Exercise 1

Problem Analysis

The objective is to find the coefficients $(\alpha, \beta, \gamma)$ for the finite difference formula:
$$
D_{h}f(\overline{x})=\frac{\alpha f(\overline{x})+\beta f(\overline{x}-h)+\gamma f(\overline{x}-2h)}{h} \quad
$$
The analysis begins by substituting the Taylor series expansions for $f(\overline{x}-h)$ and $f(\overline{x}-2h)$ around the point $\overline{x}$ into the formula.

The expansions are:
$$
f(\overline{x}-h) = f(\overline{x}) - hf’(\overline{x}) + \frac{h^2}{2}f’’(\overline{x}) - \frac{h^3}{6}f’’’(\overline{x}) + O(h^4)
$$
$$
f(\overline{x}-2h) = f(\overline{x}) - 2hf’(\overline{x}) + 2h^2f’’(\overline{x}) - \frac{4h^3}{3}f’’’(\overline{x}) + O(h^4)
$$
Substituting these into the formula and grouping terms by derivatives of $f(\overline{x})$ results in:
$$
D_{h}f(\overline{x}) = \frac{\alpha + \beta + \gamma}{h} f(\overline{x}) + (-\beta - 2\gamma) f’(\overline{x}) + \left( \frac{\beta}{2} + 2\gamma \right)h f’’(\overline{x}) + O(h^2)
$$
To approximate $f’(\overline{x})$, the coefficient of $f’(\overline{x})$ must be $1$, and the coefficient of $f(\overline{x})$ must be $0$. This yields two necessary equations:

1. $\alpha + \beta + \gamma = 0$
2. $-\beta - 2\gamma = 1$

a) Solution for First-Order Accuracy

For the formula to be of order 1, the two fundamental equations must be satisfied.

From equation (2), $\beta$ can be expressed in terms of $\gamma$:
$$
\beta = -1 - 2\gamma
$$
Substituting this into equation (1):
$$
\alpha + (-1 - 2\gamma) + \gamma = 0 \implies \alpha = 1 + \gamma
$$
Therefore, the family of coefficients $(\alpha, \beta, \gamma)$ that provides at least first-order accuracy is given by:
$$
(\alpha, \beta, \gamma) = (1+\gamma, -1-2\gamma, \gamma), \quad \forall \gamma \in \mathbb{R}
$$

b) Solution for Second-Order Accuracy

To achieve second-order accuracy, the truncation error must be $O(h^2)$. This requires the coefficient of the $h$ term in the error expansion to also be zero.

$$\frac{\beta}{2} + 2\gamma = 0$$

This creates a system of three linear equations to be solved for the unique values of $(\alpha, \beta, \gamma)$:

1. $\alpha + \beta + \gamma = 0$
2. $-\beta - 2\gamma = 1$
3. $\frac{\beta}{2} + 2\gamma = 0$

From equation (3), we find that $\beta = -4\gamma$. Substituting this into equation (2):
$$
-(-4\gamma) - 2\gamma = 1 \implies 2\gamma = 1 \implies \gamma = \frac{1}{2}
$$
With the value for $\gamma$, $\beta$ can be found:
$$
\beta = -4 \left( \frac{1}{2} \right) = -2
$$
Finally, substituting $\beta$ and $\gamma$ into equation (1) yields $\alpha$:
$$
\alpha + (-2) + \frac{1}{2} = 0 \implies \alpha = \frac{3}{2}
$$
The unique values which give a formula of order 2 are:
$$
\left(\alpha, \beta, \gamma\right) = \left(\frac{3}{2}, -2, \frac{1}{2}\right)
$$

Exercise 2

a) Midpoint Formula Calculation

The first task is to compute the integral of the function $f(x)=e^{-x}sin(x)$ on the interval $[a,b]=[0,2]$. The computation uses the composite midpoint formula with $M=10$ sub-intervals.

The resulting numerical approximation of the integral is 0.468592. The exact integral is 0.466630. So the error is 0.001962.

b) Convergence of the Midpoint Formula

This section analyzes the convergence of the composite midpoint formula for the same function and interval. The number of sub-intervals is varied as $M=10^{1},10^{2},10^{3},…10^{5}$.

The absolute error, $|I(f)-Q_{h}^{pm}(f)|$, is computed for each corresponding step size, $h=(b-a)/M$. This error is the difference between the numerical result and the exact value of the integral, $I=(1-e^{-2}(sin(2)+cos(2)))/2$.

