La complejidad selectiva, el sistema complejo de segundo orden y la ciudad

In order to match the SURF descriptors between two scenes, Lowe’s nearest neigh- bour ratio method [68] for matching SIFT descriptors is used. The use of the nearest neighbour distance ratio threshold was found to improve the robustness of SIFT and to yield acceptable matching accuracy in the performance evaluation of [75]. As SURF is basically a SIFT-based descriptor, using this threshold also benefits SURF; in fact there are even more gains in performance due to the reduced vector size of SURF (64D versus 128D for SIFT). Note that in this work, only the 64D desc component of the complete SURF descriptor is used for matching.

5.1 Encoding the salient ROIs using SURF descriptors 105

The nearest neighbour ratio method is interesting as it does not use a simple distance measure to determine if two keypoints are matching using their descrip- tors. Instead, the ratio of the Euclidean distances of the closest match, Lclose, and

the second closest match, L2close, known as the distance ratio, dratio, is used as the

measure of similarity:

dratio=

Lclose

L2close

, dratio∈ {0 · · · 1} (5.1)

As was explained in [68], using this measure allows the matching algorithm to discard keypoints that do not have good matches. The main idea is that for matches that are correct, the closest match is significantly closer than the closest incorrect match (second closest neighbour) for reliable matching. On the other hand, false matches are likely have their second closest neighbour nearer to the closest match, brining dratio closer to unity. Hence a dratio threshold that is near

to 1 allows for more relaxed matching at the expense of a higher false detection rate, while a small threshold only allows for very constrained matching with very low false detection rates. In this thesis, the dratio threshold is fixed at 0.83. This

value achieves good positive matches and rejects the majority of bad matches in practice.

5.1 Encoding the salient ROIs using SURF descriptors 106

Enforcing uniqueness constraint in matching

As Lowe’s original MATLAB code for SIFT descriptor matching (available on-R

line at http://www.cs.ubc.ca/∼_{lowe/keypoints/) does not guarantee against}

repeated matches (a many-to-one mapping) of the SIFT descriptors, mismatches can occur between a SIFT feature with many SIFT features. This problem is even more likely to occur for SURF due to its reduced vector dimensions that results in a possible reduction in uniqueness of the descriptor (Fig. 5.2).

Figure 5.2: Numerous many-to-one SURF matches (cyan lines) using the origi- nal Lowe’s matching algorithm (circled) result in unpredictable scene recognition: Wrong recognition (top) and correct recognition with some mismatches (bottom).

5.1 Encoding the salient ROIs using SURF descriptors 107

mapping theory to effectively solve the correspondence problem using three intu- itive local criteria to establish good global mapping between any two image frames. These three criteria are:

1. Principle of Similarity: similar features are matched;

2. Principle of Proximity: close features should be matched;

3. Principle of Mutual Exclusion: only one-to-one mappings are allowed, which is the uniqueness constraint.

In a ground breaking work, Scott and Longuet-Higgins [99] proposed an algorithm to match point features that encompasses Ullman’s second and third principles (proximity and mutual exclusion) using Singular Value Decomposition(SVD). This work was extended by Sharpiro and Brady [102] who used an eigenvalue approach to further constrain the matchings. In [90], Pilu improved the original Scott-Longuet- Higgins (SLH) algorithm by the use of SVD over a correlation-weighted proximity matrix that contains the cross-correlation values of the image features that en- forces Ullman’s first principle (the feature similarity principle) that complements the SLH algorithm. In this thesis, Pilu’s algorithm is adapted to SURF descriptors by constructing a similar correlation matrix without computing the SVD which is computationally expensive since there could be many SURF keypoints (> 1000). The justification for applying SVD is to impose the uniqueness constraint by mak- ing the proximity matrix orthogonal which is useful for perfect image registration.

5.1 Encoding the salient ROIs using SURF descriptors 108

Since this possibility is extremely unlikely in a practical SRS system on a mobile agent, the uniqueness constraint can be reliably approximated for most scenarios without performing SVD.

A brief description of the matching strategy follows. Note that the description below applies only to one particular colour space which can be extended to all the three colour spaces that the descriptors are encoded in (section 5.1.4). Denoting the two SURF descriptors as (Desc1, Desc2) containing (Nd(Desc1), Nd(Desc2))

SURF keypoints respectively as inputs:

1. Construct the dot-product matrix, mdotp of the SURF descriptors by

mdotp= Desc1• (Desc2)T (5.2)

where • is the dot product operator and T represents the transpose of the matrix. mdotpoptimises the computations since MATLAB is optimised forR

matrix operations.

2. Compute the arccos of mdotp which is a close approximation of the ratio of

Euclidean distances when the angles between the input vectors are small. This forms the proximity matrix, mprox:

5.1 Encoding the salient ROIs using SURF descriptors 109

where the negative sign is necessary for the algorithm to perform a maximum search for the closest match that has numerically the smallest distance in a non-negative mprox.

3. Using Pilu’s algorithm [90], a search for the maximum of each row and column in mproxis performed with the indices saved as separate variables. Comparing

the indices of the maximal elements, only the indices that are maximum in both the rows and columns are accepted as potential correspondences. This step ensures a one-to-one matching as shown in Fig. 5.3.

Figure 5.3: Ensuring one-to-one correspondences using mprox. The poten-

tial matches (red elements) are the maximum values in both the columns and rows of mprox.

4. Finally the potential correspondences are accepted if their values are smaller than the predefined threshold for dratio, with smaller values representing bet-

ter matches.

Using this algorithm results in better SURF correspondences due to the uniqueness constraint as can be seen in Fig. 5.4

5.1 Encoding the salient ROIs using SURF descriptors 110

Figure 5.4: The same matching examples of Fig. 5.2 are shown here after invoking the uniqueness constraint. The top scene is correctly recognised and the bottom scene has fewer mismatches.