Simona Lombardo

Thibault Behaghel

Bertrand Chambion

Wilfried Jahn

Emmanuel Hugot

Eduard Muslimov

Melanie Roulet

Marc Ferrari

Christophe Gaschet

Stéphane Caplet

David Henry

Curved detectors developments and characterization: application to astronomical instruments

Keywords: CMOS, curved detector, characterization, noise properties, dark current

come

INTRODUCTION

Many astronomical optical systems have the disadvantage of generating curved focal planes requiring flattening optical elements to project the corrected image on flat detectors. The use of these designs in combination with a classical flat sensor implies an overall degradation of throughput and system performances to obtain the proper corrected image. This is extremely harmful in case the science goal requires an instrumental PSF as compact as possible e.g. to measure the ultra-low surface brightness universe. One example for this is the space mission MESSIER which aims at measuring surface brightness levels as low as 35 mag arcsec -2 in the optical (350-1000 nm) and 38 mag arcsec -2 in the UV (200 nm). Any refractive surface must also be excluded in this design as they would generate Cherenkov emission due to the relativistic particles. This automatically eliminates the possibility of using optics to flatten curved focal plane.

With the recent development of curved detectors, 1-5 this is not an issue anymore. The introduction of curved detectors has two additional advantages: it allows to have a reduction of the size of the imaging systems, consequently making their design more compact and simpler (particularly important for space missions), and, at the same time, it improves their overall performances (e.g. higher resolution). This new technology has been gathering more and more attention from a very broad community, as the potential applications are multiple: from low-cost commercial to high impact scientific systems, to mass-market and on board cameras,defense and security, 6 and astronomical community.

A version of MESSIER with reduced dimensions is planned as its ground-based demonstrator. 7 This fully reflective Schmidt design will be installed in Tenerife and it will potentially be the first astronomical telescope simona.lombardo@lam.fr that uses a curved detector -with convex shape and a radius of curvature of 800 mm -probing the full capabilities of this groundbreaking technology.

In this paper, we describe the first concave curved CMOS detector developed within a collaboration between CNRS-LAM and CEA-LETI. This fully-functional detector of 20 Mpix (CMOSIS, CMV20000 8 ) has been curved down to a radius of 150 mm over a size of 24x32 mm 2 and it is sensitive to the visible light. Specific care was taken to keep the packaging identical to the original one before curving, in such a way that the final product is a plug-and-play commercial component.

We present here the methodology adopted for its characterization (Section 3) and describe in detail all the results obtained (Section 4). We also discuss the main components of noise, such as the readout noise, the fixed pattern noise and the dark current. Finally we provide a comparison with the flat version of the same sensor in order to establish the impact of the curving process on the main characteristics of the sensor.

CURVING PROCESS

CEA-LETI in collaboration with CNRS-LAM is in a phase of prototyping several curved detectors concepts. Such detectors have already shown promising results and demonstrated some of the improvements achievable in term of compactness and performances of the related optical designs. 9

The off-the-shelf initial flat sensor consists of a silicon die glued on a ceramic package. Wire bonding from the die to the package surface provides the electrical connections. Finally a glass window protects the sensor surface from mechanical or environment solicitations (figure 1A). The curving process of these sensors consists of two steps: firstly the sensors are thinned with a grinding equipment to increase their mechanical flexibility, then they are glued onto a curved substrate. The required shape of the CMOS is, hence, due to the shape of the substrate. The sensors are then wire bonded keeping the packaging identical to the original one before curving. The final product is, therefore, a plug-and-play commercial component ready to be used or tested (figure 1B).

i i i i i sum n n r n n Petzval 1 1 (1)
e index of the incident and refractive medium, r is the radius of curvature of the considered rrection of this aberration is done by the use of divergent elements called "field flatteners", iazzi Smyth in 1875 [START_REF] Guenter | Highly curved image sensors: a practical approach for improved optical performance[END_REF], which minimize drastically the Petzval sum to coincide with a flat rest was the discovery made by Thomas Sutton in 1860, with the monocentric system, or this system, all optical surfaces are spherical and share the same center of curvature, and with aberrations as the coma and astigmatism are cancelled out. This is an aberration free design and n starting point (section 3). And the third technology with an interest for sensor's curvature was hich provide wide field of view, limited aberrations with a curved detector [START_REF] Tekaya | Mechanical behavior of flexible silicon devices curved in spherical configurations[END_REF]. For all these urved sensors was possible but limited. In fact, the solid photographic plate used in the 19th te for a commercial purpose. And with the patent of Hannibal Goodwin in 1887 [START_REF] Dumas | Curved focal plane detector array for wide field cameras[END_REF], flexible been used. Since the launch of silicon industry, image sensors has been limited to flat and rigid. pments show a new interest for curved sensors. Firstly, there are several developments on the nsors. For the monolithic approach, the fabrication has been demonstrated on uncooled and , for very compact high-performance camera [START_REF]Andanta, curved high resolution CCD sensors -overview[END_REF], Charge Couple Device (CCD) [START_REF] Tekaya | Hemispherical curved monolithic cooled and uncooled infrared focal plane arrays for compact cameras[END_REF], or xide Semiconductor (CMOS) [START_REF] Muslimov | A fast, wide-field and distortion-free telescope with curved detectors for surveys at ultra-low surface brightness[END_REF][START_REF]CMV20000, Product details[END_REF]. None are in industrial production. Secondly, there are also ptical improvements of optical systems with curved sensors, on optical advantages, for instance astronomical applications [START_REF] Janesick | Photon Transfer DN to λ[END_REF], and also on the simplification of optical systems by reducing er of lenses [START_REF] Howell | Handbook of CCD Astronomy[END_REF]. A study on telescope combining free-form optics and curved detectors showed and f number [12].

and process

on the commercial product 1/1.8'' format 1.3Mpx global shutter CMOS sensor (Teledyne andard sensor structure basically consists of a silicon die (7.74 x 8.12 mm), glued on a ceramic ections are wire bonded from the die to the package surface and then, to the interconnection w process. On top, a glass cover is placed to prevent from mechanical or environment re is shown in figure 1.

mercial Teledyne EV76C560 flat sensor. b) Associated schematic view along A-A' axis.

t a spherically shaped CMOS sensor, changes are required to adapt this standard structure. First, ith a grinding equipment to a targeted thickness in order to make the sensor mechanically wed by a Chemical Mechanical Polishing (CMP). In this work the targeted thickness is 100µm. s of the CMOS sensor will be detailed in section 2.4. Then, that thinned chip is glued (structural d substrate. The CMOS final shape is drove by the substrate. Wire bonding process, developed s, is also optimized in order to prevent damages on thinned dies. Figure 2 shows a schematic of ed into its commercial packaging. The radius of curvature is R=65mm.

Associated mechanical modeling

Mechanical considerations of the curved sensors are an important part to develop optical system design and find curving limitation for the sensor. For an analytical analysis, the shell theory has been used to model the mechanical behavior of the semiconductor die. This analytical model is presented in [14]. The mechanical behavior has been also modeled, using ANSYS ® Finite Element Analysis (FEA) software. For this study, extensive simulation were conducted on 7.74 x 8.12 mm (100)-oriented silicon chip with specific quadrangle elements are used, defined by 4 nodes, 6 degrees of freedom each (SHELL 181 elements). Using these elements, the principle stress values on the bottom, middle and top surfaces of the plate are obtained. Based on a breakage ball on ring study and curved packaging work [START_REF]CMV20000, Product details[END_REF]15] on silicon square plates with different grinding, polishing, and sawing processes, two tensile failure strength criterions has been defined: S1max1 = 200MPa, which refers to a standard grinding and sawing, and S1max2= 500MPa, associated to CMP process and standard sawing.

The parameters are the thickness t of the plate with a range value from 25 to 125 µm and its radius of curvature. The die was discretized into approximately 5000 elements. For silicon mechanical data, the following reduced stiffness coefficient matrix is used, with only three independent elastic coefficients: With this two failure criterions methodology, three areas are defined: the safe area without any breakage risk with a standard grinding and sawing (S1< S1max1), a forbidden area where the intrinsic silicon mechanical limits are over (S1> S1max2), and a in between zone for processes optimizations. With our targeted thickness of 100 µm, and considering our thinning process, we expect a minimum allowable radius of curvature Rc min=32 mm before silicon breakage. In this paper we show the electro-optical characterization results of one of our prototype. This chip (Figure 2) is a CMV20000 global shutter CMOS image sensor from CMOSIS, with 5120×3840 pixels of 6.4 µm size. It has been curved with a spherical concave shape down to a radius of 150 mm.

CHARACTERIZATION METHODOLOGY OF CURVED CMOS

In order to know how the curving process impacts the noise properties and the performances of the detector itself, a set of characterization measurements is performed on the curved detector and on its equivalent flat version and their results are compared. Typical measured quantities are: 10, 11 detector gain, dark current, readout noise, fixed pattern noise, dynamic range and full well capacity.

The dark current is due to the thermal agitation of the electrons within the semiconductor and it is extremely sensitive to the temperature. This agitation can free the electrons which are, hence, collected in the potential wells of the pixels, becoming indistinguishable from the charges due to a direct illumination. It is usually obtained by reading out the detector at different exposure times in complete darkness conditions. As the averaged signal linearly grows as function of exposure time, the dark current is estimated by fitting these measurements. The value of the gain (which determines how the amount of charge collected in each pixel will be assigned to a digital number in the image) is instead obtain by exposing the detector to uniform and constant illumination, at different exposure times. Again the average signal, registered on the detector at each exposure time, grows linearly as function of exposure time until it reaches saturation. More details regarding the methodologies adopted to measure all the mentioned characterization values are provided in the following subsections.

Data acquisition

The following measurements were performed:

• For each exposure time, 30 frames were acquired.

• Firstly exposures to uniform light -also called flat fields -have been performed, and then exposures in complete darkness.

• The exposure time has been changed with values ranging from 0.0002 s to saturation, for flat field exposures, and to 0.96 s for the dark exposures.

• The concave (R c =150 mm) and a flat CMOS have been tested using the same set of measurements.

• The measurements were performed with camera gain set to 1.

The communication between sensor and computer was made through the CMV20000 evaluation kit. In order to achieve a uniform illumination of the sensors, we used an integrating sphere and placed it at 1.07 m distance from the sensor. The entrance port of the sphere was illuminated by a tungsten bulb placed inside another smaller integrating sphere. A series of light baffles were located all around the light path to reduce the scattered light. In order to avoid systematics due to the drifting of the light level or small (< 0.1 o C) temperature fluctuations the exposure time was sampled almost randomly, by alternating measurement sets with shorter and longer exposure time.

The measurements were performed at a room temperature of 21.9±0.1 o C for the concave sensor and 21.0±0.1 o C for the flat one. The CMOS temperature was monitored by the internal temperature sensor (calibrated for each sensor with thermocouples) and, during the full characterization, resulted to be 35.1±0.2 o C for the concave sample and 34.9 ± 0.2 o C for the flat sensor (the errors represent a variation of temperature during the measurements and not the error on the absolute value).

Methodology used for dark current and readout noise measurements

After acquiring the measurements, a series of data processing was required to obtain the desired characterization values. As previously mentioned for each exposure time a set of 30 dark exposure frames was acquired and a median image was obtained. By averaging further over the pixels, a mean signal level was obtained. By fitting the linear trend of the average signal as function of exposure time one obtains the dark current and the bias level (which is a positive offset set up by the electronics to avoid recording negative numbers).

An important noise component is the temporal noise, which is the variation in time of the output value of the pixels exposed to a constant illumination. It is typically composed of the shot noise, due to the dark current or light exposure, and of the noise generated when reading out the pixels (readout noise, RON). The temporal noise is obtained by computing the standard deviation of each pixel in the 30 frames acquired for each exposure time: an image is created, where each pixel is a standard deviation of the mean value of the pixel in the 30 frames. The temporal noise (for a specific exposure time) is, hence, the mean of these standard deviations.

From the temporal noise of the shortest exposure time dataset acquired in complete darkness, we measured the RON.

Methodology used for gain and PRNU measurements

The gain and the pixel response non-uniformity (PRNU) are measured from flat field images. As before 30 frames were acquired for each exposure time (sampling the exposure time randomly). From these, a median image was obtained for each exposure time. Then an average over all pixels was computed, thus providing a mean signal level. The exposure times were sampled alternating short and long ones (as before) and they ranged between 2.4×10 -4 s and 0.48 s, to allow the sensor to reach saturation.

Additionally, we estimated the temporal noise as in Section 3.1.1, but this time using the flat field exposures. The square of the temporal noise, σ temp , can be written as: 10

σ 2 temp = const + k(S mean -S offset ), ( 1
)
where k is the gain in units of DN/e -and S mean -S offset are the mean signal on the median image and the bias level (computed from the dark exposures) respectively. Equation 1shows the linear relation between the mean signal and the temporal noise, which leads to the estimation of the gain, k, by making acquisitions at different the exposure times.