A graph of the error versus the step size $h$ is then created using a logarithmic scale for both axes. In such a plot, the slope of the resulting line corresponds to the order of convergence.

The plot displays a straight line with a negative slope. A linear fit to the logarithmic data reveals a slope of approximately 2. This visually confirms that the method exhibits 2nd-order convergence, which aligns with the theoretical error bounds for the composite midpoint rule.

c) Convergence of Trapezoidal and Simpson’s Formulas

The analysis is repeated using the composite trapezoidal formula and the composite Simpson’s formula. The errors for both methods, $|I(f)-Q_{h}^{trap}(f)|$ and $|I(f)-Q_{h}^{simp}(f)|$, are plotted on the same graph in logarithmic scale.

The resulting graph contains two distinct straight lines, one for each method.

• Trapezoidal Rule: The error line for the trapezoidal rule has a slope of approximately 2. This confirms its theoretical 2nd-order convergence.
• Simpson’s Rule: The error line for Simpson’s rule is significantly steeper, with a slope of approximately 4. This demonstrates its theoretical 4th-order convergence.

A comparison of the results shows that for any given step size $h$, the error from Simpson’s rule is substantially smaller than the error from the trapezoidal rule, highlighting the superior accuracy of the higher-order method for smooth functions.

d) Analysis for a Non-Smooth Function

Finally, the convergence analysis is repeated for all three methods on a new function, $f(x)=\sqrt{|x|^{3}}$, over the interval $[a,b]=[-2,2]$. The exact value for this integral is $I=\frac{16}{5}\sqrt{2}$.

Comments on the Obtained Results

The function $f(x)=\sqrt{|x|^{3}}$ is not sufficiently smooth at the point $x=0$ within the integration interval. While the function and its first derivative are continuous, its second derivative, $f’’(x) \propto |x|^{-1/2}$, is unbounded at $x=0$.

The theoretical error estimates that predict 2nd and 4th-order convergence for these methods are based on the assumption that the function’s higher-order derivatives are continuous and bounded. Since this condition is violated, a degradation in the observed convergence rates is expected.

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Exercise 1

(a) Polynomial Interpolation

To approximate the function $$f(x) = \frac{1}{1 + \exp(4x)} $$on the interval [-5, 5], polynomial interpolation was performed using polynomials of degree n=6 and n=14. Two types of nodes were considered: equally spaced nodes and Clenshaw-Curtis nodes, the latter computed using the formula

$$
x_i = \frac{a + b}{2} - \frac{b - a}{2} \cos\left(\pi \frac{i}{n}\right), \quad i = 0, \dots, n.
$$

Using polyfit, the interpolation polynomials were computed based on the function values at the chosen nodes. The polynomials were evaluated using polyval on a fine grid over [-5, 5] and plotted alongside the true function f(x). The results showed that interpolation with Clenshaw-Curtis nodes provided better approximation near the boundaries, especially for n=14, reducing the Runge phenomenon that is prominent with equally spaced nodes.

(b) Piecewise Linear Interpolation

For the same values of n=6 and n=14, piecewise linear interpolation was carried out using interp1 with equally spaced nodes. The linear interpolant $$p_{1,h}$$ was constructed over n subintervals of length $$h = \frac{10}{n}$$. The resulting piecewise linear functions were evaluated on a fine grid and compared graphically with the original function f(x). While less smooth than the high-degree polynomials, the linear interpolation captured the overall trend of the function and showed more stable behavior near the interval edges.

(c) Error Analysis

To quantitatively assess the interpolation accuracy, the maximum approximation error

$$
E_n = \max_{x \in [-5, 5]} |f(x) - p_n(x)|
$$

was computed for n=2 to 40 for both equally spaced and Clenshaw-Curtis nodes. Similarly, the maximum error for piecewise linear interpolation, $$E_{1,h}$$, was evaluated for the same range of n. A loop was used to iterate over n, computing and storing the errors. The results were visualized on a log-scale plot with respect to n. The plots clearly demonstrated that Clenshaw-Curtis interpolation achieves significantly better accuracy with increasing n, while equally spaced nodes lead to instability and large errors as n grows. Piecewise linear interpolation, although low-order, exhibited stable and predictable convergence.

