本文为加拿大滑铁卢大学（作者：Salah Ameer）的博士论文，共131页。

图像压缩是以最经济的方式进行视频数据传输或存储的一种手段。尽管已经报道了许多算法，但仍然需要继续研究以满足对更高效传输或存储的持续需求。本研究探讨并实现了多项式拟合技术，以之作为有损图像压缩的区块化方法。为了研究非多项式模型，利用钟形函数实现了一种基于区域的图像拟合方案。这个想法只是简单地把图像看作一幅由山丘和山谷组成的三维地理地图，然而，该方案的计算量很大，比现有的许多图像压缩方案都要差。因此，只有多项式模型才能得到进一步的考虑。

本文设计了一阶多项式（平面）模型，使其在无乘法和无除法（MDF）环境下工作。将每个图像块的强度值拟合到一个平面上，然后对参数进行量化和编码。结果表明，在28.8db的压缩比下，标准图像PEPPER可以达到62:1的压缩比，在客观和主观评价上都优于JPEG。块间预测可以显著提高平面模型的压缩性能，在27.9db时压缩比达到112:1。然而，这种改进略微增加了计算的复杂性，并降低了流水处理的能力。虽然JPEG2000不是一个基于块的方案，但令人鼓舞的是，与JPEG2000相比，所提出的预测方案在计算和定性评估方面表现得更好。然而，需要更多的实验来进行更详细的比较。为了减少方块效应，采用了一种基于韦伯定律的后处理方案。据报道，使用该方案后处理的图像在主观上更令人满意，PSNR（小于0.3db）略有增加。本文对韦伯定律进行了修改，以执行边缘检测和质量评估任务。

这些结果激发了对高阶多项式的探索，使用三个参数来保持可比较的压缩性能。为了研究高阶多项式的影响，通过近似渐近的行为，设计了一种新的线性映射方案。高阶多项式逼近方案虽然计算量大，但其性能与平面模型相当，这清楚地证明了平面模型的强大近似能力。因此，所提出的线性映射方案是一种新的图像建模方法，值得进一步研究。

Image compression is a means to performtransmission or storage of visual data in the most economical way. Though manyalgorithms have been reported, research is still needed to cope with thecontinuous demand for more efficient transmission or storage. This researchwork explores and implements polynomial fitting techniques as means to performblock-based lossy image compression. In an attempt to investigate nonpolynomialmodels, a region-based scheme is implemented to fit the whole image usingbell-shaped functions. The idea is simply to view an image as a 3D geographicalmap consisting of hills and valleys. However, the scheme suffers from highcomputational demands and inferiority to many available image compressionschemes. Hence, only polynomial models get further considerations. A firstorder polynomial (plane) model is designed to work in a multiplicationanddivision-free (MDF) environment. The intensity values of each image block are fittedto a plane and the parameters are then quantized and coded. Blocking artefacts,a common drawback of block-based image compression techniques, are reducedusing an MDF line-fitting scheme at blocks’ boundaries. It is shown that acompression ratio of 62:1 at 28.8dB is attainable for the standard imagePEPPER, outperforming JPEG, both objectively and subjectively for this part ofthe rate-distortion characteristics. Inter-block prediction can substantiallyimprove the compression performance of the plane model to reach a compressionratio of 112:1 at 27.9dB. This improvement, however, slightly increasescomputational complexity and reduces pipelining capability. Although JPEG2000is not a block-based scheme, it is encouraging that the proposed predictionscheme performs better in comparison to JPEG 2000, computationally andqualitatively. However, more experiments are needed to have a more concretecomparison. To reduce blocking artefacts, a new postprocessing scheme, based onWeber’s law, is employed. It is reported that images postprocessed using thisscheme are subjectively more pleasing with a marginal increase in PSNR (<0.3dB). The Weber’s law is modified to perform edge detection and qualityassessment tasks.

These results motivate the exploration ofhigher order polynomials, using three parameters to maintain comparablecompression performance. To investigate the impact of higher order polynomials,through an approximate asymptotic behaviour, a novel linear mapping scheme isdesigned. Though computationally demanding, the performances of higher orderpolynomial approximation schemes are comparable to that of the plane model.This clearly demonstrates the powerful approximation capability of the planemodel. As such, the proposed linear mapping scheme constitutes a new approachin image modeling, and hence worth future consideration.

1 引言

2 项目背景与文献回顾

3 区域拟合公式

4 平面拟合公式

5 高阶多项式拟合公式

6 基于块内预测的平面拟合

7 后处理与其它应用

8 结论与未来工作展望

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