1. Introduction

2. Problem Statement and General Approach

3. Methods

Figure 1. Forming an object model based on overview trajectory images.

Figure 1. Forming an object model based on overview trajectory images.

3.2. Recognition of SPS Objects

Recognition of SPS objects with referencing of the AUV to the object’s CS is required for planning and performing the AUV’s trajectory directly in the CS of the object under inspection. The problem of coordinate reference of the AUV to the object is solved by comparing the visual data obtained at the position of the AUV’s inspection trajectory with the object model. A three-stage computational scheme is implemented using several views (aspects) of the object that represent an object model (Figure 3). The choice of views for comparison is based on the criterion of proximity of AUV’s positions in the external CS of the analyzed position of the AUV’s working trajectory and the position of the overview trajectory of the selected view.

At the first stage, the desired object is recognized for each selected kth view by matching the point features in the working and model stereo-pairs (see the description of the algorithm in section 3.2.1). With successful recognition (a threshold criterion is used), the result is (a) a set of points S_3D_POINT_V^k_CS^work matched to the set M_3D_POINT_V^k_CS^view_k (belonging to the model of kth view), and (b) the calculated matrix H_CSwork, _{CSview_k} for coordinate transformation from the CS of the inspection trajectory position into the CS of the model of the object’s kth view used.

Since for each view there is a matrix H_{CSview_k, CSobject_n} for coordinate transformations from the CS of this view into the CS of the nth object (obtained through forming a model of the kth view), the transformation from CS^work into CS^object_n sought can be obtained by sequentially performing the two above-indicated transformations. Each resulting transformation (the number of which is equal to the number of views used) can be considered as an acceptable solution to the problem set. The resulting matrix can also be used as an initial approximation in the optimization method when calculating the refined matrix for coordinate transformation from CS^work to CS^object_n, with point data for all views and all CE types taken into account (at the third stage). The obtained set of recognized “points” S_3D_POINT_V^kW_CS^work is entered into the set S_3D_ POINT_CS^object_n accumulated from all the views, which is coordinated in CS^object_n (the available matrix H_{CSview_k, CSobject_n} is used). The number of matched points is input in the Table of Recognition Results.

At the second stage, “lines” and “corners” in the working stereo-pair, matched to the elements of the kth view model, are recognized (using the resulting transformation H _CSwork, _{CSview_k}) (see section 3.2.2). The result is the sets S_3D_LINE_V^kW_CS^work and S_3D_CORNER_V^kW_CS^work. These sets are entered into the respective sets S_3D_LINE_CS^object_n, S_CORNER_CS^object_n, coordinated in CS^object_n (the available matrix H_{CSview_k, CSobject_n} is used). The matching results are input in the Table of Recognition Results. Thus, these sets, along with the set of points S_3D_ POINT_CS^object_n, are a full combination of recognized 3D CEs of all types of the object from all views in the object’s intrinsic CS. The visualization of them can provide a visual representation of how fully and reliably the analyzed object was recognized.

At the third stage, the refined matrix H _CSwork, _{CSobject_n} is calculated. The refinement (increase in the accuracy of AUV coordination in the object’s CS) is carried out by taking into account the recognized “points” in all the views used, and also by taking into account the points belonging to the recognized “lines” and “corners”. Three approaches to calculating the coordinate reference of the AUV to the SPS object are considered:

1. The reference is, as mentioned above, based on the matrix for coordinate transformation from CS^work to CS^object_n, which is calculated using a model of the kth view as a sequential execution of two available transformations:

$$H_{CSwork,CSobject\_n}=H_{CSview\_k,CSobject\_n}\cdot H_{CSwork,CSview\_k}$$

In this case, we obtain k matrices (according to the number of object’s views used). For each variant, the error of calculation accuracy is estimated on the basis of a test set of matched points.

