Down sampling and up sampling - Hybrid-Order Texture Boundary Detection

Chapter 3 Hybrid-Order Texture Boundary Detection

3.3 Down sampling and up sampling

After rectifying 2^nd –order features of different orientations have been extracted (as we have mentioned in 3.1.2.2), and the output of each channel has the same size to input images (in our experiments each texture pattern has 640×640 pixels). The amount of features is proportional to the number of channels. With the number of channels increasing, it cause heavy computational loading in following processing, and we improve this problem by down sampling feature space (in our experiments we down sample by 3).

By choosing appropriate down sampling rates we can accelerate the following processes without losing too much accuracy. After boundaries have been detected, we will up sample before output. It will map detected boundaries to the corresponding position in original input.

This mechanism is similar to human vision, and trade-off of spatial accuracy and computational loading is a common problem in human vision system and the proposed algorithm. In fact the whole visual pathway is like serial processes of information extraction and data compression.

Without attention, human vision generally has low resolution in the field of vision, and even with attention we only have high resolution in a relatively tiny proportion of the field of vision. Although in this thesis we only consider the Preattentive situation, we still have acceptable spatial-accuracy for boundary detection which can be observed after local peak detection.

Chapter 4 Experimental Results and Discussion

In this chapter we will apply our algorithm to images which consist of a number of different test pattern. Most of them are synthesized by textures from “Brodatz texture database” which is derived from the Brodatz album, and it also has become a standard for evaluating texture algorithms. It has a relatively large number of classes (112 classes), and a small number of examples for each class. Each texture pattern we used here are 640*640 pixel 8-bit gray-scale images respectively. When computing the texture features for pixels near the image boundary, we assume that the image is extended by its mirror image—often referred to as the even reflection boundary condition. In section 4-1, we first introduce parameter selection. In section 4-6, there are some important properties are introduced. In section 4-3, we will have a widely test on synthesizing different textures by the proposed algorithm. In section 4-4, the accuracy of the proposed method is discussed.

4.1 Parameters Selection

There are some parameters need to be selected:

1. The number of Gabor filters and

(

^U^,V^,^σ

)

of them which decide the shape and orientation of Gabor filters in frequency domain.

Gabor filtering is computation intensive, and increasing the number of Gabor filters will increase computation loading dramatically. On the other hand, unnecessary and useless feature extracted by wrong-designed Gabor filters may cause wrong boundaries.

2. σ_g of the post Gaussian filter, which decides the smoothing level.

Increasing σ can eliminate more noise, but the accuracy of the boundary may decrease. Because both Gabor filters and Gaussian filters have spatial information, the values of σ_g must cooperate with σ to obtain a better result.

Designing parameter above is an important but sophisticated problem. Designing center frequencies of Gabor filters is most discussed in filter-design approaches. They are including the unsupervised methods such as algorithm proposed by Jain and supervised methods such as algorithm proposed by Dunn [27]. Algorithm in this thesis belongs to unsupervised method which means that all information of input pattern is unknown. Nevertheless, the emphasis of this paper is not on optimizing the design of Gabor filter, but rather proposing a simple algorithm modeling early vision and being able to be implemented on CNN. Parameters in Table 4.1 are empirically chosen and they are all the same in the following experiments without indicating specifically

Parameters Value

Pattern size(Brodatz texture) 640*640 pixels

Orientation φ 0°,45°, 90°, 135°

Center frequency ^F 1/32, 1/16, 3/32, 1/8 cycles/pixel

σ of Gabor filter 16 pixels

Down sampling rate M 3

σg of post Gaussian filter 25 pixels

Mask sizes of Gabor and Gaussian 3σ , 3σ_g

Table 4.1 parameters of experiments in Chap. 4

(e)

Fig. 4-5 (c) coarse boundaries (d) (c) after boundary detection (e) superposition of (d) and input

From Fig. 4-1 to 4-4 we can find that there are always some boundaries remain undetected, and we can’t find all boundaries by a single band in this example. Fig. 4-5 shows the result of boundary detection by four bands simultaneously, and it can be found that all boundaries are detected.

