On the compression perception (III.): Geometrical interpretation

Main content:

2. Sparse representation model of signals
4. Compression measurement
6. RIP Properties
7. Restore rebuild
   2. A sparse representation model of signals

      The signal is sparse in a space, and can become sparse if transformed into a space.

      Sparse signals represent a very small number of non-0 coefficients.

      For example, the left side indicates that the x signal has only a non-0 coefficient in the R3 space, and the right side indicates that the x signal has only two non-0 coefficients in R3 space.

      If the signal is sparse, then there is no need to collect those values at a spatial factor of 0. Instead, only a small amount of non-0 coefficients is collected, and a little uncertainty is allowed.

      Then the sparse model is adopted to reconstruct the signal and solve the problem of uncertainty.

   4. Second, compression measurement

      Compression measurement: The sparse signal (k-sparse) is projected from the n-dimensional space through a linear projection into the M-dimensional space. M<<n

      Process: y=φ*x

      Y is the measured value after the linear projection;

      Φ is the measurement matrix;

      X is the signal;

      The properties to be satisfied by the measurement matrix:

      Necessity: Must have 2*k line

      Validity: Gaussian random matrix of 2*k rows

      Measurement process: Linear projection process from signal x to measured value y

      Geometric model of n-dimensional space to m-dimensional spatial mapping:

      To illustrate the selection of the measurement matrix, give a simple example:

      How should the measurement matrix be chosen here? Consider the following scenarios:

      The above matrix is too simple, but the main problem is: The space base vector of the measurement matrix and the sparse base vector of the signal must satisfy certain non-correlation.

      The following is a description of the properties that the measurement matrix needs to meet theoretically:

   6. Third, Rip Nature

      Restricted isometry Property (aka UUP)

      For K-sparse signal x, the measurement matrix satisfies the K-Order RIP property if the measurement matrix satisfies the following relationship.

      For K-sparse X1 and X2 signals, the measurement matrix satisfies the 2K-order rip Nature, which means:

      I do not understand the meaning of the above formulas.

      In practice, it is impossible to verify the validity of the measurement matrix by the above formula, which only provides a theoretical support.

      Practice has shown that some of the following random matrices can meet the needs of measurement with a large probability in satisfying situations:

   8. Iv. geometric model of restoration and reconstruction signals

      L0 Model:

      The sparsity of the signal corresponds to the minimization of the non-0 coefficients, so it is possible to model by L0,

      But the mathematical model established by L0 is not a micro-gradient method, so the greedy method is generally used to solve it.

      L2 Model:

      The mathematical model established by the L2 paradigm is not sparse, but a lot of small components. Therefore, in the compression sense, it is not suitable for modeling.

      L1 Model:

      Mathematicians have shown that, in a way, the L1 model is equivalent to the L0 model.

      The equivalence of L1 model and L0 model proves that:

On the compression perception (III.): Geometrical interpretation