Bs.thesis

A RESEARCH ON CHIP FLOW BOUNDARY OF MDF MILLING BASED ON HIGHSPEED CAMERA AND DIGITAL IMAGE PROCESSING METHODS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF WOOD SCIENCE AND TECHNOLOGY AND THE COMMITTEE ON GRADUATE STUDIES OF NANJING FORESTRY UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR

Bo Zhou May 2009

c Copyright by Bo Zhou 2009

All Rights Reserved

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Abstract The research on chip flow is a important thema in wood science, especially in milling of MDF. We tried to use highspeed CCD camera to capture the key frame during milling, and determined the stable boundary distribution of chip turbulent flow by CG(Computer Graphics) and numerical analysis tools with the conditions of varying feeding speed and velocity of the principal axis. We have been successful to prove that the speed of principle axis is the main reason which can effect the cutting angle, that would be valuable to guide the dust separation and protection on operation.

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Contents Abstract

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1 Introduction

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2 Digital Image Processing 2.1 Highspeed Camera . . . 2.2 Color Correction . . . . 2.3 Resampling . . . . . . . 2.4 Thresholding . . . . . . 2.5 GPGPU . . . . . . . . .

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3 Numerical Analysis 3.1 Approximation Theory 3.2 Least Squares . . . . . 3.3 Level Set Method . . . 3.4 Software . . . . . . . .

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8 8 9 10 13

4 Research on MDF cutting diffusion angle 4.1 Image capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Data fitting and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 14 15

5 Conclusions

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Bibliography

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List of Figures 2.1 2.2 2.3 2.4 2.5 2.6

Panasonic WV-CP430 . . . . . . . . . Curves of unormalized filters . . . . . Image Lenna . . . . . . . . . . . . . . Comparison of four image filters . . . Performance Comparision of GPU and Applications of OpenVIDIA . . . . . .

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2 4 6 6 7 7

3.1 3.2 3.3

Least squred fit noisy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implicit function φ(x) = x2 − 1 defining the regions Ω+ , Ω− , the boundary ∂Ω . . . . . . . . . . Level Set active contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 11 12

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Diagram of experimental devices Image proceessing workflow . . . A1 A2 A3 . . . . . . . . . . . . . A4 A5 A6 . . . . . . . . . . . . . A7 A8 A9 . . . . . . . . . . . . . A10 A11 A12 . . . . . . . . . . . A13 A14 A15 . . . . . . . . . . . A16 A17 A18 . . . . . . . . . . . Results comparison . . . . . . . .

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Chapter 1

Introduction The CG have been widely used in nearly most of scientisfic regions. Most problems from sciencfic regions and entertainment industry could be solved by CG and CV(Computer Vision), includes Mathmatics topology and differential geometry [10] [9] , numerical analysis Physics dynamics simulation [7], n-body problem [6] Chemistry quantum chemistry [33] [32] Engineering Visualization earthquake research [21], gas exploration [19] Entertainment film and video game The CG have been developed for nearly 30 years. In 1969, the ACM founded a department names A Spectial Interest Group in Graphics (SIGGRAPH) to organize activities within the field of computer graphics. Most of important early breakthoughts occured at the University of Utah, include research on BRDF(Bidirectional Reflectance Distribution Function), RT(Ray Tracing), Programmable Shading Pipeline. In 1980s, the CG had been involved into entertainment industry, the artist and graphics researcher began to design on personal computer, the SGI graphics workstation was used to make short films. The basic usage of CG is to generate or processing digital 2D image, such as drawing text, 2D geometric primitive, pixels images. In 3D CG, people use 3D representation of geometric data and perspective method to simulate the projection from 3D virtual world to 2D pixels plane. The CG is still running ahead, there will be more and more applications with help of it. At the sametime, digital image processing has benefited from CG very much. It performs image processing on digital images, it allows apply wide range of algorithms to input data compared with analog image processing to avoid problems such as noise and signal loss. It was developed since 1960s at the Jet Propulsion Laboratory, MIT, Bell Labs, with application to satelite imagery, image analysis, cinematic post-production workflow. By using PCA(Principal Components Analysis), digital signal processing, we can solve these problems, Pixel Processing generate histogram, modifiy RGB color, implement color management Pattern Recongnition human face detection Feature Extraction edge detection, optical flow We could use CG to solve lot’s of problems in wood science, such as, volumetric visualization about CT dataset [26], fast rendering forest landscape [27], wood-pulp chips analysis[20]. We will focus on the MDF chip flow diffusion angle, which is a important element during milling, capture the image by highspeed CCD camera with digital image processing methods. It’s important to measure the diffusion boundary and the diffusion angle in order to understand the vortex flow. As we known, there is no similar research subject as ours yet. 1