2. The matrix H_CSwork, _{CSobject_n} for coordinate transformation from CS^work to CS^object_n is calculated on the basis of a combined set of matched points (for all views of the object). A standard method to minimize the target function is used, based on the estimate of total discrepancy between the recognized points of the set S_3D_POINT_V^kW_CS^wok (in the working stereo-pair of frames) and the points of the set M_3D_POINT_V^k_CS^view_k matched to them (a model of the kth view of the object). The total discrepancy is calculated in the coordinate space CS^object_n. Two variants of obtaining a combined point set are compared: (a) with only CE of the “point” type taken into account in all the views used; (b) with more points belonging to the recognized CEs of the “lines” and “corners” types added to the “point” type CE in each view.

To write the target function in a compact form, introduce intermediate designations. Designate the set M_3D_POINT_V^k_CS^view_k as MP^k (mp^k₁, … mp^k _{np_k}), and designate the set S_3D_POINT_V^kW_CS^wok matched to the former set as SP^k (sp^k₁, … sp^k _{np_k}). Then the problem of minimizing the target function in variant (a) is formulated as follows:

$\min {\displaystyle {\sum }_1^k{\displaystyle {\sum }_{i=1}^{np\_k}R{1}_i^k}}$, where $R{1}_i^k=\Vert m{p}_i^k\cdot H_{CSview\_k,CSobject\_n}-s{p}_i^k\cdot H_{CSwork\_k,CSobject\_n}\Vert$where the matrix H_{CSview_k, CSobject_n} is used that was compiled at the stage of formation of the kth view model.

Variant (b) additionally takes into account, as mentioned above, the points belonging to the “lines” and “corners” CEs (these points determine the coordinate position of the CE). Similarly, simplify the writing by introducing intermediate designations. Designate the set of points M_3D_LINE_V^k_CS^view_k belonging to the “lines” type CEs of the kth view model as $M{L}^k\left(m{l}_1^k,\dots ,m{l}_{nl\_k}^k\right)$, and designate the set of recognized lines S_3D_LINE_V^kW_CS^work matched to the former set as $S{L}^k\left(s{l}_1^k,\dots ,s{l}_{nl\_k}^k\right)$. Here the elements ml^ki and sl^k_i denote the start and end points of ith segment line.

Then the points of the ith “line” taken into account in the target function can be written as follows:

$$R{2}_i^k=\Vert m{l}_i^k\cdot H_{CSview\_k,CSobject\_n}-s{l}_i^k\cdot H_{CSwork\_k,CSobject\_n}\Vert$$

Similarly, introduce simplifying designations for points belonging to the CEs of the “corners” type. Designate the set of points M_3D_CORNER_V^k_CS^view_k belonging to the “corners” CEs of the kth view model as $M{C}^k\left(m{c}_1^k,\dots ,m{c}_{nc\_k}^k\right)$, and designate the set of recognized corners S_3D_CORNER_V^kW_CS^work matched to the former set as $S{C}^k\left(s{c}_1^k,\dots ,s{c}_{nl\_k}^k\right)$.

Accordingly, the points of the ith “corner” taken into account in the target function can be written as follows:

$$R{3}_i^k=\Vert m{c}_i^k\cdot H_{CSview\_k,CSobject\_n}-s{c}_i^k\cdot H_{CSwork\_k,CSobject\_n}\Vert$$

Then the problem of minimizing the target function in variant (b) is formulated as follows:

$$\min {\displaystyle {\sum }_1^k\left({\displaystyle {\sum }_{i=1}^{np\_k}R{1}_i^k+{\displaystyle {\sum }_{i=1}^{nl\_k}R{2}_i^k+}{\displaystyle {\sum }_{i=1}^{nc\_k}R{3}_i^k}}\right)}$$

3. In this approach, unlike the previous ones, the AUV’s position in CS^object_n is calculated without calculating the single coordinate transformation matrix for all views. For each variant of H _CSwork, _{CSobject_n} (see approach 1), coordinates of the AUV are calculated (uniform coordinates of the AUV in CS^work – (0, 0, 0, 1)) in CS^object_n, i.e.

$$P_{AU{V}^{object\_n}}=\left(0,0,0,1\right)\cdot H_{CSwork\_k,CSobject\_n}$$

Thus, we have k variants of coordinate values for AUV in CS^object_n. As a solution being sought, we consider an averaged value of the coordinates. When averaging, the degree of confidence in each variant is taken into account in the form of weight coefficient α_k, where α_k is proportional to np_k (number of points matched).