Typically the noise of an image contains also the PRNU. This is due to slightly different responses to incoming light, among the pixels of a sensor, and it is directly proportional to the number of electrons detected through the PRNU factor, f PRNU . The total noise of an image can be written as: 11

σ tot = σ 2 e + σ 2 RON + σ 2 PRNU ( 2
)
where σ e is the shot noise, σ RON is the readout noise, and σ PRNU is the PRNU noise. As the PRNU does not depend on time, by subtracting one frame to another with equal exposure time, we eliminated its contribution from the noise equation, and we were able to estimate the shot noise. From Equation 2we obtained σ PRNU and estimated f PRNU , which is usually expressed as a percentage of the mean signal.

RESULTS

In Section 3 we described the acquisition methodology and the data processing applied to obtain the fundamental values for characterizing the sensors. In this Section we show the results obtained from the different datasets for the concave and flat samples.

Following the prescription in Section 3.1.1, we obtained the results in Figure 3. The linear increase of measured signal for exposure times larger than 0.048 s is due to the accumulation of charges caused by the dark current. By fitting these data we estimated the dark current in DN/s (the slope of the fit) and the bias level (the intercept of the fit). We obtained for the dark current a value of 80.68±0.70 DN/s for the concave sample (at 35.1 o C) and of 94.90±0.59 DN/s for the flat sensor (at 34.9 o C). The errors here are the 1σ errors on the linear fit.

For the shorter exposure times, the counts on the sensor increase and the response is not linear. As this feature is present in both sensors, we concluded that it is an intrinsic characteristic of the CMV20000. We definition of bias level -the mean value of the median frame with the shortest exposure time acquired in darkness -and therefore we used this higher value, specified in Table 1, as bias level in the following.

The dark exposures also allowed us to estimate the readout noise from the temporal noise of the dataset with the shortest exposure time, as explained in Section 3.1.1. The values obtained are in Table 1. In order to asses if the dark current and readout noises present any significant difference among the two sensors, we first evaluated their gain. Only after converting the DN in number of electrons we can have a fair comparison.

Before measuring the gain, another useful diagnostic way to evaluate the performances of CMOS sensors is the column temporal noise. This is done by plotting the standard deviation of each sensor column in the median image (from the 30 images acquired) of the dark exposure with the shortest exposure time (in this case). The column temporal noise as function of column number is shown in Figure 4 and does not present any particular difference between the two sensors. We notice a slightly larger scatter of the values for the concave sample but such values show a similar behaviour compared to the flat sensor case -they increase at the center and decrease at the two edges -and they are also overall smaller. The gain was measured as explained in Section 3.1.2, from a set of flat fields where the sensor is exposed to uniform and stable illumination. In Figure 5 the mean signals observed by the two sensors are shown vs the exposure times. The signal grows linearly for exposure times larger than 0.048 s until it reaches the saturation limit of 4095 DN (set by the Analog Digital Converter, as it has 12-bit per pixel) for both sensors. From this we estimated the full well capacity by subtracting the bias level to the saturation limit, and the dynamic range as DR = 20log(S max /RON) where S max is the saturation limit and RON is the readout noise. These values are shown in Table 1.

In Figure 6 are plotted the squared temporal noises of both detectors against the mean signal -offset. For values of mean signal -offset between 1000 DN and 3000 DN we see a linear trend as the one described by Equation 1. From the slope of the fit to the linear part, we obtained a gain of 0.200±0.002 DN/e -and 0.220±0.003 DN/e -for the concave and flat sensors respectively (again, the errors are the 1σ errors on the linear fit). This two values are apart by 10%, however we cannot conclude that this difference is due to the curving process as it might be within the manufacturing scatter. The gain quoted by the manufacturer is of 0.25 DN/e -, 8 12% larger than the gain of the flat sensor measured in this paper. By applying the gains to the dark current values measured previously we find 403.4±3.5 e -/s and 431.39±2.7 e -/s for the concave and flat sensors respectively. These two values were obtained at average temperatures (∼ 35 o C) that differed of 0.2 o C. However their temperatures can be considered the same within the errors. We find, therefore, a smaller (of 7%) value of dark current for the concave sample.

Temporal noise 2 Temporal noise 2

Flat Concave Finally, making use of Equation 2 as mentioned in Section 3.1.2 we measured the PRNU factor, f PRNU , and we found it to be 2.0% for the concave sensor and 1.2% for the flat sensor.

All the quantities mentioned above are averaged values across the full frame. In order to evaluate the noise behaviour of the two sensors, we also plotted the 2D map of the RON. This is shown in Figure 7 where we averaged the pixel values in a box of 2×2 to reduce the size of the images. We see that also the 2D map does not show any particular difference among the two. The feature in the lower right corner of the two images is an artificial effect caused by the Evaluation Bord used to control and readout the sensors. This is generated by a bad remapping for the very short exposure times. Note that this does not affect the results shown previously, as they consider averaged quantities computed on the full frame (hence on a large number of pixels).

CONCLUSIONS

We presented here the characterization results of the first curved CMOS detector prototype developed within a collaboration between CNRS-LAM and CEA-LETI. This fully-functional detector with 20 Mpix (CMOSIS CMV20000) has been curved down to a radius of 150 mm over a size of 24x32 mm 2 . After establishing a characterization methodology, we measured the main noise components for CMOS detectors and the gain. We performed these measurements twice: first on the curved sensor and then on a CMOSIS CMV20000 flat sensor.

From Table 1 we have an overview of the results and we find them to be homogeneous between the flat and curved case. By comparing the values obtained for the dark current at 35 o C, we see a decrease of 7% of dark current for the concave detector. This might be due to the curving process or, as the difference is not too significant, to intrinsic properties of the die. We also measure a smaller readout noise of 10 e -for the concave sensor with respect to the 11 e -for the flat sensor. This smaller RON generates a larger dynamic range of 66.26 dB, against the 64.74 dB of the flat sensor. Also the 2D map of the RON does not show any particular difference in the noise pattern among the two sensors.

We find no significant difference in the bias level, as both values match within the errors. We also find similar behaviour of the column temporal noise between the two sensors, where the concave sensor presented smaller values compared to the flat one. From the measurements, the gains show a discrepancy of 10% between each other, which might be due to an intrinsic characteristic that the chip already had before curving it.

The PRNU factor of the concave sensor shows an increase of 0.8% with respect to the flat sensor one. The difference between the two is not significant. However more investigations are required as it might be due to the curving process and it could explain the appearance of a strong 2D pattern for higher illumination levels.

From the overall performances tested in this paper, we conclude that the curving process does not impact the main characteristics of the detectors. As more detectors are now available for testing, soon we will be able to produce a more statistical analysis on a larger sample.

Figure 2 :

 2 Figure 2: a) Curved Teledyne EV76C560 sensor (R=65mm). b) Associated schematic view along B-B'' axis.

  c11 = 165.7GPa, c12 = 63.9GPa, c44 = 79.6 GPa [16]. The minimum allowable radius of curvature are extracted from the FEA results for both tensile failure strength criterions. Results of the minimum allowable radius of curvature are plotted on figure 3.

Figure 1 .

 1 Figure 1. Schematic view of the commercial flat CMOS (A) and curved (B). From.9

Figure 2 .

 2 Figure 2. CMV20000 8 CMOS image sensor from CMOSIS plugged on the CMOSIS evaluation board. This sensor has been curved with a spherical concave shape down to a radius of 150 mm.

Figure 3 .

 3 Figure 3. Mean dark exposure signal as function of exposure time. The black line is the fit to the data for exposure time larger than 0.048 s. Left: data for the flat sensor. Right: data for the concave sensor.

Figure 4 .

 4 Figure 4. Column temporal noise of a median dark exposure image at 0.0002 s vs column number for the flat (left) and the concave (right) sensors.

Figure 5 .

 5 Figure 5. Mean flat field signal as function of exposure time. Left: data for the flat sensor. Right: data for the concave sensor.

Figure 6 .

 6 Figure 6. Squared temporal noise of flat field frames as function of mean signal-offset (bias level). The black line is the fit to the data for mean signal-offset between 1000 DN and 3000 DN. Left: data for the flat sensor. Right: data for the concave sensor.

Figure 7 .

 7 Figure 7. 2D map of the RON for the flat sensor (left) and the concave one (right). The values are in numbers of electrons and are averaged in a 2×2 pixels box.

Table 1 .

 1 Characterization values measured for the flat and concave CMV20000, with a radius of curvature Rc =150 mm. Note that the same methodology has been applied to both sensors.

	Flat	Concave (R c =150 mm)

ACKNOWLEDGMENTS

The authors acknowledge the support of the European Research council through the H2020 -ERC-STG-2015 678777 ICARUS program. This activity was partially funded by the French Research Agency (ANR) through the LabEx FOCUS ANR-11-LABX-0013.


Jiali Feng
email: jlfeng@189.cn

Entanglement of Inner Product, Topos Induced by Opposition and Transformation of Contradiction, and Tensor Flow

Keywords: Entanglement of Inner Product, Attribute Topos, Qualitative Mapping Function of Conversion Degree. Tensor Flow, Fixation Image Operator, Intelligence Science, Noetic Science, Meta Synthetic Wisdom

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The question: "Can Machines Think?" not only involves the basic contradiction between "spirit" and "substance" in philosophy, but also a chain of secondary contradictions induced by it, such that the law of unity of opposites between an object u and its contradiction object v, the mechanism of mutual change of quality and quantity, and the rule of dialectical transformation, i.e. so-called "law of unity of opposites and dialectic transformation" have become the key fundamental problems that need to be addressed in Noetic Science, Intelligence Science and Theory of Meta Synthetic Wisdom.

"not mind matter" "spirit" "non Intelligence" Intelligence "no-biology' "biology" "non life" "life" "non sense""sense" "non think" "think' 思维"

Fig. 1. Fig. 1 basic contradiction between "matter " and "spirit" and a chain of secondary contradictions

If the human brain is defined as an organ processing information, then two basic questions would be raised:

(1) What information is processed by human brain?

(2) What processes are taken by human brain on the information? There are at least three answers for these two questions in cognition science:

(1) Symbol information and symbol operation;

(2) Neural signal information and neural or brain chemical operation;

(3) Stimuli information and feedback.

There are three school based on the three answers respectively in cognition science and artificial intelligence:

(1) Symbol School; (2) Connective or Structure School; (3) Behave or Control School. Could the three approaches be united?

There are three answers:

(1) M. Minsky: they can't be unified [START_REF] Minsky | The Society of Mind[END_REF];

(2) Prof. Y. Zhong proposed that the various existing Al approaches can all be united within the framework of mechanism approach, and that Intelligence Science can be created [START_REF] Zhong Yixin | A unified model for emotion and intelligence in machine[END_REF]. (3) Hsue-shen Tsien, the Famous Scientist of China, not only proposed Noetic Science and Meta Synthetic Wisdom, but also suggested an implementing Scheme for MSW, noted by Quantity Wisdom Image Wisdom Quality Wisdom(MSW) as shown in Fig. 2. [START_REF] Hsue-Shen Tsien | The Letter to Dai Ruwei on 25[END_REF], as shown in Fig. 2(c Based on the basic fact that the information received by human brain is, and only is the attributes of the objects, two basic hypothesis are proposed in this paper as follow Two Basic Hypothesis of Attribute Theory: (1) Human thinking can be constructed by some of processing attributes, such as, receiving, interpretation, and coding of attributes. (2) The mechanism of thinking construction and intelligent simulation can be considered as the mechanism of processing attribute. If the two hypothesis are true, then we should study the following three basic questions:

(1) What are the attributes?

(2) How is an attribute and its transformation received, deciphered, interpreted and coded by brain? (3) How can the operators of Human Brain be represented by Mathematics? In philosophy, an attribute is defined to be as follow: [START_REF] Feng | Four-key Inner Product Decomposition of Inner Product of a Constant[END_REF] Definition 1: An attribute is an expressing quality of an object when an interaction between the object and another object is happening.

As well-known that the spatial-time position of object 𝑢 z 1 (𝑡, 𝑢), is the attribute showing where 𝑢 existing, the distance 𝑑(𝑧 1 , 𝑧 2 ) between u and its contradiction v is the expressing relation distinguishing that u from v, based on the fact, the intrinsic quality 𝑞 𝑣 (𝑢) of 𝑢 distinguish from its contradicted object 𝑣 can be defined by the corresponding relation " 𝑢 ≠ 𝑣 ⟺ z 1 (𝑡, 𝑢) ≠ z 2 (𝑡, 𝑣) ⟺ d(z 1 , z 2 ) ≠ 0 ". Based on the mechanism of 𝑑(𝑧 1 , 𝑧 2 ) varies with time change △ t was controlled by law of the law of unity of opposites about the contradictory movements, such that the philosophy question: "when the 𝑑(𝑧 1 , 𝑧 2 ) vary with time change △ t, whether the intrinsic quality 𝑞 𝑣 (𝑢) of object 𝑢 keeps itself or not?" can be transform into a physical problem.

By mean of three equivalent definition of distance 𝑑(𝑧 1 , 𝑧 2 ), the Polarization Vector  𝑐 ,  𝑑 and  of Inner Product 𝑧 1 • 𝑧 2 , Entangled Vector, Entangled Circle ⨀𝑂, Entangled Coordinates and Clifford Algebra and so on for describing physical move of u and v, can be induced.

