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Satellite remote sensing (RS) is an important contributor to Earth observation, providing various kinds of imagery every day, but low spatial resolution remains a critical bottleneck in a lot of applications, restricting higher spatial resolution analysis (e.g., intra-urban). In this study, a multifractal-based super-resolution reconstruction method is proposed to alleviate this problem. The multifractal characteristic is common in Nature. The self-similarity or self-affinity presented in the image is useful to estimate details at larger and smaller scales than the original. We first look for the presence of multifractal characteristics in the images. Then we estimate parameters of the information transfer function and noise of the low resolution image. Finally, a noise-free, spatial resolution-enhanced image is generated by a fractal coding-based denoising and downscaling method. The empirical case shows that the reconstructed super-resolution image performs well in detail enhancement. This method is not only useful for remote sensing in investigating Earth, but also for other images with multifractal characteristics.

Super-resolution (SR) reconstruction is an attractive and promising method in digital image processing that aims at producing a detailed and spatial resolution-enhanced image from one or more low-resolution (LR) images [

Fractal theory is a very efficient method to depict chaotic, erratic, natural phenomena. After its conception in the 1970s by Mandelbrot, fractal theory was applied to numerous domains [

The remainder of the paper is organized as follows. In Section 2, we present detailed methods for reconstructing super-resolution images based on multifractal analysis and coding. We first investigate multifractal methods in order to better explore the multifractal characteristic of images. Then we propose the fractal-based super-resolution reconstruction method that consists of parameter estimation, denoising, and downscaling. An empirical study is presented in Section 3. Finally, we discuss the results and draw conclusions in Section 4.

The relationship between a high spatial resolution image

The framework of super-resolution reconstruction using multifractal analysis is shown in

Fractal dimension is a basic tool of fractal theory to quantify irregular patterns or behaviors in natural physical systems. It reflects the extent of a measure's smoothness or roughness quite well. For fractal objects, the relationship between a certain size and the number of objects can be expressed as [_{i}_{i}_{i}_{i}_{α}^{–}^{f}^{(}^{α}^{)}, which generalizes

A multifractal complex system can be decomposed into a series of subsets with different

Besides the singularity spectrum, the generalized dimension _{q}_{i}_{i}_{q}_{q}_{0}, entropy dimension _{1}, and correlation dimension _{2}, respectively. The relationship between them is _{0} ≥ _{1} ≥ _{2}, where equality occurs when the measure is mono-fractal [

Besides its high compression ability, fractal coding has some important properties for image resolution enhancement [

The information transfer function _{↓}(_{↓}(·) is the downsampling function.

It is worthwhile to explore the relationship between the information transfer function and traditional downsampling and averaging shrinking methods. Given _{k}_{j}^{2}), this is the down-sampling shrinking method (_{j}^{2}), then the shrinking method is averaging (

In practice, getting the information transfer function

Let _{i}_{j}_{j}_{ij}_{j}_{i}_{i}, D_{j}_{j}

Generally, the geometrical transformation _{ij}_{ij}_{j}_{i}_{ij}_{j}_{i}_{i}_{j}_{j}_{i}_{i}_{ij}^{(}^{k}^{)}(_{j}_{2} calculates the Euclidian distance between the transformed domain block g_{ij}^{(}^{k}^{)}(_{j}_{i}

Thus, the fractal code of _{i}_{ij}, β_{ij}_{i}_{j}_{k}_{ij}_{ij}

The process of upscaling from domain block _{j}_{i}_{j}_{i}_{i}^{2}) are density values of pixels within _{j}_{i}_{i}^{2}) are the discrete values of the shrinking function in discrete space, subjected to Σ_{i}_{i}

Ghazel _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}^{2}), then:

The set of five elements (_{ij}, β̃_{ij}) constitutes the PIFS code of a noise-free image ^{-6} could be acceptable as little difference between two images when the gray level lies in [0,255].

In the real world, different natural scenes/phenomena have different upscaling information transfer rules and formulisms. Even though some real scenes follow the rule of Gaussian upscaling information transfer model between scales, they may have different Gaussian function distribution intensities. The Gaussian upscaling function's form, then, would affect the reconstruction result directly. In the absence of

Shuttle Radar Topography Mission (SRTM) is a joint project between NASA, NGA, and the German and Italian Space Agencies to obtain a global digital topographic dataset [

The singularity spectrum ^{2} of all linear fits were equal to or greater than 0.99. Some characteristics of the spectrum were used to estimate a measure's multifractality (_{0} (_{0}), and the spectrum width is small and tends toward 0. An image with a convex and wide singularity spectrum can be considered multifractal rather than mono-fractal.

The range of the singularity strength _{max} - α_{min}_{min} and _{max} are calculated by fitting the spectrum curve and taking the point of intersection with the

Asymmetry of the _{0} - _{min})/(_{max} - _{0}) [

To estimate the noise variance of the image, a local statistics method was adopted. A moving window with a size of 2 by 2 pixels was selected to collect all possible blocks in the image, which moved left-to-right and top-to-bottom, and the displacement was one pixel at a time. Then, a histogram of the local variance distribution was generated, which approximately followed a lognormal distribution with mean 1.72, variance 1.03 (^{2}) of curve fit is 0.93 (

After estimating the AWGN noise and information transfer function

The SR image is an estimation of the real scenes. It performs quite well in recovering details. At the same time, there was also a bit of blur and block in some regions of the SR image (

Remote sensing provides an important approach to investigating the earth, but the spatial resolution is usually very low when researching at relatively small spatial scales, such as an intra-urban scale, rather than continental or national scales. In this paper, a fractal-based super-resolution reconstruction method was introduced. Self-similarity or self-affinity presented between different scales makes it possible to reconstruct details at a smaller scale than the original LR image's scale. We explored self-similarity and self-affinity characteristics with a multifractal analysis method. Singularity spectrum and generalized dimension are efficient indices to measure the self-similarity and self-affinity characteristic of the image. Multifractality is common in nature, especially geophysics. Different phenomena have different information transfer mechanisms between scales. The ITF determines how information is transferred and lost in upscaling. In the absence of

How to estimate the AWGN

This study is supported by the NSFC(40471111, 70571076), CAS (kzcx2-yw-308) and the MOST (2006AA12Z215). The authors would like to thank to Luke Driskell for his kind help and hard work on English language polishing of the article.

Self-similarity between scales.

Framework of SR construction.

Information transfer function

Work process to estimate the ITF parameter.

SRTM elevation dataset.

Multifractal spectrum of SRTM.

Probability distribution function of local variance.

Super-resolution reconstruction of SRTM image.