rule{0.25em}{0ex}}\left(\mathit{i}=1,2,\cdots ,\mathit{r}\right)$.

Step 2: Add some appropriate edges to T_{ n } to make it a spanning tree of G. Denote${\mathit{U}}_{0}^{\text{'}}={\mathit{V}}_{1}^{\text{'}},{\mathit{W}}_{0}^{\text{'}}=\mathit{V}{\mathit{U}}_{0}^{\text{'}},{\mathit{B}}_{0}^{\text{'}}={\mathit{E}}_{1}^{\text{'}}\cup \cdots \cup {\mathit{E}}_{\mathit{r}}^{\text{'}}$, if${\mathit{U}}_{\mathit{k}}^{\text{'}},{\mathit{W}}_{\mathit{k}}^{\text{'}}$, and${\mathit{B}}_{\mathit{k}}^{\text{'}}$ have been constructed, there exist w_{0} in${\mathit{V}}_{\mathit{j}}^{\text{'}}$ and u_{0} in${\mathit{U}}_{\mathit{k}}^{\text{'}}$ which are subject to$\mathit{d}\left({\mathit{u}}_{0},{\mathit{w}}_{0}\right)=\underset{\mathit{w}\in {\mathit{W}}_{\mathit{k}}^{\text{'}}}{\underset{\mathit{u}\in {\mathit{U}}_{\mathit{k}}^{\text{'}}}{\text{min}}}\mathit{d}\left(\mathit{u},\mathit{w}\right)$(7)
Let${\mathit{U}}_{\mathit{k}+1}={\mathit{U}}_{\mathit{k}}^{\text{'}}\cup {\mathit{V}}_{\mathit{j}}^{\text{'}},{\mathit{W}}_{\mathit{k}+1}=\mathit{V}{\mathit{U}}_{\mathit{k}+1}^{\text{'}}$, and${\mathit{B}}_{\mathit{k}+1}^{\text{'}}={\mathit{B}}_{\mathit{k}}^{\text{'}}\cup \left\{\left({\mathit{u}}_{0},{\mathit{w}}_{0}\right)\right\}$, keep iteration until${\mathit{W}}_{\mathit{r}}^{\text{'}}=0$. Each iteration will add one edge which connects two isolated tree. The spanning tree$\mathit{T}=\left(\mathit{V},{\mathit{B}}_{\mathit{r}1}^{\text{'}}\right)$ of G will be available when the iteration stops.
4.3 Experimental results with interferences
where mod(·) is the remainder operator, then node v_{ i } is considered to be an interference node. It should be removed. In Figure 9a, node 2 and node 3 are considered to be interference nodes and removed from the BLG. Finally, the Kruskal algorithm is used to the remaining nodes to generate a new MST which is denoted as T_{2}. This refreshed MST shown in Figure 9b represents the final interior layout of the building. Therefore, the MSTbased method can also be applied in building interior layout reconstruction even if there exist interference targets.
5. Conclusions
We have proposed a building interior layout reconstruction algorithm assuming that complete positions and orientations information of the graph nodes have been obtained. The proposed MSTbased method can also deal with the case when there exist interference targets. It is hoped that based on these preliminary results, we are able to predict the building interior layout using radar measurements. However, in practical situation, some trihedral nodes may not be detected in the radar image. Extension of this work is to solve this problem. It is presently under study and the results will be presented in our future research work.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants 61271441, 61372161 and the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201046.
Authors’ Affiliations
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