This letter proposes a representation learning framework of convolutional neural networks (Convnets) that aims to rectify and improve the feature representations learned by existing transformation-invariant methods. The existing methods usually encode feature representations invariant to a wide range of spatial transformations by augmenting input images or transforming intermediate layers. Unfortunately, simply transforming the intermediate feature maps may lead to unpredictable representations that are ineffective in describing the transformed features of the inputs. The reason is that the operations of convolution and geometric transformation are not exchangeable in most cases and so exchanging the two operations will yield the transformation error. The error may potentially harm the performance of the classification networks. Motivated by the fractal statistics of natural images, this letter proposes a rectifying transformation operator to minimize the error. The proposed method is differentiable and can be inserted into the convolutional architecture without making any modification to the optimization algorithm. We show that the rectified feature representations result in better classification performance on two benchmarks.

In this study, we propose a simple, yet general and powerful framework of integrating multiple global and local features by Product Sparse Coding (PSC) for image retrieval. In our framework, multiple global and local features are extracted from images and then are transformed to Trimmed-Root (TR)-features. After that, the features are encoded into compact codes by PSC. Finally, a two-stage ranking strategy is proposed for indexing in retrieval. We make three major contributions in this study. First, we propose TR representation of multiple image features and show that the TR representation offers better performance than the original features. Second, the integrated features by PSC is very compact and effective with lower complexity than by the standard sparse coding. Finally, the two-stage ranking strategy can balance the efficiency and memory usage in storage. Experiments demonstrate that our compact image representation is superior to the state-of-the-art alternatives for large-scale image retrieval.

Fourier transform is a significant tool in image processing and pattern recognition. By introducing a hypercomplex number, hypercomplex Fourier transform treats a signal as a vector field and generalizes the conventional Fourier transform. Inspired from that, hypercomplex polar Fourier analysis that extends conventional polar Fourier analysis is proposed in this paper. The proposed method can handle signals represented by hypercomplex numbers as color images. The hypercomplex polar Fourier analysis is reversible that means it can be used to reconstruct image. The hypercomplex polar Fourier descriptor has rotation invariance property that can be used for feature extraction. Due to the noncommutative property of quaternion multiplication, both left-side and right-side hypercomplex polar Fourier analysis are discussed and their relationships are also established in this paper. The experimental results on image reconstruction, rotation invariance, color plate test and image retrieval are given to illustrate the usefulness of the proposed method as an image analysis tool.

In this study, we introduce shift-invariant sparse image representations using tree-structured dictionaries. Sparse coding is a generative signal model that approximates signals by the linear combinations of atoms in a dictionary. Since a sparsity penalty is introduced during signal approximation and dictionary learning, the dictionary represents the primal structures of the signals. Under the shift-invariance constraint, the dictionary comprises translated structuring elements (SEs). The computational cost and number of atoms in the dictionary increase along with the increasing number of SEs. In this paper, we propose an algorithm for shift-invariant sparse image representation, in which SEs are learnt with a tree-structured approach. By using a tree-structured dictionary, we can reduce the computational cost of the image decomposition to the logarithmic order of the number of SEs. We also present the results of our experiments on the SE learning and the use of our algorithm in image recovery applications.

Noise greatly degrades the image quality and performance of image compression algorithms. This paper presents an approach for the representation and compression of noisy synthetic images. A new concept region-based prediction (RBP) model is first introduced, and then the RBP model is utilized on noisy images. In the conventional predictive coding techniques, the context for prediction is always composed of individual pixels surrounding the pixel to be processed. The RBP model uses regions instead of individual pixels as the context for prediction. An algorithm for the implementation of RBP is proposed and applied to noisy synthetic images in our experiments. Using RBP to find the residual data and encoding them, we achieve a bit rate of 1.10 bits/pixel for the noisy synthetic image. The decompressed image achieves a peak SNR of 42.59 dB. Compared with a peak SNR of 41.01 dB for the noisy synthetic image, the quality of the decompressed synthetic image is improved by 1.58 dB in the MSE sense. In contrast to our proposed compression algorithm with its improvement in image quality, conventional coding methods can compress image data only at the expense of lower image quality. At the same bit rate, the image compression standard JPEG provides a peak SNR of 33.17 dB for the noisy synthetic image, and the conventional median filter with a 33 window provides a peak SNR of 25.89 dB.

This paper introduces a new image representation method that is named the projective image representation (PIR). We consider an image as a collage of symmetric segments each of which can be well represented by its projection data of a single orientation. A quadtree-based method is adopted to decompose an image into variable sized segments according to the complexity within it. Also, we deal with the application of the PIR to the image compression and propose an efficient algorithm, the quadtree-structured projection vector quantization (QTPVQ) which combines the PIR with the VQ. As the VQ is carried out on the projection data instead of the pixel intensities of the segment, the QTPVQ successfully overcomes the drawbacks of the conventional VQ algorithms such as the blocking artifact and the difficulty in manipulating the large dimension. Above all, the QTPVQ improves the subjective quality greatly, especially at low bit rate, which makes it applicable to low bit rate image coding.