# Multiscale Analysis for Improving Texture Classification
## 2 Multiscale and Multiresolution Analysis
### Multiresolution  
Humans usually see regions of similar textures, colors, or gray levels that combine to form objects when looking at an image. If objects are small or have low contrast, it may be necessary to examine them in high resolution; a coarser view is satisfactory if they are large or have high contrast. If both types of objects appear in an image, it can be helpful to analyze them in multiple resolutions. Changing the resolution can also lead to creating, deleting, or merging image features. Moreover, there is evidence that the human visual system processes visual information in a multiresolution way [20]. Furthermore, sensors can provide data in various resolutions, and multiresolution algorithms for image processing offer advantages from a computational point of view and are generally robust.  
When analyzing an image, it can be helpful to break it down into separate parts so there is no loss of information. The pyramid theory provides ways to decompose images at multiple levels of resolution. It considers a collection of representations of an image in different spatial resolutions, stacked on top of each other, with the highest resolution image at the bottom of the stack and subsequent images appearing over it in descending order of resolution. Such a procedure generates a pyramid-like structure, as shown in Figure 1a.  
The traditional procedure for obtaining a lower-resolution image is to perform low-pass filtering followed by sampling [21]. In signal processing and computer vision, a pyramid representation is the primary type of multiscale representation for computing image features on different scales. The pyramid is obtained by repeated smoothing and subsampling of an image. Different smoothing kernels have been brought forward for generating the pyramid representation, and the binomial one strikingly shows up as valuable and theoretically well-founded [22].  
Accordingly, for a bidimensional image, the normalized binomial filter may be applied (1/4, 1/2, 1/4) in most cases twice or even more, along all spatial dimensions. The subsampling of the image by a factor of two follows, which leads to an efficient and compact multilevel representation. There are low-pass and band-pass pyramid types [23].  
To develop filter-based representations by decomposing images into information on multiple scales and extracting features/structures of interest from an image, the Gaussian pyramid (GP), the Laplacian pyramid (LP), and the wavelet pyramid are examples of the most frequently used pyramids. The GP (Figure 1) consists of a low-pass filter, a reduced density, where subsequent images of the preceding level of the pyramid are weighted down using a Gaussian average or Gaussian blur and scaled down. The original image is defined as the base level. Formally speaking, assuming that \(I(x,y)\) is a two-dimensional image, the GP is recursively defined as presented in (1).  
\[G_{0}(x,y)=\left\{\begin{aligned} & I(x,y),&\text{for level, }I=0\\ &\sum\limits_{m=-2}^{2}\sum\limits_{n=-2}^{2}w(m,n)G_{l-1}(2x+m,2y+n ),&\text{otherwise,}\end{aligned}\right. \tag{1}\]  
where \(w(m,n)\) is a weighting function (identical at all levels) termed the generating kernel, which adheres to the following properties: it is separable, symmetric, and each node at level \(n\) contributes the same total weight to nodes at level \(l+1\). The pyramid name arose because the weighting function nearly approximates a Gaussian function. This pyramid holds local averages on different scales, leveraged for target localization and texture analysis.  
Moreover, assuming the GP \([l_{0},\,l_{1},\ldots,\,l_{k}]\), the LP is obtained by computing \(b_{k}\,=\,l_{k}\,-\,El_{k+1}\), where \(El_{k+1}\) represents an upsampled, smoothed version of \(l_{k+1}\) of the same dimension. In the literature, LP is used for the analysis, compression, and enhancement of images and graphics applications [23].  
The GP is used in this work for the multiresolution representation of the images right before the feature extraction.