Table 2. Results of matching the CEs of the input stream (“points”, “lines”, and “corners”) to the object model for all views.

Table 2. Results of matching the CEs of the input stream (“points”, “lines”, and “corners”) to the object model for all views.

	S_3D_POINT_CSwork (number of points matched to model / number of points in model)	S_3D_LINE_CSwork (number of lines matched to model / number of lines in model)	S_3D_CORNER_CSwork (number of corners matched to model / number of corners in model)
View 1 in CS1 *	np_1 / mp	n1_1 / ml	nc1 / mc
…	…	…	…
View k in CSk	np_k / mp	nlk / ml	nck / mc
…	…	…	…
View last in CSlast	np_last / mp	nllast / ml	nclast / mc

Upon processing all views and inputting the results in Table 1, a decision is made on reliability of the object identification according to the criteria set up previously.

The AUV’s movement in the coordinate space of the object is calculated using the obtained matrix of reference to the object’s CS.

3.2.1. Subsubsection

Consider the algorithm for matching the point features in stereo-pairs of frames of the AUV’s working trajectory position and a model of certain object’s view from the overview trajectory. As noted above, at the stage of object model formation:

(a) the object’s intrinsic CS was built, for which the direction of the Z-axis of the external CS was used as the Z'-axis direction;

(b) a set of 3D points M_3D_ V^kV^k_CS^view_k was obtained in the coordinate system CS^view_k from the matched 2D sets (see 3.1). The matrix H_{CSview_k, CSobject_n} for coordinate transformation from CS^view_k to CS^object_n was calculated. Here object_n is the identifier of nth object.

Then the action of the Algorithm for object recognition on the basis of points and coordinate reference of the AUV to the object’s CS is described through performing the following sequence of steps (Figure 4):

Selection of the kth view in the model object for which the object’s field of visibility best fits the potential object’s field of visibility for the AUV’s camera at the considered position of the inspection trajectory.

Generation and matching, using the SURF detector, of special points in the left frames of the model and working stereo-pairs. The result of successful matching is the set M_2D_POINT_WV^k_LL in the image im_L_object_iew^k and the set S_2D_V^kW_LL in the image im_L_object_work.

If the matching in the previous step is successful, then the special points in the frames of the working stereo-pair are matched using the SURF detector. The result is the set S_2D_POINT_WW_LR in the image im_L_object_work and the set S_2D_POINT_WW_RL in the image im_R_object_work.

The set of 3D points S_3D_POINT_WW_CS^work is built in the coordinate system CS^work from the 2D sets matched at the previous step.

Selection of a subset of points, matched also in the left frames of the “model” and “working” stereo-pair, from the set of points matched in the frames of the “model” stereo-pair (a model of this view): subM_2D_POINT_V^k = M_2D_POINT_V^kV^k_LR $\cap$ M_2D_POINT_ WV^k _LL (if there are fewer than three points, then we assume that the camera “does not see” the object). This operation is necessary to subsequently form a 3D representation of that subset of points, from the model of this object’s view, for which matching with the recognized points of the object was obtained in the CS of the working trajectory position.

Selection, from the set of points matched in the frames of working stereo-pair, of a subset of points that was also matched with the respective subset of the model of this object’s view (subM_2D_POINT_V^k):

subS_2D_POINT_W = S_2D_ POINT_WW_LR $\cap$ S_2D_ POINT_ V^kW _LL.