4.2.2 Effects of hybrid-order features

In this experiment we will demonstrate the reason why we should consider 1^st –order and 2^nd-order features simultaneously. Fig. 4-6(a), (b), (c), (d) demonstrate the coarse boundaries and boundaries after peak detection by 1^st –order and 2^nd –order features respectively. In Fig. 4-6(a), (b) it only detects the boundaries in lower part which is obvious different in 1^st -order features, and in Fig. 4-6(c), (d) it only detects 2^nd –order boundaries. In these four images it is found that it is insufficient to detect all boundaries by a single order feature. In fig. 4-6(e), (f) hybrid-order features are considered simultaneous and all boundaries are detected.

Input(Brodatz texture: D20D110D109D76D18)

(a) (b)

(e) (f)

(g)

Fig. 4-6 (a)coarse boundaries detected by 1^st-order features (b)(a)after detection (c)coarse boundaries detected by 2^nd-order features (d)(c)after detection (e)coarse boundaries detected by

hybrid-order features (f) (f)after detection (g)superposition of (f) and input

There is still one thing that we can observe in this experiment. As fig. 4-6 shows, the boundaries detected by 1^st –order features are thinner than those detected by 2^nd –order. The reason of this property is that 2^nd –order feature is extracted by Gabor filter before Gaussian convolution, and this process would blur the boundaries we detected at last. The second order feature processed by Gabor and Gaussian filters can be regarded as being blurred twice, so 2^nd –order boundaries would be thicker. This property will also appear in Chap. 4.3 where we will demonstrate more experimental results.

4.2.3 Saturation effects

Fig. 4-7 is am example illustrate the effect of saturation. Fig. 4-7 (a) is the coarse boundary detected after Gaussian filter before thresholding without taking natural log transformation, and 4-7 (b) is 4-7 (a) after natural log transformation. Fig. 4-7 (c), (d)

are (a), (b) after thresholding respectively. Comparing Fig. 4-7 (a) and (b), we can notice that contrasts of strong difference (response) in (a) are compressed and contrasts of weak difference are raised after natural log transform. In Fig. 4-7 (b) there are strong and weak interdistances between different textures and intratdistances reflecting the nonuniform property of the right-up texture (Brodatz texture: D62).

These strong responses may raise the threshold (mean of responses) and eliminate weak ones which we should keep.

Fig. 4-7 (e) is the 2D version of Fig. 4-7 (c) and from both of them we can find that weak boundary between D6 and D109 are eliminated. On the other hand, all boundaries are kept in Fig. 4-7 (f).Fig.4-7 (g) and (h) shows the result of Fig.3-9(f) after local maximum detection, and the output is approximately consistent to our visual perception.

Input(Brodatz texture:D109D6D62D34)

(a) (b)

(e) (f)

(g) (h)

Fig. 4-7 (a) coarse boundaries without log transform; (b)(a) after log transform; (c)(a)after thresholding; (d)(b) after thresholding; (e)2D version of (c); (f) 2D version of (d); (g)(f) after

peak detection; (h)superposition of input and (h)

4.3 Collection of Testing Results by Hybrid-Order Boundary Detection

In this section the proposed algorithm is tested by a variety of textures randomly chosen from “Brodatz texture database”. For saving the space, we synthesize five textures in each image. There would be eight boundaries between two attached textures. We will show coarse boundaries, boundaries after peak detection, and superposition of boundaries and tested image in order. There are totally 57 testing results in the follow, and all parameters we use here are the same as we have mentioned in section 4.1.

In this section we classify the results of our experimental results into three categories roughly. In section 4.3.1 we collect the results which all boundaries are detected. In section 4.3.2 we collect the results which there are some boundaries missing. In section 4.3.3 we collect the results which have the poorest results.

4.3.1 Results which all Boundaries are Detected

In this section we focus on uniform texture which consists of uniform gray value

or similar texture elements. Natural textures usually have some nonuniform part, but textures we use in this section are uniform in most part. In this section, all boundaries between different textures are detected and small edges within single texture are also detected. These results are consistent to our visual perception.

4.3.2 Results which some Boundaries are not Detected

In this section we demonstrate results that some boundaries aren’t detected.

Some of textures in this section are not so uniform such that boundaries within single

texture are more obvious than boundaries between different textures. In some cases, two textures have similar features, and the boundaries can’t be detected as we can’t distinguish them at our first sight.

4.3.3 Worst Results

In this section we synthesize all textures which are not in our definition of uniform texture, and the results are poorest among all test images. Although it can’t detect boundaries as testing images in 4.3.1 and 4.3.2, the results still can reflect some meaningful boundaries. In fact there are not obvious boundaries between two different textures if we see by our eyes. From the image of coarse boundaries and input we can find that this algorithm detected the most obvious edges in these nonuniform textures, and this result is still consistent to our perceptional experience.

