# A New Adaptive Spatial Filtering Method in the Wavelet Domain for Medical Images
## 1 Introduction  
### State of the Art  
Several statistical models for images have been proposed in the wavelet specialized literature. The simplest and most widespread model considers wavelet coefficients to be independent and identically distributed according to a generalized Gaussian distribution [1; 2; 3; 4], a model successfully used in noise removal and image restoration applications. This model approximates the first-order statistics of wavelet coefficients, but does not take into account higher-order statistics, which leads to some limitations. The existing dependencies between the wavelet coefficients have been studied in order to implement efficient compression algorithms and they were explicitly formulated in [5] or implicitly in [6]. Most statistical models in the wavelet domain of images can be grouped into two categories: models that exploit inter-scale dependencies and models that exploit intra-scale dependencies. Despite the processing complications, it has been shown [7; 8; 9; 10] that using both types of dependencies leads to better results in white noise removal, compared to the cases where only one of the two dependencies are operated. Adaptive spatial filters are easy to be hardware implemented as proven by [PERSON] and [PERSON] [11]. They used this type of filter in order to enhance block coded images during the decoding stage. [PERSON] et al., used in their work [12] the same type of filter combined with PCA (Principal Component Analysis) in order to reduce noise in photo-acoustic signals. The noise sources are considered to be coming from auxiliary equipment, such as a stepping motor or a power supply. Adaptive spatial filter was, in this case, significantly more effective than, for example, a band-pass one. Two variants of adaptive spatial filtering with selectable ROIs (Region of Interest) were studied and compared by [PERSON] et al., in [13]: one where the ROIs were chosen based on known information and one where the ROIs were optimized numerically. The filters were applied for Electro-Encephalo-Graphy (EEG) data type and the latter one achieved an efficiency of 97.7% compared to 93.7% obtained by applying the first variant. The implementation of this filter in the medical and biological field was also of interest for [PERSON] et al., in their study [14] related to electromyogram signals. For example, in the case of weak contact between skin and electrode, this filter achieved 19 dB improvement, compared to other filters. Applications related to spinal cord injury were also studied in [15], giving clear information about muscle paralysis and neuromuscular changes. In [16], [PERSON] and [PERSON] took advantage of the filter's capability and used it for noise reduction in speech recognition for hearing-deficient persons, with a good preservation of the original signal and without distortions. The application fields of the adaptive spatial filter are numerous. [PERSON] used such techniques in his work [17] in order to automatically detect targets in infrared images. Again, like in [12], the filter is compared against a band-pass filter which does not prove to be as efficient as the adaptive spatial filter. [PERSON] et al., conducted in [18] a study related to the use of the filter in EEG (Electro-Encephalo-Graphy) signals for the preservation of useful neural information during the processing stage. Thus, they were able to achieve accuracies of over 86%. The use of the adaptive spatial filter in the neurology domain was also deeply studied by [PERSON] and [PERSON] in their book [19]. They cover topics like brain imaging, neural signals, and the effects that different parameters may have on the filter's efficiency. They consider adaptive spatial filters to be a real breakthrough for the neural imaging techniques. In [20], [PERSON] and [PERSON] emphasize the importance of image enhancement in order to obtain a suitable image for certain usage.