It is shown that if Fourier Orthogonal Base is necessary for the measure of Inner Product, and the requirement of value of distance 𝑑(𝑧 1 , 𝑧 2 ) is an integer is the foundation for orbit quantization. Let △ 𝑡 𝑖𝑡 = 𝑡 -𝑖 be the time change, △ 𝑧(△ 𝑡 𝑖𝑡 ) = (△ 𝑥(△ 𝑡 𝑖𝑡 ),△ 𝑦(△ 𝑡 𝑖𝑡 )) the displacement of 𝑢 and 𝑣 vary with △ 𝑡 𝑖𝑡 ,then z t (𝑢, 𝑣) = (𝑥 t (𝑢), 𝑦 𝑡 (𝑣)) = z i (𝑢, 𝑣) + △ 𝑧(△ 𝑡 𝑖𝑡 ) = (△ 𝑥(△ 𝑡 𝑖𝑡 ),△ 𝑦(△ 𝑡 𝑖𝑡 )), and a category E whose object collection is [𝑖, 𝑇) × [𝑧 𝑖 , 𝑧 𝑇 ), the arrow collection is {(△ 𝑡 𝑖𝑡 ,△ 𝑥 𝑖𝑡 ,△ 𝑦 𝑖𝑡 )} can be constructed.

The mechanism that the quality 𝑞 𝑣 (𝑢) is maintaining the same, during the quantity varies with time change in its qualitative criterion, can be abstracted as the Qualitative Mapping 𝜏(𝑥 𝑡 , [𝑥 i , 𝑥 T )) from a quantity into its corresponding quality, and the regulation of different quantity convert into different quality of new quality, can be represented by the Degree Function of Conversion 𝜂(𝑥 t ), such that a Cartesian Closed Category can be gotten by 𝜏(𝑥 𝑡 , [𝑥 i , 𝑥 T )) and 𝜂(𝑥 t ). A subobject classifier can be induced by the mechanism of a quality is changing to a simple (or non-essential) quality, so an Attribute Topos can be achieved by them.

Because a Hilbert Space ℋ can be expanded by the Family of Qualitative Mappings, such that not only the degree of conversion function can be represented by a linear combination of base, but also a Tensor Flow can be induced by a functor F from the Base of H to Base of H' with the time stream. A Fixation Image Operator, and an approach for Image Thought has been presented, some of applications in Noetic Science and Intelligent Science are discussed.

Entanglement Vectors induced by Polarization Identity of Inner

Product of Both Vectors 𝒛 𝟏 and 𝒛 𝟐 Let Z be the plan determined by vectors 𝑧 1 and 𝑧 2 , and 𝑧 1 = (𝑥 1 , 𝑦 1 ) and 𝑧 2 = (𝑥 2 , 𝑦 2 ), the distance is defined the square root of inner product 𝑧 1 ), B = B(

𝑥 1 +𝑥 2 2 , 𝑦 1 +𝑦 2 2
), ⨀A the circle which diameter equal to 𝑑 ⨀A =|𝑂𝐵|, ⨀B the circle which center is B, its radius equal to

𝑟 ⨀B = 𝑧 1 -𝑧 2 2
. It is shown in Fig. 3(a) that there exist two Intersection Points of ⨀A and of ⨀B, 𝐶(𝑐 1 , 𝑐 2 )

and 𝐷(𝑑 1 , 𝑑 2 ), their 𝑐 1 and 𝑐 2 are the horizontal coordinate of 𝐶 and thevertical coordinate of 𝐶, respectively, 𝑑 1 and 𝑑 2 of 𝐷(𝑑 1 , 𝑑 2 ) as well. Since 𝐵𝐶 ⊥ 𝑂𝐶, we have

Fig. 3. (a) Polarization Circle ⨀𝑂 of Inner Product 𝑧 1 • 𝑧 2 (b) Entangled Coordinates 𝒮(𝑧 0 (𝑥 0 , 𝑦 0 ), 𝐸, 𝐸 ′ ) { OC 2 = OB 2 -BC 2 = ( 𝑧 1 +𝑧 2 2 ) 2 -( 𝑧 1 -𝑧 2 2 ) 2 = 𝑧 1 • 𝑧 2 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 OD 2 = OB 2 -BD 2 = ( 𝑧 1 +𝑧 2 2 ) 2 -( 𝑧 1 -𝑧 2 2 ) 2 = 𝑧 1 • 𝑧 2 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 (3)
But 𝑐 1 and 𝑐 2 of 𝐶(𝑐 1 , 𝑐 2 ) are the solution of following two circles equations:

{ (𝑐 1 - 𝑥 1 +𝑥 2 2 ) 2 + (𝑐 2 - 𝑦 1 +𝑦 2 2 ) 2 = ( 𝑥 2 -𝑥 1 2 ) 2 + ( 𝑦 2 -𝑦 1 2 ) 2 (𝑐 1 - 𝑥 1 +𝑥 2 4 ) 2 + (𝑐 2 - 𝑦 1 +𝑦 2 4 ) 2 = ( 𝑥 2 -𝑥 1 4 ) 2 + ( 𝑦 2 -𝑦 1 4 ) 2 (4)
And the equation of chord 𝐶𝐷 is the follow

𝑐 1 = 2(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )-𝑐 2 (𝑦 2 +𝑦 1 ) 𝑥 1 +𝑥 2 (5)
By substituting ( 5) in (4), we get the solution of (4) as follow

{ 𝑐 1 = 2(𝑥 1 +𝑥 2 )(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )+(𝑦 1 +𝑦 2 )√(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )[(𝑥 2 -𝑥 1 ) 2 +(𝑦 2 -𝑦 1 ) 2 (𝑥 1 +𝑥 2 ) 2 +(𝑦 1 +𝑦 2 ) 2 𝑐 2 = 2(𝑥 1 +𝑥 2 )(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )-(𝑥 1 +𝑥 2 )√(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )[(𝑥 2 -𝑥 1 ) 2 +(𝑦 2 -𝑦 1 ) 2 (𝑥 1 +𝑥 2 ) 2 +(𝑦 2 +𝑦 1 ) 2 ( 6)
{ 𝑑 1 = 2(𝑥 1 +𝑥 2 )(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )-(𝑦 1 +𝑦 2 )√(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )[(𝑥 2 -𝑥 1 ) 2 +(𝑦 2 -𝑦 1 ) 2 (𝑥 1 +𝑥 2 ) 2 +(𝑦 1 +𝑦 2 ) 2 𝑑 2 = 2(𝑥 1 +𝑥 2 )(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )+(𝑥 1 +𝑥 2 )√(𝑥 1 𝑥 2 +𝑦 1 𝑦 2 )[(𝑥 2 -𝑥 1 ) 2 +(𝑦 2 -𝑦 1 ) 2 ] (𝑥 1 +𝑥 2 ) 2 +(𝑦 2 +𝑦 1 ) 2 (7)
Let ⨀𝑂 be the Circle whose center is the origin 𝑂(0,0), its radius

𝑟 ⨀O = ||, || = | 𝑐 | = √𝑐 1 2 + 𝑐 2 2 = √𝑑 1 2 + 𝑑 2 2 = | 𝑑 |, then a Polar Coordinates 𝑊(, 𝜑) can be
induced by Polarization Identity, such that the points 𝐶(𝑐 1 , 𝑐 1 ) and 𝐷(𝑑 1 , 𝑑 1 ) can be written as two vectors 𝑤 𝑐 (, 𝜑 𝑐 ) and 𝑤 𝑑 (, 𝜑 𝑑 ) satisfying the follow

{ 𝐶(𝑐 1 , 𝑐 1 ) = 𝑤 𝑐 (, 𝜑 𝑐 ) = 𝑒 𝑖𝜑 𝑐 =  cos 𝜑 𝑐 + 𝑖 sin 𝜑 𝑐 𝐷(𝑑 1 , 𝑑 1 ) = 𝑤 𝑑 (, 𝜑 𝑑 ) = 𝑒 𝑖𝜑 𝑑 =  cos 𝜑 𝑑 + 𝑖 sin 𝜑 𝑑 (8) ! " + ! $ ! " + ! $ iy iy g g ! '( g $ ' , & 2 ) g g ! '( g $ ' , & 2 )
g g ! " = (% " , ' " ) ! " = (% " , ' " ) ! " = (%" , ' " ) ! " = (% " , ' " ) ! " #" , % "

! ( g # , %#) ! ( g # , %#) ! g ! , # ! ! g ! , # ! ! ! ! ! ! g g ! g g (0,0) ! ( g # , 0) x (0,0) ! ( g # , 0) x (a) Polarization Circle ⨀" (b) Entangled Coordinates
From the view of mathematical point, a Coordinates Transformation from product 𝑍 × 𝑍 to 𝑊-Plan has been constructed by ( 6), ( 7) and ( 8), 𝑤: 𝑍 × 𝑍 → 𝑊, for any 𝑑 𝑖𝑗 (𝑧 𝑖 (𝑥 𝑖 , 𝑦 𝑖 ), 𝑧 𝑗 (𝑥 𝑗 , 𝑦 𝑗 )) ∈ 𝑍 × 𝑍, such that 𝑤(𝑑 𝑖𝑗 (𝑧 𝑖 (𝑥 𝑖 , 𝑦 𝑖 ), 𝑧 𝑗 (𝑥 𝑗 , 𝑦 𝑗 ))) = 𝑤(, 𝜑) { [START_REF] Jiali | Qualitative Mapping Orthogonal System Induced by Subdivision Transformation of Qualitative Criterion and Biomimetic Pattern Recognition[END_REF] show us that both polarization vectors 𝑤 𝑐 (, 𝜑 𝑐 ) and 𝑤 𝑑 (, 𝜑 𝑑 ) is entangled together each other by identity:

 2 = 𝑧 1 • 𝑧 2 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 𝜑 = { 𝑎𝑟𝑐𝑡𝑔 𝑐 2 𝑐 1 𝑜𝑟 𝑎𝑟𝑐𝑡𝑔 𝑑 2 𝑑 1 (9)
 2 =  𝑐 2 =  𝑑 2 among  𝑐 ,
| 2 = |𝑜E ′ | 2 + |𝑧 1 E ′ | 2 , |𝑧 2 | 2 = |𝑜E| 2 + |𝑧 2 E| 2 , addition that |𝑧 1 -𝑧 2 | 2 = |𝑧 1 | 2 + |𝑧 2 | 2 -2|𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃, then we have |𝑧 1 -𝑧 2 | 2 = |𝑜E ′ | 2 + |𝑧 1 E ′ | 2 + 𝑜E| 2 + |𝑧 2 E| 2 -2|𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 (10)
It is shown that between the pair contradicted object 𝑢 and 𝑣, there is not only a polarization relation  2 =  𝑐 2 =  𝑑 2 , but also there exist an entangled relation [START_REF] Mac | Category for The Working Mathematician[END_REF].

It is obvious to see that by using a coordinates transformation, the intersection of ⨀𝑂 and vector 𝑧 1 -𝑧 2 , note by 𝑧 0 = 𝑥 0 + 𝑖𝑦 0 , can be regarded to be the Origin point of the Entangled Coordinates 𝒮(𝑧 0 (𝑥 0 , 𝑦 0 ), 𝐸, 𝐸 ′ ), if let 𝑚 𝑧 1 and 𝑚 𝑧 2 be the mass of 𝑢 and 𝑣, respectively, and

𝑚 𝑧 0 = 𝑚 𝑧 1 𝑚 𝑧 2 𝑚 𝑧 1 +𝑚 𝑧 2
, the mass at the Electric Dipole Point 𝑧 0 , because 𝐸 and 𝐸 ′ represent Force 𝑓 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 action on 𝑚 𝑧 1 and 𝑚 𝑧 2 respectively, such that the Oscillation of EDP 𝑧 0 not only could be described by the 𝒮(𝑧 0 , 𝐸, E ′ ), and some of Entangled reaction between 𝑢 and 𝑣 can be revealed out too.

In other hands, if let 𝑧 0 be the Harmonic Oscillation with mass 𝑚 𝑧 0 , then the law of Oscillation of 𝑢 with mass 𝑚 𝑧 0 can be represented in 𝒮(𝑧 0 , 𝐸, E ′ ). In special, when the distance 𝑑(𝑧 1 , 𝑧 2 ) ≤ ℎ 𝑝𝑐 , because we have not the unit for accurate measure position of 𝑧 0 (𝑢), such that what the object 𝑢 is become an uncertain problem, but the entangled relation between 𝑢 and 𝑣 is existing stile there.

In fact, we show from Fig. 3(a) that because 𝑆 ⨀𝑂 = 𝜋 2 , we get  2 = 𝑧 From above analysis for Fig. 3, we have shown some of conceals information of Entangled Vector in bottom. One will naturally to ask: what more hidings are there the Entangled Vector? In order to get the answer, let us discuss it in detail in following Let E be the intersection point of ⨀𝑂 and vector

| 𝐸 | 2 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 = |𝑧 𝐹 ||𝑧 1 | = 𝑧 1 • 𝑧 2 (10)
Similarly, It is easy to see that because √|𝑧 𝐹 | possible are an integer number, a rational number or an irrational number, whether |𝑧 𝐹 | is commensurable by using the unit 1 𝑧 1 ? Become a key problem. For example, √|𝑧 𝐹 | = √2, because √2 can't be measured by using the unit 1 𝑧 1 , such that | 𝐸 | can't be described in precision.

| 𝐸 ′ | 2 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 = |𝑧 𝐹 ′ ||𝑧 2 | = 𝑧
In order to discuss simple, first of all, let Z-Plan and W-Plan rotate an angle 𝜃, such that the direct of vector 𝑧 1 turns into that of 𝑥 1 , such that 1 𝑧 1 = 1 𝑥 1 .