Formation of the subset M_3D_POINT_V^k_CS^view_k (belonging to the set M_3D_POINT_V^k V^k_CS^view_k) corresponding to the subset subM_2D_POINT_V^k

Building of the subset S_3D_POINT_V^kW_CS^work belonging to the set S_3D_POINT_WW_CS^work using the 2D subset subS_2D_POINT_W obtained in paragraph 6.

Since the subsets M_3D_POINT_V^k_CS^view_k and S_3D_POINT_W_CS^work were matched through performing the above operations, the matrix for coordinate transformation between the above-indicated CSs (H _CSwork, _{CSview_k}) is calculated from the set of these points. A standard method is used: minimizing the sum of discrepancies by the least squares method.

The matrix for coordinate transformation from CS^AUV_work into CS^object_n is calculated that provides direct calculation of AUV’s movement in the coordinate space of the object:

$$H_{CSwork,CSobject\_n}=H_{CSview\_k,CSobject\_n}\cdot H_{CSwork,CSview\_k}$$

To calculate the AUV’s movement in the coordinate space of SPS, a matrix for coordinate transformation from CS^object_n to CS^SPS is calculated using the matrix H_{CSobject_n , CS_SPS.}

3.1.2. Recognition and Calculation of 3D Coordinates of “Segment Line” and “Corner” Type CEs (Stage 2)

The coordinate transformation matrix H _CSwork, _{CSview_k} obtained at the first stage, linking the coordinate system CS^work with the coordinate system of the kth view of the overview trajectory CS^view_k (kth view model) substantially simplifies the solution of the problem of recognizing the “segment line” type CEs in the images of the inspection trajectory, since it does not require direct selection of segment lines in vectorized images.

When recognizing segment lines belonging to the object in the input video stream of the inspection trajectory, two related problems are solved sequentially:

(1) matching the segment images in the left (right) frame of the model stereo-pair (of the kth view) and the left (right) frame of the stereo-pair of the inspection trajectory’s position; (2) 3D reconstruction of the matched 2D segment images in the working stereo-pair of frames.

The first problem is solved by the below-described algorithm for matching segments, which uses: (a) the coordinate transformation matrix H_CSwork, _{CSview_k}, calculated from the points at the first stage, linking CS^work and CS^view_k; (b) a set of 3D segments M_3D_LINE_V^kV^k_CS^view_k (the kth view model). Each segment is defined by two spatial end points (in CS^view_k), and for each end point there are pointers at its 2D images in the stereo-pair of frames. The advantage of this technique for defining a segment is that, as mentioned above, no image vectorization is required to match the segment line image in the model with the segment line image selected in the working stereo-pair of frames. In other words, the 3D problem of segment line matching is reduced to the 2D problem of matching the segment end points with calculation of correlation estimates (Figure 5). It should be taken into account that the segment line is an edge of the object’s face. Thus, when calculating the correlation of the images of the end points compared between the images, one use the rectangular arrays of pixels on the side from the segment line where the object is located (this information is saved while forming the model). Eventually, the proposed matching technique without using the vectorized form of images substantially reduces computational costs.

Algorithm for recognizing and calculating 3D coordinates of “line” type CEs. Segment lines are sequentially selected from the 3D set M_3D_LINE_V^kV^k_CS^view_k; for each segment line, its 2D image is recognized in the stereo-pair of frames (im_L_object_work, im_R_object_work) of the analyzed position of the working (inspection) trajectory. Thus, for each segment line l_i (defined by the end points p1_i and p2_i) (Figure 5):

Pointers at 2D images of its two points in the stereo-pair of frames of the model are extracted.

The segment line is placed in the coordinate space CS^work using a matrix ${\left(H_{CSwork,CSview\_k}\right)}^{-1}$ (calculated in step 1) and is projected onto the stereo-pair of frames of the working trajectory. The expression “segment line is projected” means the projection of the segment’s end points.