Chapter 2

Digital Image Processing 2.1

Highspeed Camera

A high speed camera is a device used for recording slow-motion playback films, or used for scientific study of transient phenomena. A normal motion picture is filmed and played back at 24 frames per second, while television uses 25 frames/s (PAL) or 30 frames/s (NTSC), IMAX use 48 frames/s. High speed cameras can film up to 250,000 frames/s by running the film over a rotating prism or mirror instead of using a shutter, thus reducing the need for stopping and starting the film behind a shutter which would tear the film stock at such speeds. Using this technique one can stretch one second to more than 10 minutes of playback time (super slow motion). The fastest cameras are generally in use in scientific research, military test and evaluation, and industry. An example of an industrial application is crash testing to better document the crash and what happens to the automobile and passengers during a crash, or bullet power testing. We use Panasonic WV-CP430 camera to capture detailed milling progress here,

Figure 2.1: Panasonic WV-CP430

Its key features are, • 480 lines of horizontal resolution • DSP noise reduction • 50 dB signal-to-noise ratio • F1.4 maximal aperture

2.2

Color Correction

Because most of digital color capture devices(digital camera, highspeed camera) work in device-dependent RGB color space, so before we convert the pixels data into absolute luminance, we have to know the color space 2

which the pixels data is in and the others conditions such as device white balance, illumination environment, etc. In cinematic post-production, there are two stages about color correction: the first is to modify the integral appearance, the second is to correct the object’s color one by one according the artistic requirement. We could also borrow this idea to our application. To get the illumination graylevel map precisely, we need to transform the all color data into one uniform color space, eg., sRGB [23](Standard RGB), AdobeRGB [15]. We choose sRGB as our uniform color space because nearly most of monitor and digital camera use this colorspace as their standard. The chromaticity values for the sRGB primaries are as follows: Red x = 0.6400 Green x = 0.2100 Blue x = 0.1500 W hite x = 0.3127

y y y y

= 0.3300 = 0.7100 = 0.0600 = 0.3290

(2.1)

The color in sRGB could be converted to CIE XYZ tristimulus by this matrix, 

  Rlinear 3.2410    Glinear  = −0.9692 Blinear 0.0556

−1.5374 1.8760 −0.2040

  −0.4986 X   0.0416   Y  1.0570 Z

(2.2)

and the reverse transformation      X 0.4124 0.3576 0.1805 Rlinear       Y  = 0.2126 0.7152 0.0722 Glinear  Z 0.0193 0.1192 0.9505 Blinear

(2.3)

Before applying any operation on color data in a prescribed colorspace, we should apply CAT(Chromatic Adaptation Transform) on CIE XYZ tristimulus values by the von Kries transform from one viewing condtion to another,      0.4002 0.7076 0.0808 X L      (2.4) M  = –0.2263 1.1653 0.0457  Y  Z 0.0 0.0 0.9182 S

L2 = 2L1 /Lwhite2 M2 = (M1 /Mwhite ) · Mwhite2 S2 = (S1 /Swhite ) · Swhite2

   Xadapted 1/Lwhite   −1   Yadapted  = M ·  0.0 Zadapted 0.0

0.0 1/Mwhite 0.0

   0.0 X    0.0  · M ·  Y  1/Swhite Z

(2.5)

(2.6)

L2 , M2 , and S2 are the predicted cone responses of the perceptual match for the original LMS responses, though under the second viewing conditions. Lwhite , Mwhite , and Swhite are the cone responses of the white point in the original viewing condition, while Lwhite2 , Mwhite2 , Swhite2 are cone responses of the.The M is the Hunt–Pointer–Estevez transformation matrix in 2.4.