Second, assume √|𝑧 𝐹 | can be measured by 1 𝑥 1 , then there exits an unit system:

{ 1 𝑥 1 𝑚 𝑘 = 1 𝑥 1
𝑘 , 𝑘 = 0, … , 𝑁}, such that

|𝑧 𝐹 | = 𝑛 0 1 𝑥 1 0 + 𝑛 1 1 𝑥 1 1 + ⋯ . +𝑛 𝑁 1 𝑥 1 𝑁 = ∑ 𝑛 𝑘 1 𝑥 1 𝑘 𝑁 𝑘=0 = 𝑛 0 . 𝑛 1 … 𝑛 𝑁 (13) Since 1 𝑥 1 𝑘 = 1 𝑥 1
𝑘-1 × 𝑚, and

1 𝑥 1
𝑚 𝑁 is the minimum unit in this system, norm of |𝑧 𝐹 | can be written as follow:

|𝑧 𝐹 | = 𝑛 0 . 𝑛 1 … 𝑛 𝑁 = 𝑛 0 × 1 + 𝑛 1 × 𝑚 1 … 𝑛 𝑁 × 𝑚 𝑁 = 𝑛 (in 1 𝑥 1 𝑚 𝑁 = 1) (14)
In other hands, if there are s prime factor of 𝑛, 𝑝 𝑙 , 𝑙 = 1, … 𝑠, such that 𝑛 = 𝑝 1 × 𝑝 2 × … × 𝑝 𝑠 , then we get another unit system: { = 𝑡| 0 𝑥 = 𝑥 (16) Because 𝑥 is the norm of the interval [0, 𝑥], the solution of (16) enlightens us that integration (16) is just the measure for norm of the interval [0, 𝑥] using 𝑑𝑡 as the unity (or rule) of measure. In other words, 𝑑𝑡 can be considered to be the unit

1 𝑛 𝑝 𝑙 = 1 𝑛 𝑝 𝑙 , 𝑙 = 0, … , 𝑠}, such that |𝑧 𝐹 | = 𝑛 = (𝑝 0 × … × 𝑝 𝑙-1 × 𝑝̂𝑙 × 𝑝 1+1 × … × 𝑝 𝑠 )1 𝑛 𝑝 𝑙 (
1 𝑥 1 𝑘 , when 𝑘 → ∞, i.e. 𝑑𝑡 = lim 𝑘→∞ 1 𝑥 1 𝑘 = 1 𝑥 1 ∞ .
In other hands, for the interval [0, 𝑥], we have that 𝑥(𝑡) = (1 -𝑡)𝑥 0 + 𝑡𝑥 1 , let ∆𝑥 = 𝑥 -𝑥 0 , then 𝑡 = ∆𝑥 𝑥 1 -𝑥 0 , and 𝑑𝑥(𝑡) = (𝑥 1 -𝑥 0 )𝑑𝑡, so we have

𝑑𝑡 = lim ∆𝑥→0 ∆𝑥 𝑥 1 -𝑥 0 = 𝑑𝑥 𝑥 1 -𝑥 0 (17) Integral for (17) we get ∫ 𝑑𝑡 𝑡 𝐸 0 = 1 𝑥 1 -𝑥 0 ∫ 𝑑𝑥 𝑥 𝐸 𝑥 0 = 𝑥 𝑥 1 -𝑥 0 | 𝑥 0 𝑥 𝐸 = 𝑥 𝐸 -𝑥 0 𝑥 1 -𝑥 0 = 𝑥 𝐸 𝑥 1 | 𝑥 0 =0 𝑥 𝐸 (18)
Since

|𝑥 1 | = |𝑧 1 | = 1 𝑧 1 , we have the follow: 𝑑𝑡 = 𝑑𝑥 𝑥 1 -𝑥 0 | 𝑥 0 =0 𝑥 𝐸 = 𝑑𝑥 |𝑥 1 | = 𝑑𝑥 |𝑧 1 | = 𝑑𝑥 |1 𝑥 1 | (19)
It is hiding from ( 17) and ( 18) that ( 19) could be assumed to being the minimum unit of measure. + 𝑖 (-

𝑖𝜆𝑦 𝐸 ′ 2 2 ) = 𝜆 2 (𝑥 𝐸 2 + 𝑦 𝐸 ′ 2 ) = 𝜆 2 (23) 𝑈(𝑧 𝑡 ) = 𝜑(𝑧 𝑡 ) + 𝑖𝜙(𝑧 𝑡 ) = 𝜑(𝑥 0 + 𝜆𝑡) + 𝑖𝜙(𝑦 0 -𝑖𝜆𝑡) = 𝜆 2 (24)
Then we have a wave equation:

𝑑 2 𝑈 𝑑𝑡 2 - 1 𝜆 2 𝑑 2 𝑈 𝑑𝑧 2 = 0 (25)
It is shown from above discuss that object 𝑢 and 𝑣 are entangled each other by (24). If ∆𝑡 < 𝑡 𝑝𝑙 , 𝑡 𝑝𝑙 is Planck Time, the entangled relation (24) between 𝑢 and 𝑣 will be remained yet, then it could be called the quantum entanglement. An encryption and decryption algorithm based on the entangled vector  𝑐 and  𝑑 of 𝑧 1 and 𝑧 2 , has been proposed by Jiali Feng[5]. The encryption and decryption algorithm based quantum entanglement among  𝑐 , 𝑑 and 𝛾 would be discussed in other paper.

The Attribute Topos Induced by Mechanism of Mutual Change between Quality and Quantity

Because the position changes with time △ 𝑡 𝑖𝑡 , △ 𝑧(△ 𝑡 𝑖𝑡 ) = (△ 𝑥(△ 𝑡 𝑖𝑡 ),△ 𝑦(△ 𝑡 𝑖𝑡 )) induces a cone and a limit among arrows {(△ 𝑡 𝑖𝑡 ,△ 𝑥 𝑖𝑡 ),△ 𝑦 𝑖𝑡 )}, as shown in Fig5, 6 and 7, such that a finitely complete category can be achieved. Let[𝑖, 𝑇) and [𝑥 i , 𝑥 T ) be the time interval and position interval in which 𝑞 𝑣 (𝑢) maintains itself, then𝑞 𝑣 (𝑢) must be fixed, as long as 𝑥 t = 𝑥 t +△ 𝑥(△ 𝑡 𝑖𝑡 ) of 𝑢 does not over out [𝑥 i , 𝑥 T ), i.e. 𝑥 t ∈ [𝑥 i , 𝑥 T ),while the displacement△ 𝑥(△ 𝑡 𝑖𝑡 )vary with △ 𝑡 𝑖𝑡 .It is obvious that the mechanism of the conversion from quantity 𝑥 t into a quality 𝑞 𝑣 (𝑢)can be described by a Qualitative Mapping as follow

𝜏(𝑥 𝑡 , [𝑥 0 , 𝑥 T )) = { 𝑞 𝑣 (𝑢) 𝑥 𝑡 ∈ [𝑥 i , 𝑥 T ) ¬𝑞 𝑣 (𝑢) 𝑥 𝑡 ∈ [𝑡 0 , 𝑡 𝑖 ] (26)
Let 𝑥 t , 𝑥 s ∈ [𝑥 i , 𝑥 T ) be two quantities belonging to a same qualitative criterion[𝑥 i , 𝑥 T ), even they are conversing into the same quality 𝑞 𝑣 (𝑢), their degrees of conversion are different. To describe this case, a function of conversion degree 𝜂(𝑥 t , [𝑥 i , 𝑥 T )) = 𝜂(𝑥 t ) has been proposed by Attribute Theory. In other words, let [𝑖, 𝑇) × [𝑥 𝑖 , 𝑥 𝑇 ) be the product of [𝑖, 𝑇) and [𝑥 𝑖 , 𝑥 𝑇 ), [𝑦 i , 𝑦 T ) codomain of 𝜂(𝑥 t ), we get the follow

𝜂: [𝑖, 𝑇) × [𝑥 𝑖 , 𝑥 𝑇 ) → [𝜂 i , 𝜂 T ) (27)
It is obvious that there is an equivalent relation, between the function of conversion degree 𝜂:

[𝑖, 𝑇) × [𝑥 𝑖 , 𝑥 𝑇 ) → [𝜂 i , 𝜂 T ) and the 𝜂′: [𝑖, 𝑇) → [𝑥 𝑖 , 𝑥 𝑇 ) × [𝜂 i , 𝜂 T ) , this is said that in category hom([𝑖, 𝑇) × [𝑥 𝑖 , 𝑥 𝑇 ) , [𝜂 i , 𝜂 T )) ≈ hom ([𝑖, 𝑇), [𝑥 𝑖 , 𝑥 𝑇 ) × [𝜂 i , 𝜂 T )) (28
) Therefore, we get a Cartesian Closed Category induced by the func-tion𝜂(𝑡, [x i , x T ))and τ i (𝑥 𝑘 , [x i , x T )). In other words, there is an evaluation morphism induced by 𝜂(𝑡, [x 0 , x T )) and τ i (𝑥 𝑘 , [x i , x T )), ev: [𝑖, 𝑇) × [𝜂 i , 𝜂 T )

[𝑖,𝑇) → [𝜂 i , 𝜂 T ), such that ev(𝑡, τ i (𝑥 𝑘 )) = τ(𝑥 𝑘 )(𝑡) = 𝜂(𝑡, 𝑥 𝑘 ) , here the exponential morphism is τ i (x k , [x i , x T )), as shown in Fig. 5(a). If let 𝑥 j ∈ [𝑥 j , 𝑥 𝑗+𝑙 ) ⊆ [𝑥 i , 𝑥 T )be the point that quality 𝑞 𝑣 (𝑢) transform to its non-essential quality 𝑞 𝑗 (𝑢)

The mechanism of a non-essential transforms from the quality 𝑞 𝑖 (𝑢) to its sub-quality q j (u), notation as

τ 𝑖 𝑗 (𝑥) = τ j (𝑥, [𝑥 j , 𝑥 T )) -τ 𝑖 (𝑥, [𝑥 𝑖 , 𝑥 j )) = { 0 𝑗 -1 𝑖 = -1 𝑖 𝑗 𝑥 ∈ [𝑥 𝑖 , 𝑥 j ) 1 𝑗 -0 𝑖 = 1 𝑖 𝑗 𝑥 ∈ [𝑥 j , 𝑥 T ) (30)
The ( 14) show that a Harr wavelet induced by the difference of τ j (𝑥, [𝑥 j , 𝑥 T ))τ 𝑖 (𝑥, [𝑥 𝑖 , 𝑥 j )) or τ 𝑖 𝑗 (𝑥), when [𝑥 𝑖 , 𝑥 j ), [𝑥 j , 𝑥 T ) ⊆ [x i , x T ), and [𝑥 𝑖 , 𝑥 j ) ∩ [𝑥 j , 𝑥 T ) = ∅.

Let φ: [x i , x j ) ↪ [x 0 , x T ) be the monic function induced by the inclusion relation [x i , x j ) ⊆ [x 0 , x T ), the characteristic function τ φ (x t , [x i , x j )) induced by φ is just the qualitative mapping which criterion is [x i , x T ). This means that, not only a truth map true: {0} → Ω = {0,1}, but also a pullback square as shown in Figure 10 can be induced by τ φ (x t , [x i , x T )). In particular, if ψ: [x j , x T ) ↪ [x 0 , x T ) is a monic morphism induced by the relation [x j , x T ) ⊆ [x 0 , x T ), such that ψ(x k ) ∈ [x 0 , x T ), then there exists a function induced by ψϕ: [x j , x T ) ↪ [x i , x T ), for ∀x k ∈ [x j , x T ), ϕ(x k ) ∈ [x i , x T ), such that

φ ∘ ϕ(x k ) = φ(x ℓ ) = ψ(x k ).
In other word, because 1={0}, and there is the inclusion mapping 1 = {0} ⊆ {0,1}, for any subset [x j , x T ) ⊆ [x 0 , x T ) and the morphism [x j , x T ) → 1, the pullback square can be gotten, but any subset [x j , x T ) ⊆ [x 0 , x T ) is true, if and only if, τ ϕ (x k , [x j , x T )) = 1. This means thatthere is only one predication "for ∀x k ∈ [x j , x T ) ⊆ [x 0 , x T ), the true mapping [x j , x T ) → 1 → Ω", and Ω is a subobject classifier. Therefore, we get an Attribute Topos.