The coefficient of correlation between the compared images of the point in the model’s stereo-pair and the working stereo-pair of frames is calculated for each end point. When comparing the end points of the segment lines, the orientation of the vehicle by the compass of both the model and the working trajectory is used to provide the same orientation of the surrounding terrain compared.

In case of incomplete matching of the images, an attempt is made to find a match within a small radius, assuming that the projection (step 2) was performed with some error (step 3).

A decision is made about match or no-match of the images.

Figure 6. Recognition of rectilinear segments.

Figure 6. Recognition of rectilinear segments.

The second problem, 3D reconstruction of the matched 2D images of segment lines in the working stereo-pair of frames, is solved in a traditional way: by matching the end points in the stereo-pair of frames using epipolar constraints, then by the ray triangulation method with calculating the 3D coordinates of the segment line’s end points in CS^work.

The result of recognition of all segment lines is the set S_3D_LINE_W_CS^work_match matched to M_3D_LINE_V^k_CS^view_k_match. This result is input in the Table of Recognition Results, and the set is entered in S_3D_ LINE_CS^object_n.

Recognition and calculation of 3D coordinates of “corner” type CEs. CEs of the “corner” type are selected from the obtained set of “lines”. The selection algorithm is quite simple, since a “corner” CE is constructed from two segment lines lying in the same plane in space. By indicating whether the segment line belongs to a specific “corner”, ambiguity is eliminated in case of similar geometries of corners.

The result of recognition of all corners (the number of corners in the set S_3D_CORNER_W_CS^work_match matched to M_3D_CORNER_V^k_CS^view_k_match, with pointers at these sets) is stored in the Table of Recognition Results, and the set is entered in S_3D_ CORNER_CS^object_n.

Object identification. Upon identifying the inspection trajectory of all three types of CEs at this position, corresponding to the object model, the data of the Table of Recognition Results is analyzed, and an algorithmic decision is made about identification of the object.

Calculation of the refined coordinate transformation matrix H_{CSwork, CSobject_n} is carried out by the standard technique with minimization of the total coordinate deviation when matching of all points used. In this case, the matrix is calculated using all points belonging to CEs in the sets S_3D_ POINT_CS^object_n, S_3D_LINE_CS^object_n, S_CORNER_CS^object_n, since these points have a coordinate representation both in CS^work and in CS^object_n. As a result, the matrix H_{CSwork, CSobject_n} (4 × 4) is calculated that links the CS of the AUV’s camera and the object’s CS. Then the AUV’s coordinates at each position of the trajectory are recalculated from the CS of the AUV or its camera to the object’s CS using this matrix $P^{CSobject\_n}=P\left(0,0,0,1\right)\cdot H_{AUV,object\_n}$, where $P\left(0,0,0,1\right)$ are the uniform AUV’s coordinates in the CS of the AUV’s camera at this trajectory position. It should be noted that the CS of each position of the AUV’s trajectory is linked to the CS of the initial trajectory position using the author’s method for visual navigation [24].

The calculation of the AUV’s movement in the object’s coordinate space is performed using the matrix H_{CSwork, CSobject_n} for transformation from CS^work into CS^object_n. As shown above, it is calculated as follows:

$$H_{CSwork,CSobject\_n}=H_{CSview\_k,CSobject\_n}\cdot H_{CSwork,CSview\_k}$$

Figure 7. Overview and inspection trajectories on the seabed plane. The views close to the object are indicated on the overview trajectory; for the position indicated on the inspection trajectory, the coordinate reference of the AUV to the object is calculated.

Figure 7. Overview and inspection trajectories on the seabed plane. The views close to the object are indicated on the overview trajectory; for the position indicated on the inspection trajectory, the coordinate reference of the AUV to the object is calculated.

4. Experiments

5. Discussion

6. Conclusions

References

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