The *Local Invariant Order Pattern* (LIOP) descriptor [wang11local]} is a local image descriptor based on the concept of local order pattern*. An order pattern is simply the order obtained by sorting selected image samples by increasing intensity. Consider in particular a pixel $$ and $n$ neighbors $,,,$. The local order pattern at $$ is the permutation $$ that sorts the neighbours by increasing intensity $I({(1)}) I({(2)}) I({(2)})$.
An advantage of order patterns is that they are invariant to monotonic changes of the image intensity. However, an order pattern describes only a small portion of a patch and is not very distinctive. LIOP assembles local order patterns computed at all image locations to obtain a descriptor that at the same time distinctive and invariant to monotonic intensity changes as well as image rotations.
In order to make order patterns rotation invariant, the neighborhood of samples around $$ is taken in a rotation-covariant manner. In particular, the points $,,$ are sampled anticlockwise on a circle of radius $r$ around $$, as shown in the following figure:
Since the sample points do not necessarily have integer coordinates, $I()$ is computed using bilinear interpolation.
Intensity rank spatial binning
Once local order patterns are computed for all pixels $$ in the image, they can be pooled into a histogram to form an image descriptor. Pooling discards spatial information resulting in a warp-invariant statistics. In practice, there are two restriction on which pixels can be used for this purpose:
- A margin of $r$ pixels from the image boundary must be maintained so that neighborhoods fall within the image boundaries.
- Rotation invariance requires the pooling regions to be rotation co-variant. A way to do so is to make the shape of the pooling region rotation invariant.
For this reason, the histogram pooling region is restricted to the circular region shown with a light color in the figure above.
In order to increase distinctiveness of the descriptor, LIOP pools multiple histograms from a number of regions $R_1,,R_m$ (spatial pooling). These regions are selected in an illumination-invariant and rotation-covariant manner by looking at level sets: \[ R_t = :{t} I() < {t+1}