Exercise 2

(a) Equally Distributed Nodes

The function $f(x) = \sin(x) + x$ was interpolated on the interval [0,10] using Lagrange polynomial interpolation. Two degrees were considered: $n = 4$ and $n = 15$, corresponding to $n + 1$ equally spaced nodes. For each value of n, two datasets were created: one using the exact function values $y_i = f(x_i)$, and the other using perturbed values $z_i = y_i + \epsilon_i$, where $\epsilon_i \sim \text{Uniform}(-0.1, 0.1)$. Using polyfit and polyval, the interpolation polynomials were computed and evaluated on a dense grid for visualization.

The results show that for n = 4, both the unperturbed and perturbed interpolants approximate the true function reasonably well, though the perturbed one deviates slightly. For n = 15, the interpolant using exact values exhibits oscillatory behavior near the boundaries, a clear manifestation of the Runge phenomenon. The polynomial interpolant using noisy data becomes highly unstable, demonstrating extreme sensitivity to small perturbations in the input values.

(b) Clenshaw-Curtis Nodes

The same procedure was repeated using Clenshaw-Curtis nodes computed using the cosine-based formula. For both n = 4 and n = 15, interpolation polynomials were fitted using the exact and perturbed data values. When visualized, the interpolants using Clenshaw-Curtis nodes performed significantly better near the boundaries compared to those using equally spaced nodes. Particularly for n = 15, the polynomial interpolant with exact values closely follows the true function, with reduced oscillations. Even when perturbations are introduced, the polynomial remains relatively stable and does not exhibit the extreme sensitivity observed in the equally spaced case. This confirms that Clenshaw-Curtis nodes enhance numerical stability in high-degree interpolation.

(c) Piecewise Linear Interpolation

Piecewise linear interpolation was implemented using interp1 on equally spaced nodes for both n = 4 and n = 15. As before, both the exact and perturbed datasets were used to construct the interpolants. The linear interpolants followed the general shape of the original function and remained visually close to the true curve even when noise was added. This behavior was consistent across both values of n. While less accurate than high-degree polynomial interpolation in smooth regions, the piecewise linear approach showed strong robustness to noise and did not exhibit oscillations or instability, making it a reliable method for interpolating noisy data.

Exercise 3

(a) Least Squares Polynomial of Degree 4

The file data1.mat was loaded to obtain the dataset $(x_i, y_i)$, where $y_i = \sin(x_i) + x_i + \epsilon_i$, and $epsilon_i$ represents Gaussian noise with zero mean and standard deviation $σ=0.1$. A degree-4 least squares polynomial $p^{LS}_4$ was fitted using polyfit(x, y, 4). On the same plot, the original function $f(x) = \sin(x) + x$, the noisy data points $(x_i, y_i)$, and the polynomial $p^{LS}_4(x)$ were displayed. The polynomial provided a smooth approximation that followed the general trend of the function, mitigating the effect of noise.

(b) Higher-Degree Least Squares Polynomials (m = 7, 15)

The procedure was repeated for polynomial degrees $m = 7$ and $m = 15$. For $m = 7$, the approximation improved slightly, with the polynomial capturing more curvature of the underlying function. However, at $m = 15$, overfitting became evident—the polynomial began to follow the noise, especially near the edges of the interval. This illustrates the typical variance-bias tradeoff in polynomial regression: higher degrees reduce bias but increase sensitivity to noise.

(c) Approximation Error vs. Degree

To assess the accuracy of the least squares fit, the error $E(p^{LS}m, f) = \max{x \in [0, 10]} |f(x) - p^{LS}_m(x)|$

was computed for degrees m = 1 to 1515, evaluated over a fine grid. The error was then plotted as a function of m on a semi-logarithmic scale. The plot showed a decreasing trend in the error for small m, reaching a minimum at some intermediate degree. Beyond that, the error increased, reflecting overfitting. This confirms that, for noisy data, the optimal degree is not necessarily high.

(d) Noise Variance Estimation and Error Comparison

The variance of the noise was estimated using the formula

$\hat{\sigma}^2 = \frac{1}{n - m} \sum_{i=1}^{n+1} \left( y_i - p^{LS}_m(x_i) \right)^2,$

with m = 4 and n + 1 = 20. The estimated variance $\hat{\sigma}$ was then used to compute the expected approximation error scale

$\hat{\sigma} \cdot \sqrt{\frac{m + 1}{n + 1}}.$

The previously observed minimal error matched this estimate in magnitude, validating the analytical approximation. A comparison with the theoretical value $\sigma \cdot \sqrt{(m+1)/(n+1)}$ was also performed, showing good agreement.