3

2.3

Resampling

In fact, the 2D image could be considered as a 2D discrete function. The most common resampling operation is to scale the pixels plane with filter. The filters are the most important in generating the output data because they owns different frequency response [31]. The most common resampling filters used in digital image processing are Box, Triangle, Gaussian, Sinc, etc. Here is their shape within the range x ∈ [−2, −2], the difference

1.2

Box Sinc Gaussian Triangle

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.42.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

Figure 2.2: Curves of unormalized filters

between shapes of these filters is distinct. Most of resampling operations are Convolution. In mathmatics and functional analysis, convolution is a operation about two function f, g. It has applications that includes statistics, signal processing, eletrical engineering, differential equations, and here image processing. The convolution could be written f ∗ g, that means a integral of the production of these two functions, and it is a particular kind of integral transform: Z

∞

(f ∗ g)(t) ≡

Z

∞

f (τ )g(t − τ )dτ = ∞

f (t − τ )g(τ )dτ

(2.7)

∞

It could be also discreted, if gN is a periodic function with peroid N, (f ∗ gN )[n] =

N −1 X

(

∞ X

f [m + kN ])gN [n − m]

m=0 k=−∞

In image processing, this operation could be done easily. The pseudo-code could be like this [29], 4

(2.8)

Require: Image I, Filter f r = f .radius nx = I.width ny = I.height S[0, . . . , nx − 1] ⇐ 0 Iout [r, . . . , nx − r − 1][r, . . . , ny − r − 1] ⇐ 0 for y = r to ny − r − 1 do {convolution on x-axis} for x = 0 to nx − 1 do {convolution on y-axis} S[x] ⇐ 0 for i = −r to r do S[x] = S[x] + f [i]I[x][y − i] {process on y-axis in the radius} end for end for for x = r to nx − r − 1 do {process on x-axis in the radius} for i = −r to r do Iout [x][y] = Iout [x][y] + f [i]S[x − i] end for end for end for return Iout Algorithm 1: Image downsampling pseudo code Downsampled the Lenna 2.3 picture from 5122 to 2562 , we noticed that the Sinc and Box filter could improve the signal intensity on boundary, these pixels forms a high frequency regions there 2.3.

2.4

Thresholding

Sometimes we need to divide the RGB or luminance image into a two-valued image in order to break mark ”object” pixels from ”background” pixels. We have to choose a threshold value, the pixel values which are greater than this threshold value it turns to ”1” otherwise ”0”, that means,  1 if I(x, y) ≥ t f (x, y) = (2.9) 0 if I(x, y) < t We choose the Otsu’s algorithm [25] to perform this image processing progress. It is used to automatically perform histogram shape-based image thresholding, reduction of a graylevel image into a two-valued image. It marks ”foreground” and ”background” by choosing the optimum threshold value that minimizeds the intra-class variance, defined as a weighted sum of variances of two classes, σω2 (t) = ω1 (t)ω12 (t) + ω2 (t)ω22 (t)

(2.10)

The goal turns to maximize inter-class variance, 2

2 σb2 (t) = σ 2 − σw (t) = ω1 (t)ω2 (t) [µ1 (t) − µ2 (t)]

We will use this algorithm to divide the diffusion angle from background.