The Fixation Image Operator Induced by Orthogonal Expanded of Function

Let

[x i , x T ) = ⋃ [x i , x i+k ) m k=1
, we get a collection of qualitative mapping {τ j (x k , [x j , x j+1 )) , j = i, i + 1, … , i + m, } , and for ∀τ j 1 (x ℓ , [x j 1 , x j 1 +1 )) , τ j 2 (x k , [x j 2 , x j 2 +1 )) ∈ {τ j (x k , [x j , x j+1 ))} , ∃x ℓ ∈ [x j 1 , x j 1 +1 ) , for∀x k ∈ [x j 2 , x j 2 +1 ), their inner product satisfying the following

∫ τ j 1 (x ℓ , [x j 1 , x j 1 +1 )) j 2+1 j 2 τ j 2 (x k , [x j 2 , x j 2 +1 )) dx k = { 1 j 1 = j 2 0 j 1 ≠ j 2 (31)
This shows that not only the collection {τ j (x k , [x j , x j+1 ))} constructs an orthogonal base and by which a Hilbert Space ℋ can be expanded, but also any function of conversion degreef(x t , [x i , x T )) can be expanded into a linear combination of qualitative mappings as follow

f(x t , [x i , x T )) = ∑ f k (x t , [x i , x i+k )) m k=1 τ j (x k , [x i , x i+k )) (32)
In fact, because a generalized Fourier Transformation of f(x t , [x i , x T ))F(f) can be defined by the inner product of f(x t , [x i , x T )) and τ i (x ℓ , [x j , x j+w )) as following:

F(f) = (f(x t , [x i , x T )), τ i (x ℓ , [x j , x j+w ))) = { f k (x t , [x i , x i+k )) i = j 0 i ≠ j (33)
By (33) the degree of conversion function f(x t , [x i , x T )) could be homomorphically (or similarly) mapped from the function space into a vector or the point

(f 1 (x t , [x i+1 , x i+2 )), … , f m (x t , [x i+(m-1) , x i+m ))
in the Hilbert space ℋ , so (33) is called the "Fixation-Image" operator of function f(x t , [x i , x T )).

In other words, this shows that a Hilbert Space ℋ is expanded by an Orthogonal Base {τ j (x k , [x j , x j+1 ))} which is induced by the Subobject Classifier of Attribute Topos. In the next, let us discuss its application.

Let {f s } be the collection of the approximation f s of function f, F the "Fixation

Image" operator F: {f s } → ℋ, F({f s }) = N(F(f), δ(s)) the F-Image of {f s } in ℋ, and is called the neighbors of F(f). It is obvious, under the "Fixation Image" operator F, a (classification or) recognition algorithm for the function f t , i.e. τ f (F(f t ), N(F(f), δ(s)))

Þ

The axis of third sampling y|t=t3

The axis of second sampling y|t=t2 ) , , ( It is obviously that, if the function f(x) instead by a n-dimensional pattern Min n-Hilbert space ℋ, then under the (17), M can be homomorphically (or similarly) mapped as a point F(M) in n-Hilbert space ℋ, and the approximation of the cluster with the model is mapped to the nearest neighbor N(F(M), δ(s)), then a qualitative mapping τ M (M t , N(F(M), δ(s)) for recognition of pattern M can be given.

The Tensor Flow Induced by Restriction Morphism F and Image Thinking

Let two Hilbert Space ℋ and ℋ′be expanded by base {τ j (x k , [x i , x i+k ))} and

Fig. 2 .

 2 Fig. 2. (a)Quantity WisdomQuality Wisdom Mind Wisdom (by Dai Ruwei) (b) Quantity Wis-domImage Wisdom Qualitative Wisdom (Mate Synthetic Wisdom) (by Hsue-shen Tsien) (c) Quantity MethodFixation ImageQualitative Method (Mate Synthetic Method (by Jiali Feng)

𝑧 1 , 2 .

 12 𝐸(| 𝐸 |, 𝜑 𝐸 ) and 𝐸(| 𝐸 |, 𝛼) the E's W-Plan coordinate and Z-Plan coordinate, respectively, and 𝐸(| 𝐸 |, 𝜑 𝐸 ) = 𝐸(| 𝐸 |, 𝛼), then  𝐸 = √ 𝑧 1 • 𝑧 2 = √|𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃, and 𝜑 𝐸 = 𝛼. Let 𝐸 ′ be the intersection point of ⨀𝑂 and vector 𝑧 2 , its W-Plan coordinate and Z-Plan coordinate are 𝐸 ′ (| 𝐸 ′ |, 𝜑 𝐸 ′ ) and 𝐸 ′ (| 𝐸 |, 𝛼 + 𝜃) , respectively, and 𝐸 ′ (| 𝐸 ′ |, 𝜑 𝐸 ′ ) = (𝐸 ′ | 𝐸 ′ |, 𝛼 + 𝜃) , then  𝐸 ′ = √ 𝑧 1 • 𝑧 2 = √|𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃, and 𝜑 𝐸 ′ = 𝛼 + 𝜃. Let F be the projection on vector 𝑧 1 of 𝑧 2 ,𝐹(| 𝐹 |, 𝜑 𝐹 ), | 𝐹 | = |𝑧 2 |𝑐𝑜𝑠𝜃, and 𝜑 𝐹 = 𝛼. F' the projection on vector 𝑧 2 of 𝑧 1 , F ′ (| 𝐹 ′ | , 𝜃 + 𝛼), | 𝐹 ′ | = |𝑧 1 |𝑐𝑜𝑠𝜃,𝜑 𝐹 ′ = 𝜃 + 𝛼. Let G be the intersection point of ⨀𝐴 and 𝑥-axis, its W-Plan coordinate and Z-Plan coordinate are 𝐺(| 𝐺 |, 0) and 𝐺(| 𝐺 |, 𝛼 = 0) , respectively, and 𝐺(| 𝐺 |, 𝜑 𝐺 ) = 𝐺(| 𝐺 |, 𝛼), 𝜑 𝐺 = 𝛼. 𝐺 ′ be the intersection point of ⨀𝐴 and 𝑦-axis, its W-Plan coordinate and Z-Plan coordinate are 𝐺 ′ (| 𝐺 ′ |, 𝜑 𝐺 ′ = 𝜋 2 ) and 𝐺 ′ (| 𝐺 ′ |, 𝜋 2 ) , respectively, and 𝐺 ′ (| 𝐺 ′ |, 𝜑 𝐺 ′ ), 𝜑 𝐺 ′ = 𝜋 Since |𝑧 𝐹 | = |𝑧 2 |𝑐𝑜𝑠𝜃, we get the follow

= 1 )

 1 (20) Let ∆𝑥(∆𝑡) = 𝜆𝑡 then 𝑥 𝑡 = 𝑥 0 + ∆𝑥(∆𝑡) = 𝑥 0 + 𝜆𝑡, 𝜑(𝑥 𝑡 ) = ∫ 𝑥 𝑡 𝑑𝑡 ∆𝑦(∆𝑡) = -𝑖𝜆𝑡 then 𝑦 𝑡 = 𝑦 0 + ∆𝑦(∆𝑡) = 𝑦 0 -𝑖𝜆𝑡, 𝜙(𝑦 𝑡 ) = ∫ 𝑦 𝑡 𝑑𝑡 ∆𝑧(∆𝑡) = 𝜆(1 -𝑖)𝑡 and 𝑧 𝑡 = 𝑧 0 + ∆𝑧(∆𝑡) = (𝑥 0 + 𝑖𝑦 0 ) + 𝜆(1 -𝑖)𝑡 Then we get 𝑈(𝑧 𝑡 ) = 𝜑(𝑥 𝑡 ) + 𝑖𝜙(𝑦 𝑡 ) = 𝜆𝑥 𝐸 2 2

Fig. 4 .

 4 Fig. 4. (a) Category induced by arrows {(∆𝑡, ∆𝑥, ∆𝑦)} Fig.4(b) Cone of arrows 𝐹 = {(𝑎 𝑘 , 𝑏 𝑘 , 𝑐 𝑘 )}, Fig.4(c) Pullback (𝑎 𝑘 , 𝑏 𝑘 ) of position mapping (𝜑 𝑘 ,𝜑 𝑘 )

Fig. 5 . 1 of

 51 Fig. 5. (a) the Cartesian Closed Category induced by function 𝑓(𝑥 𝑡 , 𝑥 𝑘 ) and 𝜏 𝑖 (𝑥 𝑘 ) (b) Quality

O

  a1 y1Î[a1,b1] b1 The axis of first sampling y|t=t1 (a) the Top and down of sampling tj of EGR [aj,bj] (b) qualitative Criterion [a,b]=[a1,b1]´…´[am,bm] in m dimension coordinate System whose axes are {t=tj}

Fig. 6 .

 6 Fig. 6. (a) "Fixation-Image" operator of EGR. (b) Recognition Algorithm of EGR Based on Qualitative Mapping A human faces recognition based on Attribute grid computer is developed by Gangxiao Lv, as shown in Fig.7.

Fig. 7 .

 7 Fig. 7. Human Face recognition based on Qualitative Mapping

  x k , [x i , x i+k )) the restriction function of function 𝑔 ((y t , [y i , y i+k )k , [y i , y i+k )), and 𝐹: 𝑓(𝑥 𝑡 ) → 𝑔(𝑥 𝑡 ) the restriction morphism between the function 𝑓(𝑥 𝑡 ) and 𝑔(𝑥 𝑡 ). Consider that time from t to t+1, F could be dived tow partial, one is for the function f k (x t )𝐹 𝑓 (f k (x t )) → g k (y t ), and second for the bases𝐹 𝜏 (τ j (x k , [x i , x i+k ))) → τ 𝑗+1′ (y k+1 , [y i+1 , y i+k+1 )), in this case, it should be satisfying as follow𝐹 (f(x t , [x i , x T ))) = 𝐹 (∑ f k (x t ) m k=1 τ j (x k , [x i , x i+k ))) = ∑ 𝐹 𝑓 (f k (x t )) m k=1 𝐹 𝜏 (τ j (x k , [x i , x i+k ))) = ∑ g k+1 (y t+1 ) m k=1 τ 𝑗+1 ′ (y k+1 , [y i+1 , y i+k+1 )) = 𝑔 ((y t+1 , [y i+1 , y i+k+1 ))) (34)Let (𝑇 𝑘,(𝑘+1) 𝑗,(𝑗+1) ) be the the transformation tensor from τ j (x k , [x i , x i+k )) to τ 𝑗+1 ′ (y k+1 , [y i+1 , y i+k+1 )), then the base morphism 𝐹 𝜏 can be represented as follow 𝐹 𝜏 (τ j (x k , [x i , x i+k ))) = τ 𝑗+1 ′ (y k+1 , [y i+1 , y i+k+1 )) = (𝑇 𝑘,(𝑘+1) 𝑗,(𝑗+1) ) τ j (x k , [x i , x i+k )) Recognition of human face by the attribute grid computer

  ). A Quantitative Method Fixation Image Method Qualitative Method(MSW) for MSW is proposed by Attribute Theory.[START_REF] Feng | Four-key Inner Product Decomposition of Inner Product of a Constant[END_REF]

   𝑑 and 𝑑 𝑖𝑗 (𝑧 𝑖 (𝑥 𝑖 , 𝑦 𝑖 ), 𝑧 𝑗 (𝑥 𝑗 , 𝑦 𝑗 )). 𝐸(| 𝐸 |, 𝜑 𝐸 ) be the projection of 𝑧 2 on vector 𝑧 1 , | 𝐸 | = |𝑧 2 |𝑐𝑜𝑠𝜃, and 𝜑 𝐸 = 𝛼. E ′ (| 𝐹 ′ | , 𝜃 + 𝛼) the projection of 𝑧 1 on vector 𝑧 2 , | 𝐸 ′ | = |𝑧 1 |𝑐𝑜𝑠𝜃, 𝜑 𝐸 ′ = 𝜃 + 𝛼, as shown in Fig.3(b), so we have |𝑧 1

	Let

  1 • 𝑧 2 = |𝑧 1 ||𝑧 2 |𝑐𝑜𝑠𝜃 = 𝑆 ⨀𝑂 𝜋 . And Fig.3(b) show us that since |𝐸𝑧 2 ̅̅̅̅̅ | = |𝑧 1 ||𝑧 2 |𝑠𝑖𝑛𝜃,the Exterior Product 𝑧 1 ∧ 𝑧 2 can be got by𝑧 1 ∧ 𝑧 2 = |𝑧 1 ||𝑧 2 |𝑠𝑖𝑛𝜃 = |𝑧 1 ||𝐸𝑧 2 ̅̅̅̅̅ |, and the Geometric Product 𝑧 1 𝑧 2 can be defined by 𝑧 1 𝑧 2 = 𝑧 1 • 𝑧 2 + 𝑧 1 ∧ 𝑧 2 too, so a Geometric Algebra or Clifford Algebra as well.

  𝑧 2 , then we have from (12) that the norm |𝑧 𝐹 | of 𝑧 𝐹 in 1 𝑧 1 equal to the norm |𝑧 𝐹 ′ | of 𝑧 𝐹 ′ in 1 𝑧 2 .