(e) Large-Scale Data: n + 1 = 20000

The analysis from parts (a) to (d) was repeated using the dataset in data2.mat, which contains n + 1 = 20000 samples. The results demonstrated a dramatic improvement in the stability and accuracy of least squares fitting. Overfitting effects were less pronounced even at higher polynomial degrees due to the large number of samples. The error curve in (c) showed a more gradual and consistent decrease, and the noise influence was better averaged out. Variance: 0.04814, Bound ≈ 0.00347

(f) Error Comparison Between Small and Large Datasets

The minimal approximation error achieved using the larger dataset was significantly smaller than that obtained from the 20-point dataset. This reflects the benefit of larger sample sizes in reducing noise impact and improving model robustness. Moreover, the estimated noise error scale $\hat{\sigma} \cdot \sqrt{(m+1)/(n+1)}$ decreased accordingly due to the much larger denominator, reinforcing the statistical intuition.

Exercise 4

the Lagrange interpolation error at any point x in the interval containing the nodes is given by

$f(x) - p_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{k=0}^n (x - x_k)$
Taking the absolute value yields the error bound
$|f(x) - p_n(x)| \leq \frac{M}{(n+1)!} \left| \prod_{k=0}^n (x - x_k) \right|$

where $p_n(x)$ is the interpolating polynomial of degree n through n+1 points. For both functions, the interpolation degree was set to n=4, which requires the fifth derivative of the target function.

For the function $f_1(x) = \cosh(x)$, the interpolation nodes were defined as $x_k = -1 + \frac{k}{2}$for $k = 0, 1, \dots$, giving equally spaced points in the interval [−1,1]. The fifth derivative of $\cosh(x)$ is $\sinh(x)$, since derivatives of hyperbolic functions follow a repeating pattern. The maximum of $|\sinh(x)|$ over[−1,1] occurs at x = 1, where $\sinh(1) \approx 1.175$. Substituting into the error formula, the upper bound depends on this derivative value and the absolute value of the product of linear terms $(x - x_k),$ divided by $5! = 120$.

For the function $f_2(x) = \cos(x) + \sin(x)$, the interpolation nodes were chosen as $x_k = -\frac{\pi}{2} + \frac{\pi k}{4}$ for $k = 0, \dots, 4$, covering the interval [−2π,2π]. The fifth derivative of $f_2$ is $-\cos(x) + \sin(x)$, based on the periodic differentiation cycle of sine and cosine. The maximum of the absolute value of this expression was found numerically over the interpolation interval. This value was then used in the same error formula as above, with the factorial and the product of distances from the interpolation nodes.

In both cases, the error bound reflects how the function’s higher-order smoothness and the choice of interpolation nodes influence the potential deviation between the true function and its polynomial approximation. The approach confirms that even for smooth functions, the interpolation error can grow significantly away from the nodes if the derivatives are large or if the node distribution leads to large oscillations in the interpolating polynomial.

Repository:
https://github.com/PSGBOT/pixtral-12B-Inference

本地图片上传

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def encode_image(image_path):
    """Encode the image to base64."""
    try:
        with open(image_path, "rb") as image_file:
            return base64.b64encode(image_file.read()).decode('utf-8')
    except FileNotFoundError:
        print(f"Error: The file {image_path} was not found.")
        return None
    except Exception as e:  # Added general exception handling
        print(f"Error: {e}")
        return None

Prompt

VLM物体描述的prompt:

核心需要：准确定位物体所在方位，不把远景识别为物体，降低False Positive

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Focus on the area highlighted in green in the image.

Step 1: Determine if the highlighted area represents a distinct, identifiable object or instance:
- If the highlighted area is clearly a distinct object, proceed to Step 2.
- If the highlighted area is abstract, ambiguous, or you cannot confidently identify it as a specific object (e.g., part of background, texture, partial view), respond with "Valid: No".

Step 2: If the highlighted area is a distinct object, provide:
1. The specific name of the object (be precise and use technical terms when appropriate)
2. The primary function or purpose of this object
3. Any notable features visible in the highlighted area (no color description)
4. If there is text visible on the object, include what it says

Remember, if you're uncertain about the highlighted area being a distinct object, respond only with "Valid: No".

输出结果：

• Valid

  1 2 3 4 5 6 7 8 9

  Valid: Yes 1. The specific name of the object: Soap dispenser 2. The primary function or purpose of this object: To dispense liquid soap or hand sanitizer. 3. Notable features visible in the highlighted area: - The dispenser has a pump mechanism at the top. - The body of the dispenser is cylindrical. - The material appears to be translucent plastic. 4. There is no visible text on the object.

