Texture analysis plays an important role in applications like automated pattern inspection, image and video compression, content-based image retrieval, remote-sensing, medical imaging and document processing, to name a few. Texture Structure Analysis is the process of studying the structure present in the textures. This structure can be expressed in terms of perceived regularity. Our human visual system (HVS) uses the perceived regularity as one of the important pre-attentive cues in low-level image understanding. Similar to the HVS, image processing and computer vision systems can make fast and efficient decisions if they can quantify this regularity automatically. In this work, the problem of quantifying the degree of perceived regularity when looking at an arbitrary texture is introduced and addressed. One key contribution of this work is in proposing an objective no-reference perceptual texture regularity metric based on visual saliency. Other key contributions include an adaptive texture synthesis method based on texture regularity, and a low-complexity reduced-reference visual quality metric for assessing the quality of synthesized textures. In order to use the best performing visual attention model on textures, the performance of the most popular visual attention models to predict the visual saliency on textures is evaluated. Since there is no publicly available database with ground-truth saliency maps on images with exclusive texture content, a new eye-tracking database is systematically built. Using the Visual Saliency Map (VSM) generated by the best visual attention model, the proposed texture regularity metric is computed. The proposed metric is based on the observation that VSM characteristics differ between textures of differing regularity. The proposed texture regularity metric is based on two texture regularity scores, namely a textural similarity score and a spatial distribution score. In order to evaluate the performance of the proposed regularity metric, a texture regularity database called RegTEX, is built as a part of this work. It is shown through subjective testing that the proposed metric has a strong correlation with the Mean Opinion Score (MOS) for the perceived regularity of textures. The proposed method is also shown to be robust to geometric and photometric transformations and outperforms some of the popular texture regularity metrics in predicting the perceived regularity. The impact of the proposed metric to improve the performance of many image-processing applications is also presented. The influence of the perceived texture regularity on the perceptual quality of synthesized textures is demonstrated through building a synthesized textures database named SynTEX. It is shown through subjective testing that textures with different degrees of perceived regularities exhibit different degrees of vulnerability to artifacts resulting from different texture synthesis approaches. This work also proposes an algorithm for adaptively selecting the appropriate texture synthesis method based on the perceived regularity of the original texture. A reduced-reference texture quality metric for texture synthesis is also proposed as part of this work. The metric is based on the change in perceived regularity and the change in perceived granularity between the original and the synthesized textures. The perceived granularity is quantified through a new granularity metric that is proposed in this work. It is shown through subjective testing that the proposed quality metric, using just 2 parameters, has a strong correlation with the MOS for the fidelity of synthesized textures and outperforms the state-of-the-art full-reference quality metrics on 3 different texture databases. Finally, the ability of the proposed regularity metric in predicting the perceived degradation of textures due to compression and blur artifacts is also established.