				1 • 𝑧 2	(11)
		Then we get	|𝑧 𝐹 ||𝑧 1 | = |𝑧 𝐹 ′ ||𝑧 2 |	(12)
		We know that |	𝑧 1 |𝑧 1 | | and |	𝑧 2 |𝑧 2 | | are the unit of vector 𝑧 1 and 𝑧 2 , respectively, note by
	|	𝑧 1 |𝑧 1 | | ≜ 1 𝑧 1 and |	𝑧 2 |𝑧 2 | | ≜ 1

  15) here, symbol 𝑝̂𝑙represents the lack of the prime factor 𝑝 𝑙 , It is mean from (14) that by using prime factor 𝑝 𝑙 of 𝑛 as unit for the measurement of the norm of |𝑧 𝐹 |, its value equal to (𝑝 0 • 𝑝 1 • 𝑝 𝑙-1 • 𝑝̂𝑙 • 𝑝 1+1 • 𝑝 𝑠 ). Therefore, prime factor𝑝 𝑙 can be called an eigenvector of |𝑧 𝐹 |, and (𝑝 0 • 𝑝 1 • 𝑝 𝑙-1 • 𝑝̂𝑙 • 𝑝 1+1 • 𝑝 𝑠 ) is called the eigenvalues of the eigenvector 𝑝 𝑙 of |𝑧 𝐹 |.If √|𝑧 𝐹 | is an infinite rational number, for instant 1/3, then If √|𝑧 𝐹 | can't be measured by 1 𝑥 1 , then we need an unit system whose value of unit is continuously variable, otherwise, the norm of √|𝑧 𝐹 | can't be measured.

	We all know that	𝑥 ∫ 𝑑𝑡 0

  , note by "𝑞 𝑣 → 𝑞 𝑗 " or 𝑞 𝑣 𝑗 , [𝑥 j , 𝑥 𝑗+𝑙 ) the qualitative criterion of𝑞 𝑗 (𝑢), in which 𝑞 𝑗 (𝑢) will remain itself, as long as 𝑥 t varies with time change △ 𝑡 𝑖𝑡 but still in [𝑥 j , 𝑥 𝑗+𝑙 ), and let𝑥 T ∈ (𝑥 T -, 𝑥 T + ) be the point at that the quality 𝑞 𝑣 (𝑢)transform to the quality 𝑝 𝑢 (𝑣) of 𝑣 , the contradiction of 𝑢 , or when the distance |(𝑥 T -, 𝑥 T + )| ≤ ℎ 𝑝𝑙 , the Planck constant, a system 𝑆 (𝑢,𝑣) can be constructed by 𝑢 and 𝑣, i.e. 𝑥 t ∉ [𝑥 i , 𝑥 T ), 𝑥 t ∈ (𝑥 T -, 𝑥 T + ) ∧ |(𝑥 T -, 𝑥 T + )| ≤ ℎ 𝑝𝑙 . Then we have a Qualitative Mapping as following

		𝑆 (𝑢,𝑣) 𝑥 t ∈ [𝑥 T , 𝑥 T + ] ∧ |[𝑥 T , 𝑥 T + ]| ≤ ℎ 𝑝𝑙
		𝑝 𝑢 (𝑣)	𝑥 t = [𝑥 T -, 𝑥 T ]
	𝜏(𝑥 𝑡 , [𝑥 0 , 𝑥 T )) =	𝑞 𝑗 (𝑢)	𝑥 𝑡 ∈ [𝑥 j , 𝑥 𝑗+𝑙 )
		𝑞 𝑣 (𝑢)	𝑥 𝑡 ∈ [𝑥 i , 𝑥 T )
		{ ¬𝑞 𝑣 (𝑢)	𝑥 𝑡 ∈ [𝑡 0 , 𝑡 𝑖 ]

Acknowledgement

The authors specially thank Professor Pei-zhuang Wang for his introduction of the Factor Space Theory [5], Dr. He. Ouyang for the discussion in category and Topos and Prof. Zhongzhi Shi for his help in many time.

Let (τ j (x k , [x i , x i+k )), (𝑇 𝑘,(𝑘+1) 𝑗,(𝑗+1) ) , τ 𝑗+1 ′ (y k+1 , [y i+1 , y i+k+1 ))) be the triple of bases and tensor, since ℋ and ℋ ′ are orthogonal Hilbert space expanded by


Weina Zhou
email: wnzhou@shmtu.edu.cn

Xinwei Lin
email: xwlin@shmtu.edu.cn

Zhijing Xu
email: zjxu@shmtu.edu.cn

Xiangyang Xue
email: xyxue@fudan.edu.cn

Patch Image based LSMR Method for Moving Point Target Detection

Keywords: Point target detection, Patch image, Low rank and sparse matrix recovery, Frame difference, Local threshold

The quick and high accuracy detection of point targets is a difficult but important technique in infrared surveillance. In this work, we proposed a patch image based low-rank and sparse matrices recovery (LSMR) method to detect moving point target in infrared marine surveillance. After analyzing the relationship between detection speed and image size, we wisely use frame difference and local threshold to efficiently exclude majority non-target patches before LSMR, which greatly accelerated the detection speed. Then integrated with the powerful classification ability of LSMR, the method proposed in this paper finally gains a high accuracy in point target detection in a marine environment. The experiment results show that the proposed method could effectively enhance the system's signal-to-clutter ratio gain and background suppression factor while keeping a high detection speed compared to other similar algorithms.

Introduction

Moving point target detection is the most difficult and significant task in infrared target surveillance. Although a lot of methods have been proposed to detect point target during the past few years, their performance is still unsatisfactory due to many reasons. The main reasons resulting in the problems lie in two aspects: infrared point targets always have less texture, color and shape information, and the detection environment is commonly very harsh.

Recently, low-rank and sparse matrices recovery (LSMR) theory was proposed and proved to be much more effective than many traditional methods in a lot of fields [START_REF] Filho | Low-rank Decomposition based on Disjoint Component Analysis with Applications in Seismic Imaging[END_REF][START_REF] Guo | GoDec+: Fast and Robust Low-rank Matrix Decomposition based on Maximum Correntropy[END_REF][START_REF] Hao | Landing Cooperative Target Robust Detection via Low Rank and Sparse Matrix Decomposition[END_REF][START_REF] Yang | Shape-Constrained Sparse and Lowrank Decomposition for Auroral Substorm Detection[END_REF][START_REF] Dai | Reweighted Infrared Patch-Tensor Model with both Nonlocal and Local Priors for Single-Frame Small Target Detection[END_REF][START_REF] Gao | Infrared patch-image model for small target detection in a single image[END_REF][START_REF] Li | Infrared Small Target Detection in Compressive Domain[END_REF][START_REF] Zheng | Small Infrared Target Detection based on Harmonic and Sparse Matrix Decomposition[END_REF][START_REF] Wen | Joint Video Frame Set Division and Low-rank Decomposition for Background Subtraction[END_REF][START_REF] Yang | Structured Sparse Coding Method for Infrared Small Target Detection in Video Sequence[END_REF][START_REF] Kim | Low-rank Representation-based Object Tracking Using Multitask Feature Learning with Joint Sparsity[END_REF]. And it is also applied in target detection and tracking to further improve the performance. They also can be classified into two kinds. The first kind prefers to detect target in a single image [START_REF] Dai | Reweighted Infrared Patch-Tensor Model with both Nonlocal and Local Priors for Single-Frame Small Target Detection[END_REF][START_REF] Gao | Infrared patch-image model for small target detection in a single image[END_REF][START_REF] Li | Infrared Small Target Detection in Compressive Domain[END_REF][START_REF] Zheng | Small Infrared Target Detection based on Harmonic and Sparse Matrix Decomposition[END_REF]. It can solve the problems when imaging backgrounds change quickly due to rapid relative motion between the imaging sensor and the target. And the other kind of methods makes use of information of multi-frames to predict the target position more quickly and accurate [START_REF] Wen | Joint Video Frame Set Division and Low-rank Decomposition for Background Subtraction[END_REF][START_REF] Yang | Structured Sparse Coding Method for Infrared Small Target Detection in Video Sequence[END_REF][START_REF] Kim | Low-rank Representation-based Object Tracking Using Multitask Feature Learning with Joint Sparsity[END_REF]. However, both of their performance could degrade rapidly when the signal-to-clutter ratio (SCR) is low, or the computation is too time-consuming to be realized in real-time.

In this paper, we fully exploited the advantage of LSMR algorithm, and proposed a patch-image based LSMR method to detect moving point targets in infrared marine surveillance. Different from previously proposed LSMR based algorithms, our method takes full advantage of the information from intra and inter frames, and greatly reduces computation time while obtaining a good performance on signal-to-clutter gain (SCRG) and background suppression factor (BSF).

This paper is organized as follows. In section II, preliminary work including the LSMR algorithm and its application in small target detection will be introduced briefly. In section III, the relationship between the computation amount and image size is analyzed, thus a patch image based LSMR method is proposed and discussed in detail. In Section IV, experiments are executed to validate the performance of the proposed approach. And in final section, a conclusion and our future work is presented for further research.

2

LRMR Theory and Small Target Detection

Low-rank and Sparse Matrices Recovery (LRMR) Algorithm

As a theory stretching from compressive sensing (CS) and sparse represent algorithm, the LRMR supposes an ideal image or video as a low-rank matrix. Take image signal as an instance, its rank is usually much smaller than its size.

) , min( n m   (1)
In [START_REF] Filho | Low-rank Decomposition based on Disjoint Component Analysis with Applications in Seismic Imaging[END_REF], r is the rank of an ideal image, m, n are image's width and height respectively. In fact, the entries of the matrix are often corrupted by errors or noises, making the image or video not a low-rank matrix any more. Consequently, every image/video can then be regarded as the combination of a low-rank matrix and a noise matrix in LRMR theory, which could be represented as

S L I   (2)
I is an usual image/video, L is an ideal image/video which is low-rank, S represents random noise and errors, or other unregular signal caused by various of factors, which can be regarded as sparse matrix. It has been proved that, under surprisingly broad conditions, L and S can be exactly recovered from I via Robust Principal Component Analysis (RPCA) by solving the following convex optimization problem. λ is a positive weighting parameter, which could be utilized to fine tune L and S for optimum results.

LSMR in Point Target Detection

Generally speaking, there are three components in an infrared image (II): the background (B), the point targets (T), and various kinds of noises (N). And the infrared image can be expressed as follow:

N T B II    (4)
When detecting point target in a single image, we found out that small targets have no concrete shape and texture, and are commonly not larger than 10×10 pixels because of the long imaging distance. They can be considered as "sparse" with respect to the extensive background. However, background patches are always "low-rank" and approximately context correlated with each other, even though the pixel distance between two patches may be large. As for detecting targets in multi-frames, background in each frame is also 'low-rank' due to the consistency between adjacent frames. And target is "sparse" because of its moving characteristics.

Thus the point target detection could be regarded as a typical problem of recovering a sparse component from the infrared image in LSMR theory. This assumption has been proved valid in [START_REF] Zheng | Small Infrared Target Detection based on Harmonic and Sparse Matrix Decomposition[END_REF], under the assumption that the random noise is i.i.d. and its Frobenius norm is smaller than some σ (σ> 0). Positive weighting parameter λ plays a promoting role in enhancing the detection stability and suppressing random noise.

  F T - B - II (5)
Nevertheless, the recovery of low-rank and sparse matrix usually leads to high computational cost due to demanding convex optimization of RPCA especially with multiframes. And the method of detecting in a single frame always cannot ensure the detection accuracy.

Patch based LSMR Algorithm

How to improve the detection speed while keeping a high accuracy? A figure was drawn up to show the factor that may increase the detection time. In Fig. 1, we can see the relationship between the processing time of five randomly selected infrared marine images and their downscaled versions. Accelerated proximal gradient (APG) algorithm is used to solve the convex optimization problem in detection. From Fig. 1, we can find out that it is hard to meet the real-time requirements of most applications for detecting targets even in a single frame. However, at the same time, we also find out that the processing time is closely related to the image size. As the image shrinks, the processing time falls sharply. In that case, the processing time could be decreased when the image size is downscaled. In view of the hugeness of the computation amount in detection with all frames in video, we use image difference to extract information between frames and divide difference image into patches to further smaller the candidate region to be processed by LSMR. These methods restrict the processing time to a reasonable range. Fig. 2 is the schematic diagram of the proposed method, which can be described in detail as the following steps.

Step 1: Frame difference The frame difference is used to narrow the candidate region with little computation cost by utilizing information between adjacent frames. It could not only extract target position information from adjacent frames but also exclude the stationary disturbances for obtaining the candidate moving target regions which we are most interested in. Suppose C as the candidate area including moving point target, it can be obtained by [START_REF] Gao | Infrared patch-image model for small target detection in a single image[END_REF].

1) - II(t - II(t) C  (6)
Here, II(t)is the t th frame, II(t-1) is the frame before the I(t). | * | is the absolute operation.

Step 2: Divided into patches and processed by a local threshold After obtaining the frame difference image C, we divide the image into partially overlapped patches, which further decreases the data to be processed at a time.