5

(2.11)

Figure 2.3: Image Lenna

(a) Result of Box filter

(b) Result of Sinc filter

(c) Result of Gaussian filter

(d) Result of Triangle filter

Figure 2.4: Comparison of four image filters

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Require: Graylevel image I {Compute histogram and probabilities of each intensity level} ωi (0) ⇐ 0 for t = 1 to maximum intensity do {Update ωi and µi } {Update σb2 (t)} end for return the t corresponds to the maximum σb2 (t) Algorithm 2: Pseudo-code of Otsu’s algorithm

2.5

GPGPU

When NVIDIA released Geforce FX graphics card in 2002, the programmable shading pipeline has been involved into realtime rendering. Until today, more and more applications have used GPGPU(General Proposed Graphics Processing Unit) to accelerate the program in parallel, most of them are computional numerical problems, such as Monte Carlo simulation [30], linear system [1] [22], computer vision [8], fluid simulation, etc,. Compared with traditional CPU, the architecture of GPU could supply high parallel power to achieve amazing FLOPS(FLoating point Operations Per Second).

Figure 2.5: Performance Comparision of GPU and CPU

(a) Canny edge filter

(b) Corners Locator

Figure 2.6: Applications of OpenVIDIA We could map nearly the all digital image processing into GPGPU, such as color space transform, reshape, resampling, crop, etc. In fact, there have been some commercial cinematic post-production products using c c GPGPU to accelerate the progress, such as SideFX Houdini [14], da vinci Resolve 4K [11].

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Chapter 3

Numerical Analysis 3.1

Approximation Theory

For given data sites x1 < x2 < . . . < xN and function values f1 , . . . , fN , we could find exactly one polynomial pf ∈ πN −1 (R1 ) that interpolate these data sites. We neeed to process the scatterd data by interpolation or fitting. This kind of problems are very common in applied mathmatics and computer science, with with quantitatively characterizing the errors introduced thereby. There are two major problems, they are 1. approximate known target function by a specific class of functions, for example, polynomials or others rational functions 2. approximate unknown target function by known data sites, for example, fitting scattering data as a curve The Weierstraß proved his famous theorem on approximation of continuous functions by algebraic polynomials, Lemma 1 (Weierstraß approximation theorem [2]). Suppose we want to approximate function f which are continuous on an interval [a, b]. For every ε > 0, there exisits a polynomial function p over C such that for all x in [a, b], the supremum norm | f − p |< ε. A popular approximation solution is orthogonal nomials, T0 (x) T1 (x) Tn+1 (x) P∞ n n=0 Tn (x)t U0 (x) U1 (x) Un+1 (x) P∞ n n=0 Un (x)t

polynomials, eg. first and second kind of Chebyshev poly= 1 = x = 2xTn (x) − Tn−1 (x) 1−tx = 1−2tx+t 2

= 1 = 2x = 2xUn (x) − Un−1 (x) 1 = 1−2tx+t 2

(3.1)

(3.2)

or generialized Fourier series for periodic function. Once the domain and degree of the polynomial are chosen, the polynomial itself is chosen in such a way as to minimize the worst-case error, we need to minimize the maximum value of | P (x) − f (x) |, P (x) is the polynomial, f (x) is the actual function.

8

3.2

Least Squares

Least squares is a powerful often applied in statistical context, eg. regression analysis. This method was first destribed by Carl Freidrich Gauß in 1790s, Legendre published this method firstly. It grew out of astronomy and geodesy during the age of exploration. Supposed we have a data set consists of n points (xi , yi ), i = 1, . . . , n, where xi is independent variable and yi is dependent variable whose value is found by experiment. This method would find parameters for which the function model ”best” fits the data. The least squares method defines the squared residuals S is a minimum, ri S

= =

ˆ yi − f (xi , β) Pn 2 i=1 ri

(3.3)

For example, if we think some measured data follow a sinusoidal pattern, f (x, β) = β0 sin(β1 2πxi + β2 )