• invalid

  1

  Valid: No

VLM输出->Structured Output

使用另一个LLM来对VLM输出的内容进行parse，转化成json文件, 通过mistral ai 提供的接口实现:

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class Instance(BaseModel):
    valid: str
    name: Optional[str] = None
    feature: Optional[List[str]] = Field(default_factory=list)
    usage: Optional[List[str]] = Field(default_factory=list)

def parse_description_msg(msg):
    message = [
        {"role": "system", "content": "Extract the description information."},
        {
            "role": "user",
            "content": msg,
        },
    ]
    return message

chat_response = self.client.chat.parse(
	model=self.llm,
	messages=msg,
	response_format=Instance,
	max_tokens=self.llm_max_tokens,
	temperature=self.llm_temperature,
)
return json.loads(chat_response.choices[0].message.content)

https://pan.sjtu.edu.cn/web/desktop/personalSpace?path=Disney

https://pan.sjtu.edu.cn/web/share/268e334a73ed7a82daca26aecf2bfd67

使用mpm安装(https://wiki.archlinux.org/title/MATLAB)
Download mpm from https://www.mathworks.com/mpm/glnxa64/mpm and make it executable.

安装：

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./mpm install --release=R2024b --destination=/home/cyl/matlab MATLAB

安装后启动完成lisense注册后使用patch(https://bbs.archlinux.org/viewtopic.php?id=303177)

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patchelf --clear-execstack /home/user/.MathWorks/ServiceHost/-mw_shared_installs/v2024.13.0.2/bin/glnxa64/libmwfoundation_crash_handling.so
patchelf --clear-execstack /home/user/.MathWorks/ServiceHost/-mw_shared_installs/v2024.13.0.2/bin/glnxa64/mathworksservicehost/rcf/matlabconnector/serviceprocess/rcf/service/libmwmshrcfservice.so # may not needed

如果出现空白窗口左下显示ready，那么参考(www.reddit.com/r/matlab/comments/1dhejp5/matlab_gui_not_loading_properly_on_arch/)，设置环境变量

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export _JAVA_AWT_WM_NONREPARENTING=1

Sources

Single Instance

• Image Classification - 32 Classes - Fourniture: About 10K

Complicated Scene

• Real
  • Indoor Training Set (ITS) [RESIDE-Standard]: 1.4K
  • MIT Indoor Scenes: 15K
• Synthetic
  • InteriorVerse

chair_dataset

• image_1~image_3000: kaggle furniture image dataset
• image_3001~image_4887: DeepFurniture

desk_dataset

• image_1~image_700: pix3d

home_appliance_dataset

• image_1~image_3000: kaggle furniture image dataset fridge only
• image_3001~image_3429: DeepFurniture home-appliance category

shelves_dataset

• image_1~image_3000: kaggle furniture image dataset
• image_3001~image_3243: pix3d wardrobe category
• image_3244~image_3604: pix3d bookcase category

sofa_dataset

• image_1~image_1947: pix3d
• image_1498~image_3888: DeepFurniture

table_dataset

• image_1~image_3000: kaggle furniture image dataset
• image_3001~image_4870: pix3d
• image_4871~image_7293: DeepFurniture

tool_dataset

• image_1~image_115: pix3d
• image_116~image_1441:kaggle mechanical tool dataset hammer
• image_1442~image_1812:kaggle mechanical tool dataset plier
• image_1813~image_3138:kaggle mechanical tool dataset screw driver
• image_3139~image_4469:kaggle mechanical tool dataset wrench

tv_dataset

• image_1~image_3000: kaggle furniture image dataset

PSR Dataset

Cabinet

from shelves_dataset 3000 images -> ? train & ? val samples

Desk (Processing)

from desk_dataset 699 images -> ? train & ? val samples

Tool (Processing)

from tool_dataset (simplified) 1335 images -> 2729 train & 911 val samples

Furniture (Processing)

from 130k Images/furniture 1983 images -> ? train & ? val samples

Coco (Processing)

from coco2017 2212 images -> ? train & ? val samples

current total:

• train: 12,415-686(bg)=11729
• val: 3,028-182(bg)=2846

https://www.youtube.com/watch?app=desktop&v=HVcTPeitxmw

Download pkg from https://apps.cloud.blackmagicdesign.com/davinci-resolve

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git clone https://aur.archlinux.org/davinci-resolve.git
cd davinci-resolve
mv ~/Downloads/DaVinci_Resolve_19.1.4_Linux.zip ./
makepkg -si