2D adaptive filters
The main difference between 1D and 2D adaptive filters is that the former usually take as inputs signals with respect to time, what implies in causality constraints, while the latter handles signals with 2 dimensions, like x-y coordinates in the space domain, which are usually non-causal.
The topic of 2D adaptive filters is very important in electrical engineering and signal processing since these filters have the ability to take into account the nonstationary statistical properties of 2D signals.
Adaptive filters find applications in areas such as Noise cancellation, Signal prediction, Equalization and Echo cancellation.
Examples of applications of 2D adaptive filters include Image Denoising,[1] Motion Tracking,[2] OFDM channel estimation,[3] magnetic recording equalization [4] 2D Adaptive Filters can be used to identify systems.
is the system function of the 2D adaptive filter when its output comes to steady.
, is minimized if the unknown system and known 2D adaptive filter have the same input, and if the resulting outputs are similar.
In digital signal processing, any linear shift invariant system can be represented by the convolution of the signal with the filter's impulse response, given by the expression:
, the adaptive system can be obtained by continuously adjusting the weight values according to some cost function
The weight matrix at the next iteration is equal to the present weight matrix plus a change proportional to the negative gradient of the mean square error.
Advantages: The TDLMS adaptive filter can be implemented without any form of matrix operations or any averaging or differentiation.
The algorithm convergence does not depend on the initial conditions and it will converge for any arbitrarily initial value, hence, it provides good performance in nonstationary images.
Disadvantages: The exact values of the expectations of the TDLMS adaptive filter will not converges to a fixed value, if we need to maintain its tracking ability.
Therefore, the design choice of μ depends on the particular application and it involves a tradeoff between the convergence speed, tracking ability, and steady-state MSE.
[6] The two-dimensional IIR filter`s difference equation can be written as
are, respectively, the output and input of the adaptive filter.
Advantages: IIR filters can satisfy the prescribed frequency response because it can reduce the filter`s order requirements.
Disadvantages: The performance surfaces of adaptive LMS IIR Adaptive filters are nonquadratic and may have local minima.
2D Recursive Least Square Adaptive Filters [7] can be developed by applying 1D recursive least squares filters along both horizontal and vertical directions.
The RLS adaptive is an algorithm which finds the filter coefficients recursively to minimize the weighted least squares cost function.
The RLS algorithm is different to the least mean squares algorithm which aim to reduce the mean square error, its input signal is considered deterministic.
For this reason, the RLS algorithm has fast convergence characteristic.
Advantages: The RLS algorithm has fast convergence property.
The accuracy of image denoising based on RLS algorithm is better than 2D LMS adaptive filters.
Disadvantages: The RLS algorithm needs a large amount of computations, especially in two-dimensional and multidimensional case.
[5] This simplifies the implementation and makes it possible to benefit from the extensive literature that is available for 1D adaptive filters and utilize all of the existing 1D algorithms.
Compared to the direct approach, this system has the advantages of a lower computational complexity and a faster convergence rate.
However, in order to work properly, it needs some a priori information about the system to correctly select the transformation function parameters, making the system pre-constrained.
Block Diagonal 2D Adaptive Filters is an alternative approach [10] that scans the signal through blocks and applies weight adjustments for each block, instead of for each sample as in the traditional adaptive filters.
The advantage of this kind of system is that it takes into account signal correlations along both dimensions.
On the other hand, it assumes a higher local stationarity of the signal.
