Multi-resolution Sampling

Multi-resolution sampling [22], proposed by DeBonet, also constructs an analysis and a synthesis Guassian/Laplacian pyramids in texture synthesis. But it has two major improvements over the previous method [46]; First, multi-resolution sampling extracts a set of more detailed and sophisticated image features by applying a filter bank onto each pyramid level. Second, multi-resolution sampling explicitly takes the joint occurrence of texture features across multiple pyramid levels into account, while the previous method processes each pyramid level separately.

The method synthesises a texture by first generating a synthesis pyramid on a top-down, level-by-level basis and then collapsing the pyramid to obtain a synthetic texture. At each level of the synthesis pyramid, an image is generated pixel by pixel, each pixel being directly sampled from the corresponding level of the analysis pyramid. The sampling process, e.g., selection of pixels, is based on a similarity measure defined in terms of a parent structure, associated with each image location $(x,y)$ at the level $i$ of the pyramid for an image ${g}$, as follows [22],

$$\overrightarrow{\mathcal{S}}_i({g},x,y) = [ \mathbf{F}_{i+1}^0(\f... ...^1(\frac{x}{2}, \frac{y}{2}),..., \mathbf{F}_{i+1}^M(\frac{x}{2}, \frac{y}{2}),$$

$$\mathbf{F}_{i+2}^0(\frac{x}{4}, \frac{y}{4}), \mathbf{F}_{i+2}^1(\frac{x}{2}, \frac{y}{2}),..., \mathbf{F}_{i+2}^M(\frac{x}{4}, \frac{y}{4}),$$

$$......$$

$$\mathbf{F}_{N}^0(\frac{x}{2^N}, \frac{y}{2^N}), \mathbf{F}_{N}^1(... ...c{x}{2^N}, \frac{y}{2^N}),..., \mathbf{F}_{N}^M(\frac{x}{2^N}, \frac{y}{2^N}) ]$$ (5.3.4)

where $N$ and $M$ are the number of pyramid levels and the number of selected filters respectively; $\mathbf{F}_i =\{\mathbf{F}_{i}^m: m=0..M\}$ denote a vector of $M$ filter responses for the pyramid level $i$.

A parent structure represents the joint occurrence of multi-resolution features at a particular image location. The distance between parent structures provides a similarity measure for different image locations, i.e. if two locations have similar parent structures, they are considered indistinguishable [22]. To synthesise a pixel, the algorithm searches the same level of the analysis pyramid for candidate locations that have the most similar parent structures to the current one, i.e. the square distance of each component $\mathbf{F}_i$ between two structures is below a predefined threshold. Then a randomly selected pixel from the candidate list is used as the synthesised value.

This method introduces a novel constrained sampling technique that generates a new texture using pixels selected from the training image. In fact, a training texture is considered as an exemplar and the synthetic texture as a rearrangement of image signals randomly and coherently sampled from the training one. This technique is also known as non-parametric sampling [29]. Based on the same idea, a variety of methods based on non-parametric sampling techniques have been developed, ranging from pixel-based to the latest patch-based sampling [29,104,65,59,28,78]. Compared to model-based image generation, these techniques provide much faster texture synthesis.