Obviously, the size of the patch and the overlapping area will influence the detection capability. Since point target is usually smaller than 10×10 pixels, the vertical and horizontal steps are set as P-10 to ensure no loss of targets with minimum redundant calculation. P is the width of patches, which will be discussed in the following section. The patch is then processed by a local self-adaptive threshold to further narrow the detection region. The local self-adaptive threshold is described in [START_REF] Li | Infrared Small Target Detection in Compressive Domain[END_REF].

       others M M M y x if y x C 0 ) ( & ) ) , ( C ( 1 ) , ( min max max   (7)
Where max M and min M are respectively the maximum and minimum pixel values of the local region.  is the threshold coefficient and set as 0.6,  is the threshold for judging the existing of the target. For the targets brightness is usually relative high in a local region, no target is expected to exist in the patches whose difference result of max M and min M is smaller than  . In the paper, we choose  = 40.

Step 3: LSMR and segment Last but not least, we recover the target region T by LSMR in the left candidate region C and then use a simple segment method to extract targets. APG algorithm is used to solve the convex optimization problem of (3). It can achieve dramatically better performance among all the RPCA algorithms [START_REF] Gao | Infrared patch-image model for small target detection in a single image[END_REF]. The algorithm is explained as follows:

Algorithm: RPCA via APG Input: infrared difference image with target  , n m R C   1. . 1 0 ; ; 0 ; 1 , 0 ; 0 B 0 0 1 0 1 0 1 0                  t t T T B While not converged do 2.
). ( 1),

( 1 Y 1 1 1 1 k             k k k k k T k k k k k k B T T t t T Y B B t t B 3. . C - Y Y 2 1 - Y G T k B k B k B k ) (   4. '. ] [ US B G svd V S U 2 1 k B k k V S     ), ( ) , , ( 5. . C - Y Y 2 1 - Y G T k B k T k T k ) (   6. ]. G [ S T T k 2 1 k k    7.
). , max( ; 2

1 t 4 1 t 1 k 2 k 1 k    k      

k=k+1. End while

Output:

k k T T ; B B   4
Experimental Results and Analysis

The Performance Comparison with Different Weight of Sparse Error Term (λ) and Patch Size

To verify the capability of the proposed method, a real and representative cluttered marine surveillance video sequence with 48 frames is used to execute experiments. The resolution of the frames is 498×696, and the moving point targets in the frames occupy about 5×5 pixels. All experiments were implemented by Matlab software on a PC with 2-GB RAM and 2.60-GHz Intel-i5 processor. Average detection time (ADT), SCRG, BSF are employed for objective evaluation. SCRG, BSF are same to the metrics presented in [START_REF] Hilliard | Selection of a Clutter Rejection Algorithm for Real-time Target Detection from an Airborne Platform[END_REF]. They are defined by ( 8) and [START_REF] Wen | Joint Video Frame Set Division and Low-rank Decomposition for Background Subtraction[END_REF]. Where, S is the target amplitude and C is the clutter standard deviation within the original frame or the processed frame. The patch sizes range from 20×20 pixels to 490×490 pixels with an increasing of 10 pixels. And the average results of all frames in the above mentioned video sequence are used for comparison. Fig. 3 shows the performance comparison with different weight of sparse error term (λ), which is an important parameter in APG algorithm. The ADT has been enlarged by 5, 10, 20 times in Fig. 3(b), Fig. 3(c), Fig. 3(d) respectively for convenient observation. And it means the loss of targets when the SCRG and BSF are zero. In addition, it can be seen from Fig. 3 that, optimum detection time always exists with each determined λ, which is directly related to the patch size. Patches with too small or too large sizes always consume more time to be processed. The least detection time is obtained with patch size of 280×280 pixels in our experiment. The SCRG and BSF roughly increase with the patch size under determined λ, and increase with λ under determined patch size. Taking every aspect into consideration, we finally choose the optimum patch size as 280×280 pixels with λ= 0.3 in the proposed method. Its ADT is 3.53s. And SCRG, BSF reach 107.0 and 641.1 respectively.

in out S ) C / ( S/C SCRG ) (  (8)

Evaluation Comparison

The detection capability of the proposed method is also compared with the result of the same algorithm with non-partitioned image and IPI model [START_REF] Gao | Infrared patch-image model for small target detection in a single image[END_REF]. From Table I, we can see that, the proposed method has better performance than other two methods under the same circumstance. Compared to IPI, our method reduces the detection time by 97.4% while improving SCRG, BSF by 180.1% and 389.4% respectively. What's more, the detection speed could be further accelerated by parallelism due to independence of patches in our method.

By analyzing the results in above figure and table, we can see that: the proposed method could really improve the detection capability. And the reasons are as follows. First, the difference between adjacent frames and local threshold excluded a large amount of non-target patches before LSMR algorithm, and greatly decreased the computation load of LSMR. Since the frame difference and threshold process need much less computation than LSMR, the total computation time could be reduced tremendously. Second, the LSMR is still effective in detection as long as the point targets are "sparse" in each patch. Third, the proposed method integrates methods of frame difference, local threshold, image segment and LSMR, which takes full advantage of multi-methods to exactly distinguish targets from complex background.

Conclusion and Future Work

In this paper, we proposed a patch-image based LSMR method for moving point target detection in infrared coastal surveillance. It is the first time the patches of frames are independently processed by LSMR algorithm, and the optimum patch size is also analyzed with metrics of ADT, SCRG and BSF. Experiment results show that the propose method prompts the detection speed while keeping a high detection capability. However, detection results of the proposed method would be affected by dynamic noises which are also irregular. Thus, more research work should continually focus on suppressing noise and improving detection speed and accuracy in future. A comprehensive method which can both handle the detection velocity and accuracy parameters is essential for actual industrial applications.

  the nuclear norm of a matrix, 1 || ||  denotes the norm-1, and

Fig. 1 .

 1 Fig. 1. Processing time of different sized images

Fig. 2 .

 2 Fig. 2. Schematic diagram of the proposed method framework

Fig. 3 .

 3 Fig. 3. Proposed Performance comparison with different weight of sparse error term (λ). (a) Performance with different patch sizes (λ=0.1); (b) Performance with different patch sizes (λ=0.2) ;(c)Performance with different patch sizes(λ=0.3); (d)Performance with different patch sizes(λ=0.4).

Table 1 .

 1 Performance comparison of point target detection methods

	Detection methods	IPI	Non-partitioned im-	Proposed
			age
	SCRG	38.2	85.6	107.0
	BSF	131.0	141.5	641.1
	ADT(s)	136.67	21.29	3.53

Acknowledgments

. This research was financially supported in part by the National Natural Science Foundation of China under Grant No. 61404083 and 51579143, the China Postdoctoral Science Foundation under Grant No. 2015M581527, the Innovation Program of Shanghai Municipality Education Commission under Grant No. 15ZZ080.


Qiming Li
email: qimingli@shmtu.edu.cn

Liqing Hu

Yaping Gao

Yimin Chen

Lizhuang Ma

A Two-Step Pedestrian Detection Algorithm based on RGB-D Data

Keywords: pedestrian detection, RGB-D, down-sample, PFH, HOG, SVM

A two-step pedestrian detection algorithm based RGB-D data is presented. Firstly, down-sample the depth image and extract the key information with a voxel grid. Second, remove the ground by using the random sample consensus (RANSAC) segmentation algorithm. Third, describe pedestrian characteristics by using Point Feature Histogram (PFH) and estimate the position of pedestrian preliminarily. Finally, calculate the pedestrian characteristics based on color image by Histogram of Oriented Gradient (HOG) descriptor and detect pedestrian using Support Vector Machine (SVM) classifier. The experimental results show that the algorithm can accurately detect the pedestrian not only in the single-pedestrian scene with pose variety but also in the multi-pedestrian scene with partial occlusion between pedestrians.

Introduction

Pedestrian detection is a hot research topic in computer vision. It plays an important role in vehicle assistant driving, intelligent video surveillance, human abnormal behavior detection, and so on. Pedestrians are both rigid and flexible, and the appearance is susceptible by wearing, occlusion, scale zoom, perspective and other factors, so the pedestrian detection is still a challenging issue. Pedestrian detection technology based on RGB image has been mature, but the RGB feature is unitary. With the popularity of depth acquisition device, more and more researchers have begun to study pedestrian detection based on depth image. The classical HOG features used in color image is used to depth image by Spinello [START_REF] Spinello | People Detection in RGB-D Data[END_REF][START_REF] Luber | People Tracking in RGB-D Data With On-line Boosted Target Models[END_REF], the Histogram of Oriented Depth (HOD) descriptor is proposed. Based on the algorithm presented by Spinello, Wongun Choi et al. [START_REF] Choi | Detecting and tracking people using an rgb-d camera via multiple detector fusion[END_REF] proposed a framework of multiple detector fusion, including skin color detection, face detection, trunk detection and so on. Shengyin Wu et al. [START_REF] Wu | An attempt to pedestrian detection in depth images[END_REF] proposed the histogram of depth difference feature. Shiqi Yu et al. [START_REF] Yu | Sltp: A fast descriptor for people detection in depth images[END_REF] proposed simplified local ternary patterns features based on depth image. Lu Xia et al. [START_REF] Xia | Human detection using depth information by kinect[END_REF] detect human targets by using two-dimensional contour model and 3D facial model. Wang et al. [START_REF] Wang | A new depth descriptor for pedestrian detection in RGB-D images[END_REF] presented a pyramid depth self-similar feature extraction algorithm according to the strong similar depth information of the human local body. Ikemura et al. [START_REF] Ikemura | Real-time human detection using relational depth similarity features[END_REF] think that the frequency distribution of the depth values of different objects has a great diversity. They divide the depth map into nonoverlapping blocks and carry on histogram statistics of depth values for each block. The relational depth similarity Features of all blocks in series is used as the feature of the image. In recent years, Chen et al. [START_REF] Cheng | Semi-supervised Learning for RGB-D Object Recognition[END_REF] proposed a semi supervised learning framework. Wang et al. [START_REF] Wang | Multi-modal Unsupervised Feature Learning for RGB-D Scene Labeling[END_REF] extract the fusion features of RGB and depth information, and reduce the dimension of the feature by using sparse automatic encoding. Gupta et al. [START_REF] Gupta | Aligning 3D models to RGB-D images of cluttered scenes[END_REF] reselect exactly after primary select. Liu et al. [START_REF] Liu | Detecting and tracking people in real time with RGB-D camera[END_REF] use synthetic image to detect pedestrian. Linder et al. [START_REF] Linder | Real-time full-body human attribute classification in RGB-D using a tessellation boosting approach[END_REF] detect pedestrian by using the local multi feature.

In summary, most of the existing pedestrian detection techniques are based on RGB images, so the feature is single, while the pedestrian detection based on depth images is prone to drift. Therefore, a pedestrian detection method based on RGB-D image is proposed in this paper. The algorithm is executed in two steps. Based on the depth image detection, the initial detection results are given first, and then the final detection results are obtained based on the RGB image.

Two-Step Pedestrian Detection Algorithm

As shown in Fig. 1, firstly, the point cloud expression of the scene is constructed based on depth data, and the down sampling is performed to the point cloud by using the voxel grid method in order to reduce the data size. Then the random sampling consistency segmentation algorithm is adopted to remove the ground plane from point cloud to reduce point cloud data further. Finally the PFH and SVM are used to screen out the candidate pedestrians.

Further detection needs to combine with RGB image. At present, the HOG feature is still the most effective and widely used in the pedestrian detection method based on RGB image, so the HOG feature and the SVM classifier are used in the further accurate detection to get the final pedestrian detection results.

Down-sampling to Depth Point Cloud

The point cloud expression of the scene is constructed based on depth image. However, the data size of point cloud is relatively large. In order to improve the detection speed, the point cloud data must be down-sampled on the premise of keeping the shape of the cloud. In each frame, the spatial data is divided into a set of voxels. The coordinates of all points within each voxel are approximated to the coordinates of the voxel's center. A 3D voxel grid is created based on the point cloud data by using the point cloud library (PCL), and the center of gravity is approximately expressed by the coordinates of all points in voxel. Voxel can reduce the order of magnitude of data processing, reduce the calculation of pedestrian characteristic, and achieve the realtime performance. In addition, this operation is also conducive to constant point cloud density, makes the cloud data volume consistent in the unit space, and makes the point cloud density dependent on the distance from the sensor no longer. In this paper, the selected size of voxel is 0.08m, and the original point cloud and the filtered result are shown in Fig. 2a and Fig. 2b respectively.

Ground Segmentation

We make a hypothesis that people walk on the ground in this paper. The ground plane is estimated by several points marked in advance on the ground plane. According to the estimated ground plane, we can remove the corresponding voxel point cloud, and further reduce the irrelevant data. The RANSAC segmentation algorithm is adopted to design a model for segmentation judging, and the input data is iteratively extracted by the judgment criterion. First, set thresholds for all points of the point cloud dataset, and randomly select points from the grid point cloud data and calculate the parameters of the given judgment model. If the distance between the point and the determining model is smaller than the threshold, then the point is internal, otherwise it is external out of points. The external points will be removed. As shown in Fig. 2c, the voxels of the ground plane are removed and the volume of the target image is greatly reduced.

