# Interpolation-Filtering Method for Image Improvement in Digital Holography
## 2 Image Quality Enhancements in Digital Holography
### Noise Reduction Methods in Digital Holography
#### 2.3.3 Digital Methods  
Digital methods are the most common group of noise reduction methods in digital holography [45; 46; 47; 48; 49]. Digital filters are based on various operations: simple mathematical operations (averaging [42], finding the median [43]), more complex weighted local averaging (Lee and Frost filter [46]), nonlocal estimation (NLM [47]), and complex combinations of filters and transformations (BM3D [48]).  
Median filtering [50] is one of the simplest approaches to image filtering. The filtering window is the only input parameter. This window passes through a certain size of all image values. At each step, the signals in the window are sorted and lined up from smaller to larger. The output for this window is the median signal.  
A more complex filter relying on statistics is the Wiener filter [91]. It has many variations for solving different problems. The filter aims to minimize the root mean square error. The filtering is realized inside a window that traverses the whole image. The input parameters are the noise variance and the size of the window.  
The Lee filter [92] relies on local statistical quantities as well as a multiplicative speckle model. The filter also traverses the image with a window, the size of which is an input parameter. Special weighting factors for each of the summands are used to calculate the final signal value in a pixel. Thus, the filter uses local weighted averaging. Therefore, at low values of deviation, smoothing is performed, and at high values, it is not, which allows the preservation of image details. Such filters are called adaptive filters.  
The Frost filter [93] is an adaptive filter based on weighted averaging. It is a convolution filter and is designed to suppress multiplicative noise. The algorithm is based on working with coefficients of variation, which are the ratios of the local standard deviation to the local mean. At each step, the center pixel of the window is replaced by a weighted sum over its nearest neighbors. The weight of the contribution decreases with distance from the center pixel and increases with increasing deviation.  
The above filters relied on local statistics. Nonlocal means (NLM) [47] is also applied. The pixel signal is calculated based on weighted averaging over the whole image. During implementation, the averaging radius can be selected, which will be the filter input parameter. The similarity of signal levels of two pixels depends on the similarity of signal levels of their nearest neighbor vectors and decreases as a function of weighted Euclidean distance. The efficiency of operation is ensured by the dependence of similarity on the nearest neighborhood of the compared pixels.  
One of the most advanced [94] filters is BM3D [95]. 3D structures are formed from similar parts of the image, and then a threshold or Wiener filter is applied to them, which is a filter parameter. The estimates obtained during filtering are used as a reference for re-filtering the 3D blocks of the original image. Local averaging is used as a final stage.