LPQ (local phase quantization) study notes

I feel that there are very few articles explaining this concept. Here I will sort out the (few) data I found to give my understanding:

1. LPQ related:
   1. The LPQ descriptor is used to deal with blurred images, using the short-time Fourier transform: (u for frequency) (sift, more can refer to Gabor filter study notes_Wsyoneself's blog-CSDN blog) Extracted local phase information to analyzeAround the MxM area of ​​the target pixel x, Fu(x) is the sift output of pixel x using the 2D spatial frequency u.
   2. In the LPQ descriptor, only 4 complex frequencies are considered: u0=(α,0), u1=(α,α), u2=(0,α), u3=(-α,-α),α is a small scalar frequency (much less than 1, a = 1/M (region side length)), corresponding to directions 0, 45, 90, 135, respectively.(in turn into the sift formula)
   3. The underlying LPQ feature at pixel position x is given by the vector:

      Fx = [Re{Fu0(x), Fu1(x), Fu2(x), Fu3(x)},Im {Fu0(x), Fu1(x), Fu2(x), Fu3(x)}]

      where Re {....} and Im {....} are the real and imaginary parts of the complex number.
   4. The elements of this vector are then quantized using the delta function:
      1. If x>=0, δ(x)=1;
      2. Otherwise δ(x)=0.rcandrn(n=1,...,P), the intensity value of the center pixel (x,y) and its P_neighbourhood pixel at the circle of radius R (R>0).(that is, take the similar pixels as the upper bound, and collect the generated number)
   5. Finally, the resulting binary quantized coefficients are represented as integer values ​​in [0-255] and collected into a histogram.
   6. In order to make LPQ statistically independent, a whitening transform based decorrelation step can be applied before the quantization process.

Supplementary knowledge:

Whitening Transform:

1. Steps of the whitening algorithm:
   1. PCA preprocessing: The original data obtains two eigenvectors u1, u2 through the covariance matrix, and then projects each data point to these two eigenvectors to obtain the new coordinates of x in the new feature space
   2. PCA whitening: The features of each dimension are subjected to a standard deviation normalization process (that is, each dimension is directly divided by the standard deviation of the dimension), that is, the variance normalization operation is performed on the new coordinates
   3. ZCA whitening: Based on PCA whitening, the coordinates are converted to the original coordinate system (multiplied by a coefficient matrix)
2. PCA is generally used for dimensionality reduction, but if dimensionality reduction is not performed, only PCA is used to obtain the feature vector, and then the data is mapped to a new feature space, which satisfies the first property (the correlation between features is relatively high.Low).
3. Assuming that the training data is an image, the training input is redundant due to the strong correlation between adjacent pixels in the image.​​​​​​​
4. Whitening is a linear transformation(y=ax), used to decorrelate the source signal, the purpose is to reduce the redundancy of the input data, so that the whitened data has the following properties:
   1. Remove correlation between features//Low correlation between features
   2. All features have variance 1//All features have the same variance