(3.4)

ei =| yi − β0 sin(β1 2πxi + β2 ) |

(3.5)

the residual vector is

15

Fit Noisy True

10 5 0 5 10 15 0.00

0.01

0.02

0.03

0.04

0.05

0.06

Figure 3.1: Least squred fit noisy data

Linear least squares, or ordinary least squares (OLS), is an important computational problem, that arises

9

primarily in applications when it is desired to fit a linear mathematical model to measurements obtained from experiments. The goals of linear least squares are to extract predictions from the measurements and to reduce the effect of measurement errors. It can be stated as the problem of finding an approximate solution to an overdetermined system of linear equations. In statistics, it corresponds to the maximum likelihood estimate for a linear regression with normally distributed error. Linear least square problems admit a closed-form solution, in contrast to non-linear least squares problems, which often have to be solved by an iterative procedure. The algebraic solution of the normal equations can be written as βˆ = (X> X)−1 X > y = X + y

(3.6)

where X> is Moore–Penrose pseudoinverse of X, it satisfies XX> X X> XX> (XX> )H (X> X)H

= = = =

X X> XX> X> X

(3.7)

If the matrix X> X has full rank, continue to use Cholesky decompostion R> R, the R is an upper triangular matrix, R> Rβˆ = X> y, (3.8) to solve for z R> z = X> y,

(3.9)

Rβˆ = z.

(3.10)

to get the βˆ

The errors on known data can be obained by error propagation. Let the variance-covariance matrix for the observations be denoted by M and that of the parameters by Mβ , we have Mβ = (X> WX)−1 X> WMW> X(X> WX)−1 ,

(3.11)

Mβ = (X> WX)−1 .

(3.12)

when W = M−1

When unit weights are used (W = I) it is implied that the experimental errors are uncorrelated and all equal: M = σ 2 I, the σ 2 is the variance of an observation, I is the identity matrix. Let S denote the minimum value of objective function, we have S (X> X)−1 , (3.13) Mβ = n−m the Mβii gives variance of parameter βi , Mβij gives the covariance between βi and βj , and the Mβij = Mβji .

3.3

Level Set Method

The LSM(Level Set Method)was development by Stanley Osher and James Sethian in 1980s, it’s a numerical technique for tracking interfaces and shapes by project the low-dimension function to hyper-dimension to solve. It has been popular in many research regions, Image processing PDE(Partial Differential Equation)-based image restoration [24], medical image segmentation track [18] [4]

10

Computional geometry point cloud reconstruction [17] Computational dynamics fluid dynamics [5], deformable dynamics [28] Given a function embedded in an space Rn , its interface Γ represents the space to Rn−1 , eg., in 3D space, the Γ represents a 2D space. Similarly in 2D space, Γ is a curve that represent a 1D space(a curve). We define a implicit level set function φ(x, t) with following properties, φ(x, t) > 0 φ(x, t) = 0 φ(x, t) < 0

for x ∈ /Ω for x ∈ Γ for x ∈ Ω

(3.14)

and we could define a signed distance function as d(~x) = min(| ~x − ~xI |), ~xI ∈ ∂Ω

(3.15)

1.5 1.0 0.5 0.0

φ >0,Ω +

φ >0,Ω +

φ <0,Ω−

0.5 1.02.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

Figure 3.2: Implicit function φ(x) = x2 − 1 defining the regions Ω+ , Ω− , the boundary ∂Ω

In level set formulation of active contours, the fronts are represented by the zero level set C(t) = {(x, y) | φ(x, y, t) = 0},

(3.16)

the evolution equation of level set function φ(x, y, t) can be written in the following general form, ∂φ + V · | ∇φ |= 0 ∂t

