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Texture and feature extraction is an important research area with a wide range of applications in science and technology. Selective extraction of entangled textures is a challenging task due to spatial entanglement, orientation mixing, and high-frequency overlapping. The partial differential equation (PDE) transform is an efficient method for functional mode decomposition. The present work introduces adaptive PDE transform algorithm to appropriately threshold the statistical variance of the local variation of functional modes. The proposed adaptive PDE transform is applied to the selective extraction of entangled textures. Successful separations of human face, clothes, background, natural landscape, text, forest, camouflaged sniper and neuron skeletons have validated the proposed method.

Texture is one of the important features characterizing many natural and man-made images. Texture characterization and analysis are usually performed according to the spatial as well as frequency variations of brightness, pixel intensities, color, and texture orientation in the different regions of the image corresponding to different types of textures. For example, the roughness or bumpiness of an image usually refers to variations in the intensity values, or gray levels. Texture segmentation, recognition, and interpretation are critical for human visual perception and processing. As a result, research on texture analysis has received considerable attention in recent years. A large number of approaches has been proposed for texture classification and segmentation [

In general, the total texture extraction has become a mature technique in real applications. However, despite the progress in the past few decades, selective extraction of entangled textures encounters a number of difficulties. One difficulty is due to

In this work, we propose an adaptive partial differential equation (PDE) transform approach for selective extraction of entangled textures. By using arbitrarily high-order PDEs, the PDE transform is able to decompose signals, images, and data into functional modes, which exhibit appropriate time-frequency localizations [

In the past two decades, PDE-based image processing approaches have raised a strong interest in the image processing and applied mathematical communities and have opened new approaches for image denoising, enhancement, edge detection, restoration, segmentation, and so forth. The use of PDEs for image analysis started as early as 1980s when Witkin first introduced diffusion equation for image denoising [

Arbitrarily high-order nonlinear PDEs were introduced by Wei in 1999 to more efficiently remove image noise in edge-preserving image restoration [

In general, the nonlinear PDE operators described above serve as lowpass filters. PDE-based nonlinear highpass filters were introduced by Wei and Jia [

In the PDE transform, intrinsic mode functions

The PDE transform is applied to Figure

Extraction of various embedded textures using the PDE transform. (a) shows the original image composed of various horizontal and vertical textures. (b)–(d) show the three texture patterns extracted by applying the PDE transform, one at each time. (e) shows the edge mode obtained by applying the PDE transform to (a). (f) shows the variance of the local variation of the image mode function (e). (g) and (h) show the projection, or average, of the variance in (f) along

The separation of textures that are highly entangled in spatial locations, frequency ranges, and gray scales become a challenge, and conventional segmentation techniques are in general not applicable for such cases. For example, highly oscillatory textures can be separated from slowly varying background but cannot be separated from another texture with overlapping frequency distribution purely based on frequency fingerprints. To selectively distinguish such entangled textures of high frequency, one needs a mode decomposition algorithm that is able to be highly localized in frequency. Second-order PDEs are poorly localized in the frequency domain [

Nonlinear PDEs have been widely applied to detect images with noises. However, despite better image edge protection, the nonlinear anisotropic diffusion operator may still break down when the gradient generated by noise is comparable to image edges and features [

Similar statistical analysis can be employed to perform selective texture extraction for images containing highly entangled and overlapping textures. In the present approach, we first compute the local variation of each pixel of the image mode functions obtained by the high-order PDE transform. Unlike the total variation, the local variation is still a function, of which the variance can be calculated:

Algorithm of adaptive PDE transform for entangled texture separation.

Figure

In this section, the adaptive PDE transform is applied to three different cases to illustrate its superior capability of selective texture separation. The three images feature different types of entangled textures. Figure

Extraction and separation of texts, background watermark, and textures of (a). Shown in the 3(b) and 3(c) are the image mode function and extracted texture using the proposed adaptive PDE transform.

The adaptive PDE transform method employing the variance of the local variation of the image mode functions is applied to several benchmark test cases. In particular, separation of text and texture can be regarded as a generalized type of texture analysis. In Figure

The present algorithm of selective texture extraction is also tested on one of the most widely used images, the Barbara, in Figure

Adaptive PDE transform for selective texture extraction in the Barbara image. The variance of the local variation is shown in the top chart.

PDE transform is applied on (a) to extract edges of all textures into 5(b). Adaptive PDE transform is then applied to extract different textures from 5(b). In 5(c)–5(f), all the textures are superimposed on the original image for better viewing.

Original image

Image mode function

Texture 1

Texture 2

Texture 3

Texture 4

Sniper detection by using adaptive PDE transform method. Textures 1, 2, and 3 are, respectively, from the forest, the tree trunk, and the sniper.

Original image

Texture 1

Texture 2

Texture 3

Neuron image classification by using the adaptive PDE transform.

Original neuron image

Class 1 of the selective neuron skeleton

Class 2 of the selective neuron skeleton

Class 3 of the selective neuron skeleton

In Figure

In the previous introduction to the adaptive PDE transform algorithm and applications, local variation is defined and calculated for the intensity of image mode functions to selectively extract textures beyond the total texture extraction. The selective texture extraction can be generalized to indicate any spatial parts of the image characterized with specific (and usually functionally important) spatial orientation and/or frequency oscillation, such as different parts in the neuron synapses, brain cells, and retina vasculatures. In Figure

Classification of natural neuron skeletons.

Neuron skeleton class | Physical meaning | Percentage of the total neuron surface area |
---|---|---|

Class 1 shown in Figure | Soma (neuron cell body) | 22% |

Class 2 shown in Figure | Major (root of) dendrite | 24% |

Class 3 shown in Figure | Fine (tips of) dendrite | 54% |

Selective extraction and separation of image textures involving spatial entanglement, gray-scale mixing, and high-frequency overlapping are challenging tasks in image analysis. In this work, we introduce an appropriate adaptation to our earlier partial differential equation (PDE) transform [

This work was supported in part by NSF Grants CCF-0936830 and DMS-1043034; NIH Grant GM-090208; MSU Competitive Discretionary Funding Program Grant 91-4600.