Mean and Standard Deviation of the Wavelet Coefficients

The simplest features to be extracted are the energies of the wavelet coefficients at each scale and for each subband. Generally a $L^1$-norm is used, which will be later justified. This gives a first range of features for image $I_j$:

$$E^{(j)}(n,i)= \sum_{x\in R} \vert(WI_j)(x,2^n,i)\vert$$

where $R$ (rectangle) is the domain of the image. Assuming that the textures are homogeneous, it is possible to consider features deriving from second order statistics, such as the mean value of the coefficients at each scale and each subband:

$$\mu^{(j)}(n,i)= \frac{1}{\vert R\vert^2} \sum_{x\in R} \vert(WI_j)(x,2^n,i)\vert$$

and the standard deviation:

$$\sigma^{(j)}(n,i)= \frac{1}{\vert R\vert^2} \sqrt{\sum_{x\in R} (\vert(WI_j)(x,2^n,i)\vert-\mu^{(j)}(n,i))^2 }$$

Then the signature of image $I_j$ is given by $\theta^{(j)} = \{\mu^{(j)}(n,i), \sigma^{(j)}(n,i) \}_{n=1...d, i=1,2,3}$. Usually for 512x512 images the depth $d$ of the decomposition is chosen between 3 and 5. Then the normalized Euclidean distance is chosen to measure similarity between the signatures.
The advantage of this model compared to the previous one is that few coefficients are needed to describe a texture, and this representation is quite precise, since it is a multiscale representation: it is precise until scale $2^d$.
Rotation-invariance can be achieved from this model, even with separable wavelets (see [5]): it suffices to consider translations, dilations and rotations of the mother wavelet in the wavelet transform. Then the wavelet coefficients of image $I_j$ are given by:

$$(WI_j)(x,2^n,i,\varphi) = \int_{R^2} I_j(y)\psi_{x,2^n,i,\varphi}^{*}(y)dy$$

where $\psi_{x,2^n,i,\varphi}=\psi_{x,2^n,i} \circ r_{\varphi}$, $r_{\varphi}$ is the rotation centered in 0 and of angle $\varphi$, and $\psi_{x,2^n,i}$ is the wavelet corresponding to scale $n$, subband number $i$ and centered on $x$. We define $\mu^{(j)}(n,i,\varphi)$ and $\sigma^{(j)}(n,i,\varphi)$ and the signatures

$$\theta^{(j)} = \{\{\mu^{(j)}(n,i,\frac{2\pi k}{N}), \sigma^{(j)}(n,i,\frac{2\pi k}{N}) \}_{n=1...d, i=1,2,3} \}_{k=0...N-1}$$

in a similar way, and use the normalized Euclidean distance. But since this similarity measurement is not rotation invariant, similar textures oriented in a different way may have a big similarity measurement, the retrieval task will flop. [5] proposed an efficient way to make the signature and the similarity measure rotation-invariant by a simple circular shift on the feature map: specifically we calculate the features in every orientation $\varphi=\frac{2\pi k}{N}$. The orientation corresponding to the highest mean value $\mu^{(j)}(n,i,\varphi)$ is called the dominant orientation. We then apply a circular permutation in the signature such that the dominant orientation comes first in the signature. This is computed for every image of the data base and the query image. Then the classic normalized Euclidean distance is used to compare the features of different images. This model is equivalent to rotate every image such that the main direction of the texture be vertical. It is a rotation-invariant model. The numerical results with the Gabor wavelets are good, and computational complexity of the model quite small.
This approach is now generalized to color textures.