(3.17)

where V = Vn n ˆ + Vs sˆ is the velocity of the surface (in 2D), which may be a function of both position and time. For image segmentation, the function V depends on the image data and the level set function φ. To improve the accuracy of computation, we need to initialize φ as a SDF before evolution. The re-initialization method is 11

to solve the following equation, ∂φ = sign(φ0 )(1− | ∇φ |). (3.18) ∂t In image segmentation, active contours are dynamics curves that moves toward the object boundaries. We could explicitly define an external energy that can move the zero level curve toward the boundaries. Let the I be the image, g be the edge indicator function defined by 1 , 1+ | ∇Gσ ∗ I |

g=

(3.19)

where the Gσ is the Gaussian function with standard deviation σ. The external energy for the function φ(x, y) is Eg,λ,υ (φ) = λLg (φ) + υAg (φ) (3.20) where λ > 0 and υ are constants. The Lg φ and Ag φ are defined by Z gδ(φ) | ∇φ | dxdy

Lg (φ) =

(3.21)

Ω

and

Z Ag φ =

gH(−φ)dxdy,

(3.22)

Ω

where δ is Dirac function and H is the Heaviside function,  0 if | x |>  δ (x) =  1 [1 + cos( πx )] if | x |≤ . 2 

(3.23)

The total energy functional E(φ) = µP (φ) + Eg,λ,υ (φ) And we know it will satisfie the Euler-Lagrange equation

∂E ∂φ

(3.24)

when it’s stable,

∂E ∇φ ∇φ = −µ[∆φ − div( )] − λδ(φ)div(g ) − υgδ(φ), ∂φ ∇φ ∇φ

(3.25)

combined with 3.17, the steepest descent process for minimization of the functional E is the following gradient flow, ∂φ ∇φ ∇φ = µ[∆φ − div( )] + λδ(φ)div(g ) + υgδ(φ). (3.26) ∂t ∇φ ∇φ

(a) init

(b) 450 iterations

(c) 650 iterations

Figure 3.3: Level Set active contour

12

3.4

Software

To solving these numercial problem, we will use GNU Octave [13], GNU Scientific Library[12], fit the scatterd data with linear least squared. The lsmlib [16] and the matlab code [3] has been used to experiment the LSMbased image segmentation detection.

13

Chapter 4

Research on MDF cutting diffusion angle 4.1

Image capture

PC

CCD

MDF Board

Milling Machine

Figure 4.1: Diagram of experimental devices The experimental devices included - MX160 single vertical spindle milling machine, - Leitz carbide-tipped milling cutter, D = 190mm, BO = 40mm, SB = 20mm, γ = 20◦ , α = 20◦ , - Panasonic WV-CP 430 Camera - PC We captured the image data under these conditions, - f 2.8, 1/50s - Color Temperature 5300k (midday, measured by Nikon D80 DSLR with KODAK 18% graycard) 14

Number A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18

feed speed(m/min) 7.17 7.50 8.33 9.38 10.71 12.50 15.00 18.75 22.50 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

princple axis speed (r/min) 4500 4500 4500 4500 4500 4500 4500 4500 4500 6278 6000 5400 4800 4200 3600 2000 2400 2000

Table 4.1: 2 groups of experimental plans

- Varying feed speed and velocity of the milling Our image processing workflow is

Capture Frames

Resampling

RGB To Graylevel

Mark Out Chip Flow

Thresholding

Figure 4.2: Image proceessing workflow In first stage, we resampled origin sized frames at the beginning of milling to half size with Sinc filter to enhance the boundary distribution on GPU. Then we apply CAT to convert the image to sRGB(D65) color space, continue to convert them to luminance graylevel image. To break the chip flow from background, we applied Otsu’s algorithm to the manually selection regions around the flow boundary, marked the chip part as white, residual part as black.