4.4 Discussion of Accuracy

In this section the accuracy of the proposed method is discussed. The way we estimate the error is as fellow:

1. Only the case that synthesizes two texture patterns in Brodatz texture is considered. In the algorithm every boundaries are proposed independently. It is also hard to judge accuracy if considering multi-boundaries simultaneously when some boundaries are detected and some are not.

2. The distance between the answer and the result detected by the algorithm is measured in the condition of boundary being detectable.

We define the error by dividing measured distance into the number of total pixels.

example:

(a) (b) (c)

Fig. 4-8 an example of error estimation (a) input; (b)answer(middle line); (c) output;

3. For simplicity, 70 textures of Brodatz textures (112) which are generally consistent to our definition of textures are picked. Each test image is synthesized by choosing two textures from the 70 textures randomly, and there should be about 2500 combinations. Here we test 500 combinations, and it is believed that the result can reflect the same property as testing all combinations.

Fig. 4-9 is a histogram of error estimation in our experiment, and the results with error less than 5% is account for 85% for test images. The mean of error less than 5%

is 0.76%, and it reflects the accuracy of the condition that boundaries between different textures can be detected.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

statistics of test images(synthesize two textures randomly)

error

number of test images

Fig. 4-9 histogram of error estimation

There is one thing we should notice is that images with big estimation errors are reasonable to human perception, and from Fig. 4-10 to Fog.4-13 are some of them. In these examples the boundaries between different textures (middle line) exist but are weaker than local boundaries caused by non-uniform regions. For simplicity only the biggest peaks are kept during error estimation, so the boundaries in the middle are not kept in output images. Although in these examples the outputs are consistent to human visual perception, errors measured by the way as we mentioned above are big. It is hard to define a generally “correct answer” for all test images, and the way we measure error is not suited for the test images similar to these examples. For this reason, we don’t measure the error for the input images synthesized by the rest 42 textures in Brodatz textures.

(a) (b)

Fig. 4-10 an example of test image with big estimation errors(D50D32); (a)input; (b) output;

(a) (b)

Fig. 4-11 an example of test image with big estimation errors(D105D83); (a)input; (b)output;

(a) (b)

Fig. 4-12 an example of test image with big estimation errors(D17D70); (a)input; (b) output;

(a) (b)

Fig. 4-13 an example of test image with big estimation errors(D8D16); (a)input; (b) output;

Chapter 5 Conclusions and Future Works

In this thesis, a simple framework for hybrid-order boundary detection is proposed. It mimics mechanism of early stage of human vision, and experimental results are generally consistent to human visual sensation. After post processing, the detected boundaries also have adequate accuracy for other image processing applications such as stereo, and pattern recognition. By implementing the proposed algorithm on Cellular Neural Networks (CNN), the computational time will greatly decrease. The real-time processing capability is critical in some applications such as tracking.

Although the proposed algorithm is widely tested to detect boundaries of synthetic textures successfully, there are still some problems demanding to be overcome

1. Just as other algorithms for textures analysis with Gabor filters, there are too many parameters need to be determined. Determining parameters will much more complex when the synthesized texture patterns increase, and there is still not a simple and efficient method to solve this problem until now.

Besides parameters of 2^nd-order features, the weighting of combining 1^st and 2^nd–order features is also needed to be concerned.

In addition, we use mean of difference as threshold in this thesis, and thin the coarse boundaries by peak detection. Peak detection has good performance in our experiments, but the number of peaks we should keep will be a problem when the input image is composed by complex objects. An adaptive threshold for local peaks may be more robust.

Determining parameters above automatically or even optimized would improve the results and may be the next step of this algorithm.

2. Because we want to keep the structure simple and combine hybrid-order features easily without clustering methods, all Gabor filters in this algorithm have the same resolution.

Multiresolution-based approaches which can control the trade-off between the spectral information and spatial structure have been widely used in the field of texture analysis. It is believed that the proposed algorithm can be extended to multi-resolution such that it will have better performance.

3. In this algorithm we only consider the 1^st and 2^nd-order features, and there are still some higher-order features that can be utilized. Color is one of them, and the proposed algorithm can be extended to color textures by integrating color information.

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在文檔中早期視覺下之生物啟發式混合質感邊界偵測模型 (頁 44-0)