Marine Radar Image Processing of Compressed Sensing Based on Arbitrary Block Statistical Histogram and Dynamic Dictionary

Abstract

In compressed sensing (CS), the absolute value of the inner product of signal and atom (or dictionary) is used to select atom (or dictionary); the atoms (or dictionaries) with larger absolute values of inner product are selected to decompose and reconstruct signals. This characteristic usually makes it impossible to distinguish the edges of objects (or targets) in some complex cases, especially when the density of objects is very high. For example: let signal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x_{1} =[{\cdots 00110110\cdots }]^{T}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x_{2} =[{\cdots 001110100\cdots }]^{T}$ </tex-math></inline-formula> ; <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{1} $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{2} $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{3} $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{4} $ </tex-math></inline-formula> denote several sparse basis matrices (i.e., sparse dictionaries); <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{1i} =[{\cdots 00000100\cdots }]^{T}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{2j} =[{\cdots 00110000\cdots }]^{T}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{3k} =[{\cdots 00111000\cdots }]^{T}$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{4l} =[{\cdots 00111110\cdots }]^{T}$ </tex-math></inline-formula> are the sparse bases (i.e., atoms) from these sparse basis matrices (i.e., dictionaries), respectively. For signal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x_{1} $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{4l} $ </tex-math></inline-formula> (or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{4}$ </tex-math></inline-formula> ) is selected instead of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{2j} $ </tex-math></inline-formula> (or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{2}$ </tex-math></inline-formula> ); likewise, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x_{2} $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{4l} $ </tex-math></inline-formula> (or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{4}$ </tex-math></inline-formula> ) will be selected instead of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{3j} $ </tex-math></inline-formula> (or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{3}$ </tex-math></inline-formula> ) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{1i} $ </tex-math></inline-formula> (or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi _{1}$ </tex-math></inline-formula> ). This causes the edge of the object to be unrecognizable. Another example is in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{p} $ </tex-math></inline-formula> regularization, it is not necessarily the optimal case: the smaller the value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\| {\Psi _{d}^{-1} x} \|_{0} $ </tex-math></inline-formula> , the better the sparse basis matrix (i.e., dictionary), or the larger the weighing parameter. At present, the inner product is used to process almost all signals in CS. Choosing the dictionary that matches the signal best rather than the dictionary with the maximum inner product value can reconstruct the signal more accurately. In order to overcome the above shortcoming, we propose a fully automatic radar image processing algorithm of CS based on arbitrary block statistical histogram and dynamic dictionary. We use arbitrary block statistical histogram to calculate the non-zero block numbers of different sizes, which can better choose the appropriate sparse base (or dictionary) for the signal. Furthermore, the better measurement vector y can be obtained, and the signal can be reconstructed more accurately than the state-of-the-art methods at the receiving end. To realize the proposed method, we construct objective functions and flow charts for noiseless and noisy signals, respectively. In our method, for noiseless signal, discarding the sub-images that do not contain objects can reduce running time; for noisy signal, according to the theory of wavelet, choosing the appropriate wavelet (i.e., wavelet basis) can usually suppress noise. Our proposed algorithm can overcome the above shortcoming of CS in using the absolute value of the inner product, reduce running time, suppressnoise, and improve the signal-to-noise ratio (SNR). The simulation results show that our method is superior to the state-of-the-art methods.