4.2

Data fitting and analysis

We consider the coordinates of boundary on image as the scattered data sites, use linear least squared fitting to fit the data to y = ax + b straight line mode. The angle between each pair of up and bottom boundary was calculated by k2 − k1 | (4.1) tanθ =| 1 + k1 k2

15

40 35 30 25 20 15 10

λ =3.21504964457 ◦

5 00

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λ =3.05752586835 ◦

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λ =3.67697182933 ◦

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Figure 4.3: A1 A2 A3

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λ =5.02784769423 ◦

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λ =4.57657422166 ◦

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λ =3.94347736538 ◦

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Figure 4.4: A4 A5 A6

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λ =2.16182015624 ◦

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λ =3.1504271861 ◦

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λ =4.66159186493 ◦

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Figure 4.5: A7 A8 A9

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λ =6.71009669847 ◦

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λ =8.0103744916 ◦

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λ =6.67520621659 ◦

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Figure 4.6: A10 A11 A12

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λ =5.45387442024 ◦

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λ =5.27493278342 ◦

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λ =5.51052959245 ◦

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Figure 4.7: A13 A14 A15

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λ =1.50716983512 ◦

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λ =3.42060434553 ◦

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λ =4.98015840473 ◦

5 00

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Figure 4.8: A16 A17 A18

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Number A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18

y y y y y y y y y y y y y y y y y y

Up boundary = 0.460413x + 19.321951 = 0.464728x + 21.137805 = 0.488462x + 18.100000 = 0.441557x + 21.314634 = 0.460694x + 20.141463 = 0.473546x + 18.265854 = 0.500750x + 19.135366 = 0.499719x + 19.830488 = 0.503283x + 20.060976 = 0.516792x + 19.097561 = 0.494184x + 20.463415 = 0.528330x + 19.347561 = 0.587899x + 18.260976 = 0.592026x + 17.530488 = 0.613133x + 16.893902 = 0.683396x + 14.398780 = 0.652158x + 15.682927 = 0.625610x + 15.775610

Bottom boundary y = 0.530300x + 8.734146 y = 0.531332x + 11.289024 y = 0.570638x + 7.047561 y = 0.550938x + 9.831707 y = 0.561445x + 8.926829 y = 0.560788x + 7.914634 y = 0.548874x + 9.471951 y = 0.570450x + 10.701220 y = 0.609850x + 11.582927 y = 0.675516x + 5.402439 y = 0.682364x + 7.043902 y = 0.687899x + 5.760976 y = 0.724015x + 6.981707 y = 0.723921x + 6.658537 y = 0.754221x + 6.092683 y = 0.722702x + 7.007317 y = 0.740807x + 7.579268 y = 0.753846x + 5.250000

Angle(λ) 3.21504964457◦ 3.05752586835◦ 3.67697182933◦ 5.02784769423◦ 4.57657422166◦ 3.94347736538◦ 2.16182015624◦ 3.1504271861◦ 4.66159186493◦ 6.71009669847◦ 8.0103744916◦ 6.67520621659◦ 5.45387442024◦ 5.27493278342◦ 5.51052959245◦ 1.50716983512◦ 3.42060434553◦ 4.98015840473◦

Table 4.2: Analysis formulas of fitted lines

fit λi ,i =1, ,18 to y = ax + b

9

λ1−9 λ10−18 y =0.032733x +3.594188 y = −0.520778x +7.385588

8 7 6 5 4 3 2 10

1

2

3

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5

Figure 4.9: Results comparison

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6

7

8

Chapter 5

Conclusions We have seen the experimental results distinctly : • Spindle speed is the main reason which could strongly affect the MDF chip flow diffusion angle compared with the feed speed. To improve the accuracy of fitting, we could choose some others function model, add more samplers, so the limitation of this experiment is our fitted linear mode it too coarse. We should invove more realtime video image processing methods in future research, use GPGPU to process the countious frames streamly, indicate the boundary and fit the parametric curve. In fact, to cut off scar of lumber is a intuitionistic extension, segment the scar by edge detection or level set based ”Snake” method, convert the parametric curve into CNC program source code. We could also apply multiphase fluid method to this subject. The milling is a kinds of mass-conservation progress, we could model the inter-exchange interface by the MCLS(Mass Conserving Level Set) to simulate.

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