On the compression perception (II.): Theoretical basis

Main content:

1. Sparse representation of a signal
2. Coded measurements (sampling process)
3. Recovery algorithm (non-linear)

A sparse representation of signals and images

In DSP (digital signal processing), there is a very important concept: Transform domain (a linear space: a set of basic functions supported by the space)

In general, our signals are represented in the time domain or in the airspace, in fact we can represent the signals in other transform domains by the linear combination of some orthogonal basis functions. such as: Sinusoids, wavelets, Curvelets, Gabor functions,...

For a transformation domain or space, its base function is determined, as long as this set of values of the coefficient α, the coefficient vector can be expressed by the signal.

What about the coefficient alpha?

Having said so much, why should we express the signal by changing the coefficients of the domain?

Obviously, the size of the coefficient vector α is much smaller than the original signal, which is a process of compression and dimensionality reduction (sparsity), which facilitates storage, transmission, and processing.

In the following example, we introduce the traditional image representation method DCT and the modern image representation method Wavelet transform:

Classical Image REPRESENTATION:DCT

Discrete cosine Transform (DCT)

Basically a real-valued Fourier transform (sinusoids)

As shown, the left side is the original image and the right is the DCT transformed image.

The image represents a two-dimensional frequency amplitude factor, which shows that most of the coefficients in the lower right corner are close to 0. This means that most of the energy in the image is concentrated in the low-frequency portion of the upper-left corner (sparsity),

So we can reconstruct the image on the left as long as we keep the information (compression) in the upper left corner. Lossy

This is also the basis of the JEPG image compression standard: DCT transform.

Image features of DCT reconstruction (inverse transformation): Smooth areas behave well, edges may blur or ring (because some high-frequency signals are lost)

Modern Image representation:2d wavelets

The knowledge of wavelet transform, here is not detailed, can refer to: http://www.zhihu.com/topic/19621077/top-answers

As shown, the left side is the original image, the middle is the scale image, and the right is the coefficient structure after the wavelet transform.

Coefficient frame: Very large coefficients, small coefficients (sparse)

This is also the basis of the JPEG2000 compression standard: wavelet transform.

Image features of wavelet transform reconstruction (inverse transformation): Smooth region is good, edge is more sharp (on edge processing, better than DCT)

Image reconstruction with Wavelet transform:

Distribution of wavelet coefficients:

Reconstruction of Wavelet transform:

This part mainly introduces the transformation domain and the sparse representation of the signal in the transform domain, and takes the DCT and wavelet transform of the image as an example to illustrate the sparsity of the signal in the transform domain.

Summary of the function of sparsity:

1. Compression
2. Noise removal
3. Dimension reduction

Second, the Code measurement

Unlike traditional acquisition, the compression-sensing acquisition is not a pixel, but a set of measured values in a linear combination.

The following formula represents the calculation process for each measured value Yi, and F represents the signal, φ represents the measurement matrix, and the Nei Jizhi of the two is Yi.

After M measurement, the required m measurement data y is obtained.

The question is, how should the measurement matrix be chosen?

In order to reconstruct the signal, the choice of the measurement matrix is particularly important, and the matrix needs to satisfy the sparse representation base ψ of the signal. (RIP nature, specifically not detailed)

The experimental results show that Gaussian random matrix, uniform sphere Matrix, binary random matrix, local Fourier matrix, local hada matrix and Gnas matrix can meet the above conditions in a large probability.

The measurement formula is as follows:

Third, sparse reconstruction algorithm

Assuming that the signal is k-sparse, the measurement matrix is a Gaussian random matrix, and now the m measurements are obtained by acquisition, how can we recover our signals?

Measurement process:

Reconstruction process:(Mathematical modelling: L1 minimization, of course, other methods, follow-up recount)

How many measurements do you need to be able to effectively recover the signal? One, two is obviously not good, N is obviously not the meaning of compression, then at least how much is appropriate?

The following formula gives an estimate:

Transform domain Reconstruction:

Example:

On the compression perception (II.): Theoretical basis