Similarity-invariant saliency

The first version of the Kadir–Brady saliency detector[10] only finds Salient regions invariant under similarity transformation. The algorithm finds circle regions with different scales. In other words, given ${\displaystyle H_{D}(x,s)}$, where s is the scale parameter of a circle region ${\displaystyle R}$, the algorithm selects a set of circle regions, ${\displaystyle \{x_{i},s_{i};i=1\dots N\}}$.

The method consists of three steps:

• Calculation of Shannon entropy of local image attributes for each x over a range of scales — ${\displaystyle H_{D}(x,s)=-\sum _{i\in (1\dots r)}P_{D}(d_{i},x,s)\log P_{D}(d_{i},x,s)/10}$;
• Select scales at which the entropy over scale function exhibits a peak — ${\displaystyle s_{p}}$ ;
• Calculate the magnitude change of the PDF as a function of scale at each peak — ${\displaystyle W_{D}(x,s)=\sum _{i\in (1\dots r)}|{\frac {\partial }{\partial s}}P_{D,}(d_{i},x,s)|}$ (s).

The final saliency ${\displaystyle Y_{D}(x,s_{p})}$ is the product of ${\displaystyle H_{D}(x,s_{p})}$ and ${\displaystyle W_{D}(x,s_{p})}$.

For each x the method picks a scale ${\displaystyle s_{p}}$ and calculates salient score ${\displaystyle Y_{D}(x,s_{p})}$. By comparing ${\displaystyle Y_{D}(x,s_{p})}$ of different points ${\displaystyle x}$ the detector can rank the saliency of points and pick the most representative ones.

Affine-invariant saliency

Previous method is invariant to the similarity group of geometric transformations and to photometric shifts. However, as mentioned in the opening remarks, the ideal detector should detect region invariant up to viewpoint change. There are several detector [] can detect affine invariant region which is a better approximation of viewpoint change than similarity transformation.

To detect affine invariant region, the detector need to detect ellipse as in figure 4. ${\displaystyle R}$ now is parameterized by three parameter (s, "ρ", "θ"), where "ρ" is the axis ratio and "θ" the orientation of the ellipse.

This modification increases the search space of the previous algorithm from a scale to a set of parameters and therefore the complexity of the affine invariant saliency detector increases. In practice the affine invariant saliency detector starts with the set of points and scales generated from the similarity invariant saliency detector then iteratively approximates the suboptimal parameters.

Comparison

Although similarity invariant saliency detector is faster than Affine invariant saliency detector it also has the drawback of favoring isotropic structure, since the discriminative measure ${\displaystyle W_{D}}$ is measured over isotropic scale.

To summarize: Affine invariant saliency detector is invariant to affine transformation and able to detect more generate salient regions.

Salient volume

It is intuitive to pick points from a higher salient score directly and stop when a certain number of threshold on "number of points" or "salient score" is satisfied. Natural images contain noise and motion blur which both act as randomisers and generally increase entropy, affecting previously low entropy values more than high entropy values.

A more robust method would be to pick regions rather than points in entropy space. Although the individual pixels within a salient region may be affected at any given instant, by the noise, it is unlikely to affect all of them in such a way that the region as a whole becomes non-salient.

It is also necessary to analyze the whole saliency space such that each salient feature is represented. A global threshold approach would result in highly salient features in one part of the image dominating the rest. A local threshold approach would require the setting of another scale parameter.

A simple clustering algorithm meets these two requirements are used at the end of the algorithm. It works by selecting highly salient points that have local support i.e. nearby points with similar saliency and scale. Each region must be sufficiently distant from all others (in R3) to qualify as a separate entity. For robustness we use a representation that includes all of the points in a selected region. The method works as follows:

1. Apply a global threshold.
2. Choose the highest salient point in saliency-space (Y).
3. Find the K nearest neighbours (K is a pre-set constant).
4. Test the support of these using variance of the centre points.
5. Find distance, D, in R3 from salient regions already clustered.
6. Accept, if D > scalemean of the region and if sufficiently clustered (variance is less than pre-set threshold Vth ).
7. Store as the mean scale and spatial location of K points.
8. Repeat from step 2 with next highest salient point.

The algorithm is implemented as GreedyCluster1.m in matlab by Dr. Timor Kadir^[5]

Performance under global transformation

In order to measure the consistency of a region detected on the same object or scene across images under global transformation, repeatability score, which is first proposed by Mikolajczyk and Cordelia Schmid in [18, 19] is calculated as follows:^[7][8]

Firstly, overlap error ${\displaystyle \epsilon }$ of a pair of corresponding ellipses ${\displaystyle \mu _{a}}$ and ${\displaystyle \mu _{b}}$ each on different images is defined:

${\displaystyle \epsilon =1-{\frac {\mu _{a}\cap (A^{T}\mu _{b}A)}{\mu _{a}\cup (A^{T}\mu _{b}A)}}}$

where A is the locally linearized affine transformation of the homography between the two images,

and ${\displaystyle \mu _{a}\cap (A^{T}\mu _{b}A)}$ and ${\displaystyle \mu _{a}\cup (A^{T}\mu _{b}A)}$ represent the area of intersection and union of the ellipses respectively.

Notice ${\displaystyle \mu _{a}}$ is scaled into a fix scale to take the count of size variation of different detected region. Only if ${\displaystyle \epsilon }$ is smaller than certain ${\displaystyle \epsilon _{0}}$, the pair of ellipses are deemed to correspond.

Then the repeatability score for a given pair of images is computed as the ratio between the number of region-to-region correspondences and the smaller of the number of regions in the pair of images, where only the regions located in the part of the scene present in both images are counted. In general we would like a detector to have a high repeatability score and a large number of correspondences.

The specific global transformations tested in the test dataset are:

• Viewpoint change
• Zoom+rotation
• Image blur
• JPEG compression
• Light change

The performance of Kadir–Brady saliency detector is inferior to most of other detectors mainly because the number of points detected is usually lower than other detectors.

The precise procedure is given in the Matlab code from Detector evaluation #Software implementation.

Performance under intra-class variation and image perturbations

In the task of object class categorization, the ability of detecting similar regions given intra-class variation and image perturbations across object instance is very critical. Repeatability measures over intra-class variation and image perturbations is proposed. The following subsection will introduce the definition and discuss the performance.