Preliminary Detection based on Point Cloud

By parameterized querying the difference between point and point, PFH forms a multidimensional histogram to describe the spatial geometric attributes of the K points in neighborhood. The high-dimensional hyperspace provides measurable information space for features. The six dimensional pose of a surface is invariant and robust to different sampling densities or neighborhood noises. PFH representation is based on the relationship between the point and its K neighborhood and their estimated normal, considers the interaction between the normal directions, captures the best changes of the point cloud surface, and describes the geometric features of data samples. For example, for the query point P, the radius of its affected region is r, and all its K neighborhood elements are completely connected to each other in a connected region. By computing the relation between the two connected points in the connected region, the point feature histogram is the final PFH descriptor.

After removing the ground data from the point cloud, the false positive rate strategy is adopted for preliminary pedestrian detection, and the accurate detection is performed in the next step. The advantage of this approach is that it can allow as many candidate targets as possible pass this initial screening to minimize the omission ratio. First, scan the whole image with a fixed size detection window, then describe the characteristics of pedestrian cloud by using PFH, and select the candidate pedestrians using SVM classifier at last. Kinect sensor is used to capture data in 4 different indoor scenes, including classroom, restaurant, dormitory and library. In order to make the depth information more accurate, only the data the shooting distance of which is between 3~8 meters is selected. After manually tailoring the picture, 1218 pairs of positive samples and 3000 pairs of negative samples are obtained. Among them, the negative samples are randomly extracted from pictures that contain no human body. The sample size is 64×128. Finally, extract the PFH features of positive and negative samples from the depth images and input them into SVM for training.

Accurate Detection based on RGB Image

The accurate pedestrian detection algorithm using HOG features is as following. A fixed size detection window is used, and the window is divided into a grid the unit of which is cell. The gradient direction of the pixels in each cell is computed into a onedimensional histogram. The intuitive formulation is that the local appearance and shape can be well described by the distribution of local gradients without having to know the exact locations of these gradients in cell. A set of cell is aggregated into block for local contrast normalization. The histogram of all the blocks is concatenated to form the descriptor vector of the detection window. This descriptor vector is used to train the linear SVM classifier. When detecting pedestrian, the detection window slides in different scale spaces of the image and the HOG descriptors of each position and scale are calculated. Then a trained SVM classifier is used to classify, and finally the location of the pedestrian is obtained. Compared with the depth SVM classifier trained in section 2.3, the training process of the RGB SVM classifier is basically the same, but it uses HOG features. Considering the real-time application requirement, the modified HOG [START_REF] Maji | Classification using intersection kernel support vector machines is efficient[END_REF] is adopted to reduce dimensionality. Similarly, the training set constructed in section 2.3 is also used for training the RGB SVM classifier.

Results

Based on the above idea, the related algorithms are implemented. The development environment is the Visual C++ platform, the software that drives the Kinect is Open-NI interface (Prime Sense Company), and the PCL is used for processing the 3D point clouds, deep and color images. The hardware platform is core duo CPU@2.2GHz, 4GB memory, GTX 660 graphics card and Kinect for Windows.

Experiments were carried out on 382 pairs of images containing 1201 human bodies, and the detection rate achieves an average performance of 97.75%. The experiment is divided into two types: single scene and multi person scene. The experimental results show that the pedestrian detection algorithm based on RGB-D has good detection results even in the circumstance when the pose of pedestrian is changeable in single scene. The multi person scene is divided into two cases: with occlusion and without occlusion. The algorithm has good performance in both cases. Some experimental results are shown in Fig. 3.

Conclusion and Future Work

A two-step pedestrian detection algorithm based RGB-D data is presented in this paper. Firstly, the algorithm captured the depth image and color image by using OpenNI SDK to drive Kinect camera. Then down-sampled the depth image and extracted the key information by voxel gridding method. After that, eliminated the ground by using the RANSAC segmentation algorithm and further screened the point cloud data independent with the target pedestrian. Next, described pedestrian characteristics by using PFH histograms and preliminary estimated the position of pedestrian. Finally, the pedestrian characteristics based on color image were further calculated by HOG descriptor and SVM classifier was used for pedestrian detection. The experimental results show that the algorithm can accurately detect the pedestrian not only in the single scene with pedestrian posture changing (stand, sit down, front, side and back) but also in the multi person scene with occlusion and without occlusion.

Although the algorithm has a high performance in a certain range of indoor applications, there are still some problems to be solved in the wide range scene applications. Limited by Kinect capability, the scope that can obtain the depth information is limited. When the distance between the target and the camera is more than 4m, the amount of target information will sharply reduce. So, how to detect pedestrian based on RGB-D in a large range application scene is a further research direction.

Fig. 1 .

 1 Fig. 1. Algorithm flow

Fig. 2 .

 2 Fig. 2. Point cloud grid: a. the initial point cloud; b. point cloud after grid and down-sampling; c. point cloud after ground segmentation

Fig. 3 .

 3 Fig. 3. pedestrian detection results: a. single scene(stand, front); b. single scene(stand, front); c. single scene(stand, side); d. single scene(stand, back); e. single scene(sit down, side); f. single scene(sit down, front); g. multi person scene(no occlusion); h. multi person scene(no occlusion); i. multi person scene(occlusion); j. multi person scene(occlusion).

Acknowledgement

This work is supported by Natural Science Foundation of China (Grant No.: 61472245) and Shanghai Municipal Natural Science Foundation (Grant No.: 14ZR1419700 and 13ZR1455600).


Yan Chen

Qiqi Xie

Qiulin Chen

Li Zhang

Yaodong Cui

Zuqian Meng

The Optimization Algorithm of Circle Stock Problem with Standard Usage Leftover

Keywords: standard usage leftover, recursive algorithm, sequential heuristic procedure, circle stock problem

In view of the cutting stock problem of the plate fragments and scrap cannot make full use, this paper introduces the concept of standard usage leftover, uses the recursive algorithm and the sequential heuristic procedure to solve the circle cutting problem, achieving the follow-up orders specification to use the usage leftover of the previous orders. This method can make the cutting process more simple, convenient usage leftovers inventory management, as well as conform to the requirements of the blanking for a long time. Computational experimental results show that the algorithm has a high material utilization, and playing a guiding role to the actual industrial production.

Introduce

The problem of circle stock cutting exists in many industrial processes. It is inevitable to produce leftover, which results of material waste and inventory management difficulties. [START_REF] Chen | Optimization algorithms for the two-dimensional three-staged guillotine cutting stock problem[END_REF] There are usually three models are used to solve the Cutting Stock Problem with Leftover (CSPL): minimizing bar cost; minimizing leftover and waste; minimizing the number of sheet. The model Cui [START_REF] Cui | New model and heuristic solution approach for one-dimensional cutting stock problem with usable leftovers[END_REF] proposed aims at minimizing bar cost. It discusses One-dimensional Multiple Stock Size Cutting Stock Problem (1DMSSCSP). Residual length of a cutting pattern is taken as a leftover if it is longer than a threshold; as trim loss otherwise. The model Trkman [START_REF] Trkman | One-dimensional cutting stock optimization in consecutive time periods[END_REF] proposed aims at minimizing leftover and waste. It discusses General One-dimensional Cutting Stock Problem (G1D-CSP). According to the order is cyclical or not, Trkman classifies the problem into three types: (1) all items will be produced in one production cycle; (2)all items will be produced in multiple production cycle when order and material is ensure;

(3) all items will be produced in multiple production cycle. But only the first production cycle task is ensured. The model Chen [START_REF] Chen | Optimization algorithms for the two-dimensional three-staged guillotine cutting stock problem[END_REF] proposed aims at minimizing sheet number. She pointed out two aspects: (1) Stock number is taken as the main part while usage leftover value is neglected; (2) As a part of stock, leftover has been included in production cost. In concrete solution method, Cui [START_REF] Cui | New model and heuristic solution approach for one-dimensional cutting stock problem with usable leftovers[END_REF] used integer programming(IP) and column generation algorithm to solve 1DMSSCSP. Andrade [START_REF] Andrade | Two-stage two-dimensional guillotine cutting stock problems with usable leftover[END_REF] came up with mixed integer programming to solve how to use leftover. Cherri [START_REF] Cherri | The one-dimensional cutting stock problem with usable leftover-a heuristic approach[END_REF] discussed leftover length and defined the concept of waste. Yurij [START_REF] Yurij | Φ -functions for primary 2D-objects[END_REF] proposed the concept of standard existing algorithms are restricted to 2D objects; Miyazawa [START_REF] Miyazawa | Polynomial-Time Approximation Schemes for Circle and Other Packing Problems[END_REF] presented iterative separation management which could simplified process.

In this paper, a concept of standard leftover is proposed, and Recursive Algorithm (RA) with Sequential Heuristic Procedure (SHP) is used to maximize the sum of items and leftover value.

Mathematical Model

The meaning of circle stock problem with standard usage leftover is direct cutting along the optimal line of a vertical plate on the rectangular stock. The left side is the sheet stock used for the current order and the right is the standard leftover for the subsequent order; the material for the current order only uses the sheet on the left. There are three concepts.

(1) Leftover: remained part in the cutting stock process. The leftover sheet stock won't be placed any items in this order, it will be used in the subsequent order.

(2) Standard leftover: the usage leftover generated in current order. It has the same width with stock. The standard leftover can be used in the next order. As Fig. 1 shows, the part in the lower right is standard leftover, and the cross slash area is waste. b is the demand of item type i The problem can be formulated as the following programming problem: Function ( 1) is to maximize the sum value of items. Constrains (2) constrain the number of item produced. Constrains (3) is to circulate leftover value.

In fact, the total value of stocks is the sum of items value and usage leftover value because of the introduction of standard leftover concept.

Algorithm Description (1) Generate Cutting Pattern

Standard leftover length is k l , width is k W . There will be one or multiple rows in sheets. When 0 k l  , no leftover is produced. Traversing all items for this year, making inventory management of leftover to make it usable for future. RA [START_REF] Cui | Strips minimization in two-dimensional cutting stock of circular items[END_REF] is used to solve the direct cutting problem. It will assemble a horizontal homogeneous sheet along the direction perpendicular to the width of the plate. 2 sheets are considered when calculating sheet value. One sheet length is x and width is y-1 while another sheet length is x-1 and width is y. The sheet with bigger value will be chosen. The recursive algorithm can be expressed as:

If min y d  or   min k L l d   ,   , 0 k F L l y   If min y d  and   min k L l d           1 max{ 1, , , 1 } , , , , 0 max k
k i k i i i i m F L l y F L l y F x y u F L l y w w y n                     (5)
When sheet stock length or width is less than min d , no item can be cut from the sheet. So stock value is equal to leftover value; otherwise, it is equal to the sum value of sheet and leftover.

(2) Generate Cutting Plan

The sequential heuristic algorithm is used to solve the circle stock problem with standard usage leftover. It refers to getting a series of cutting pattern with given constraints by modifying the current remaining demand and item value until the demand is 0. Finally, stocks are cut into sheets and standard leftover. The cutting plan Maximize:

  1 K ik i k k k a v x V    (1)
Subject to:

1 K ik k i k a x b    (2)
 

1 K k ik i k k V e v x    (3) max N   max 0 k ll  k xN  N   (4)
will be completed, and an optimized one which has the highest utilization ratio is obtained.

Input: LW  , m 、 i d 、 i b ; step1: input initialize 2 2 i i d v      , P  ,

Experiment

Experiments are on VS2013 platform with C# on windows10, and the computer is 2.60GHz, 8 GB RAM. Experiments are compared with literature [9]. The range of parameter values is shown in table1. Random generating 500 test instances. The upper bound of leftover length and type is 700mm and 10 respectively. Table3 shows the cutting stock plan when leftover is allowed. There are 10 type cutting patterns. When leftover is not allowed, the cutting patterns are shown in table 4. The same type cutting patterns is 10. But the average utilization and the total stocks are different. The difference shows in table 5. We can learn that the former utilization is 5.35% higher than the latter, and the former plates are 34 more than the latter.

Conclusion

In this paper, the concept of standard leftover generated, so that it can be available for the later orders. The constraints of leftover will make it convenient for inventory management, which is benefit for long period order. Experiments show the combination of sequential heuristic algorithm and recursive algorithm has a higher material utilization. It will play a guiding role in actual production.

Fig1.

  Fig1. Standard leftover and the waste(3) Waste: the remaining part in the cutting stock process. The leftover sheet stock won't be placed any items in this order, it won't be used in the subsequent order.The problem is characterized by the following data: L : stock length W : stock width K : number of cutting patterns  : number of standard leftover types; k l : length of leftover appearing in cutting pattern k , 1,..., k K 

  of stock whose length is x and width is y min d : length of the smallest item i u : value of sheet consists of item i i n :number of item i appearing in the sheet whose width is i w

3 Fig. 2

 32 Fig.2is a cutting plan of one of the random instances. Amount of item is 4, and diameters are 188, 278, 232, and 182. Demands are 2770, 1350, 700, 950. The computational experimental results show that stock number is 115, the cutting pattern is 4, and the standard leftover is 3. The cutting patterns are shown in table2. The average utilization is 74.69% while the one in paper[9] is 70.81%, which shows the average is enhanced by 3.88%.

Acknowledgments

This research is part of Projects 71371058 and 61363026 supported by National Natural Science Foundation of China.