# Conflation Optimized by Least Squares to Maintain Geographic Shapes

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

**Figure 1.**The surface distance to assess two polygons similarity, the ratio between the areas of the grey surfaces (the intersection of polygons A and B and their union).

**Figure 2.**The principles of rubber sheeting conflation (

**a**) the black geometries match with the green ones (

**b**) a vector field is computed to make the black features fit into green feature’s geometry.

## 3. Conflation by Least Squares Optimization

#### 3.1. Least Squares Based Map Generalization

**A**(Equation (1)) and its solution can be expressed with a residual vector

**v**(Equation (1)).

**P**that conveys the relative importance of each equation (e.g., it is more important to be close to solution for equation i than for equation j), the least squares adjustment finds a solution that minimizes:

#### 3.2. Least Squares Applied to Conflation

#### 3.2.1. Principles

**Figure 4.**Two sets of polygonal data to be conflated: two feature-to-feature matchings (red arrows) and two vertex-to-vertex matchings (blue arrows).

**v**, which gives a direct value for the geometrical error estimation of Adams et al. [2].

#### 3.2.2. Constraints to Preserve Shape

**P**matrix lines corresponding to this constraint is 1 in our experiments while other constraints (e.g., the conflation constraint described in Section 3.2.3) have a weight of 20.

#### 3.2.3. Constraints to Conflate Data

**Figure 7.**Computing vector displacement from the partial matching of features (the textured features are matched).

**Figure 8.**Conflation constraint 1: contribution of the displacement vector to the closest vertex of close features.

**u**that aggregates the displacement vectors contributions:

#### 3.2.4. Constraints to Maintain Data Consistency

**Figure 10.**Constrained Delaunay triangulation used to identify proximities: (

**1**) dashed edges dropped because inside objects, (

**2**) edge dropped as distance > threshold, (

**3**) black edges used for point-to-point proximity, (

**4**) grey edges used for point-to-segment proximity.

**Figure 12.**Constraint expression of the preservation of relative orientation and position relations.

#### 3.2.5. Propagation of Additional Data

**Figure 13.**Propagation for topologically connected objects: the connected points are added to the system and a propagated displacement is applied to the remaining points (P

_{3}and P

_{4}).

- Only unmatched features should be propagated.
- Small features and rigid features like should be preferred for propagation as such features often need fewer distortions.
- Features inside conflated features are good candidates for propagation, as it provides accurate propagation vectors.

## 4. Experiments

#### 4.1. Use Case: Land Use Data Conflation

**Figure 14.**Extract of the less accurate dataset of the use case, to conflate with accurate city limits.

#### 4.2. Implementation

#### 4.2.1. Least Squares Adjustment Model

- the chosen conflation constraint should have very high weights (20 in the experiment),
- key shape constraints like stiffness for parcels should have high weights (16 in the experiment),
- the movement constraint should have a minimal weight (1 in the experiment).

#### 4.2.2. Scalability Issues

- Features that share topology and spatial relations should be grouped in a partition.
- Constraints between features at the edge and features outside the partition should be included in the adjustment.

#### 4.3. Results

**Figure 16.**Conflated parcels (dashed lines for initial data) extracted from the Figure 14 (area with small distortion, the arrows show some matching vectors) (

**1**) polygon shapes are well preserved, (

**2**) small spaces between parcels are preserved, (

**3**) even curve spaces due to rivers are preserved.

**Figure 17.**Zoomed extracts of the second conflated city: despite large distortions, complex shapes are well preserved.

#### 4.4. Evaluation

**Figure 18.**Conflation results with data that require large distortions (conflated parcels are in plain blue, initial outlines are dashed and arrows show matching vectors).

**Figure 19.**Propagation of the least squares conflation to buildings inside parcels: Initial geometries drawn with dashed lines.

**Figure 20.**Other examples of propagated buildings, including buildings topologically connected to conflated parcels.

**Figure 21.**(

**1**) Comparison of the least squares based conflation (in plain colors), the rubber sheeting conflation (with dots) and the initial data (with dashes); (

**2**) a broader view of the initial data with deformation vectors.

**Table 1.**Root Mean Square errors (RMS) for Least Squares conflation (LS) and Rubber Sheeting conflation (RS) compared to initial data, for five shape comparing measures and 200 features.

RMS Error LS | RMS Error RS | |
---|---|---|

Area increase ratio | 3.39% | 5.48% |

Surface distance | 0.152 | 0.102 |

Turning function | 0.093 | 0.184 |

Polygon signature | 0.536 | 0.932 |

Hausdorff distance | 3.087 | 3.736 |

**Figure 22.**Initial data to conflate where the identified two defects (very long (

**1**) and very short (

**2**) segments with large distortions) may occur.

**Figure 23.**(

**1**) Conflation benchmark data (

**2**) least squares and rubber sheeting results compared to benchmark final data.

#### 4.5. Discussion

**A**) to process. The matrix to inverse,

**A**is 1,400 × 1,400. On a very standard desktop computer, the conflation takes approximately two minutes with the standard Java matrix API, but less than five seconds when using the C sparse matrix API. The second dataset presented in the results contains less features (70) but larger parcels, with more vertices (Figure 18). The computation time, there, is a bit faster so large and complex features do not slow conflation computation. The framework was also tested on a much larger dataset, with 930 features including very large parcels. Computation takes less than one minute with the sparse matrix API, but the increase of computation time is not due to the adjustment but to the proximities computation. Indeed, filtering the triangulation edges to define proximities has Ο(n

^{T}PA^{2}) complexity due to implementation issues. If we include the required matching process in the computation time, there is no drastic change: an automatic matching technique was tested between initial parcels and conflated parcels of the test datasets, and the processing time is negligible compared to conflation computation time. More complex matching processes may take more time but should not drastically increase the total computation time.

## 5. Conclusions

## Conflict of Interest

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**MDPI and ACS Style**

Touya, G.; Coupé, A.; Jollec, J.L.; Dorie, O.; Fuchs, F. Conflation Optimized by Least Squares to Maintain Geographic Shapes. *ISPRS Int. J. Geo-Inf.* **2013**, *2*, 621-644.
https://doi.org/10.3390/ijgi2030621

**AMA Style**

Touya G, Coupé A, Jollec JL, Dorie O, Fuchs F. Conflation Optimized by Least Squares to Maintain Geographic Shapes. *ISPRS International Journal of Geo-Information*. 2013; 2(3):621-644.
https://doi.org/10.3390/ijgi2030621

**Chicago/Turabian Style**

Touya, Guillaume, Adeline Coupé, Jérémie Le Jollec, Olivier Dorie, and Frank Fuchs. 2013. "Conflation Optimized by Least Squares to Maintain Geographic Shapes" *ISPRS International Journal of Geo-Information* 2, no. 3: 621-644.
https://doi.org/10.3390/ijgi2030621