Awakening Elementary School System in University
Chapter 20 - 19: Variational Models in Image Processing

Chapter 20: Chapter 19: Variational Models in Image Processing

Shen Yan didn’t get angry at Chen Lin’s teasing. Instead, a faint blush crept up her cheeks, but she quickly composed herself.

She took her phone out of her bag, tapped the screen a few times, then looked up at Chen Lin. "I sent you a file."

Chen Lin’s phone vibrated. He opened WeChat and saw the PDF file Shen Yan had sent.

He opened it and saw that it was a set of notes Shen Yan had taken while studying her research topic. The title was "Variational Models in Image Processing."

It was filled with dense formulas and derivations, written in neat handwriting, with key points highlighted in different colors.

’She really is a top student,’ Chen Lin sighed in admiration. ’Just look at how meticulous these notes are. It’s obvious how hard she studies.’

However, without a specific mathematical problem to solve, Chen Lin didn’t feel anything special looking at the content.

The "Little Mathematician" ability was powerful, but it needed a clear problem to be activated.

But Chen Lin knew the routine.

He calmly flipped through a few pages, then looked up at Shen Yan with a profound, inscrutable expression and raised an eyebrow slightly.

The meaning was clear: ’So? What do you want to ask?’

His pretentious act almost made Shen Yan laugh, but she held it in.

She opened her laptop, pulled up the same file, and skillfully flipped to a specific page.

"It’s like this," she explained as she navigated the file. "After you helped me with that new algorithm for the modeling competition last time, I showed it to Professor Gu, and he was very pleased with my work."

Chen Lin nodded. ’Of course he was.’

"Then Professor Gu gave me this research direction on variational models in image processing and told me to self-study the relevant foundational knowledge before the next semester starts," Shen Yan continued. "There are a few parts I don’t quite understand, particularly the material on functional analysis."

She turned her laptop screen toward Chen Lin and pointed at the content. "I’ve gone over these sections many times, but I still can’t quite grasp them."

Chen Lin followed her finger and saw several technical terms written on the screen:

"Sobolev Space Theory"

"Calculus on Banach Spaces"

"Variational Principles (Minimizing Sequences, Palais-Smale Condition)"

Seeing these terms, Chen Lin felt a little troubled.

’These are all knowledge points, sure, but the problem is, without a specific math problem, the "Little Mathematician" ability is useless.’

"This material," Shen Yan said, a bit frustrated, "is something I’m supposed to learn next semester. I’ve been studying it on my own for a while, but there are still many parts I don’t fully understand."

Chen Lin thought for a moment, then asked tentatively, "Are there any related practice problems? I think understanding the concepts by solving problems might be more intuitive."

Shen Yan’s eyes lit up as if she’d found a kindred spirit. "That’s exactly what I was thinking!"

She quickly began searching on her computer. "I found some related problems online that cover these three areas. I was having a really hard time with them, so they’re perfect for you to explain."

Chen Lin breathed a silent sigh of relief.

’Problems make things easy. Now the "Little Mathematician" finally has a chance to shine.’

Soon, Shen Yan found the file with the problems and pushed the laptop in front of Chen Lin. "Look, these problems are pretty difficult."

Chen Lin took a closer look. The first problem:

[Let the domain be Ω=(-1, 1)⊂ℝ, and let the function be u(x)=|x|.

Task:

(a) Verify that u∈ L²(Ω) and find its weak derivative u’∈ L¹_loc(Ω).

(b) For any test function φ∈ C_c^∞(Ω), verify that the equality ∫_Ω uφ’ dx = -∫_Ω u’φ dx holds.]

The second problem:

[Let {u_n}⊂ W^{1,p}(Ω) be a Cauchy sequence (where 1

As m, n→∞, ||u_m - u_n||_{W^{1,p}}→ 0.

Prove: There exists a function u∈ W^{1,p}(Ω) such that the sequence converges to u in the W^{1,p} norm, i.e.:

As n→∞, ||u_n - u||_{W^{1,p}}→ 0]

Chen Lin quickly skimmed through all the problems and realized there were quite a few of them.

He turned to Shen Yan and said, "There are a lot of problems; we definitely won’t finish them all this afternoon. How about this? We pick a few typical problems from each of the three topics and I’ll explain as we go?"

"Okay." Shen Yan nodded, clearly prepared for this.

Chen Lin stood up and sat down naturally in the seat next to Shen Yan.

The distance between them closed in an instant. Chen Lin could even smell the faint, fresh scent on her.

He took a scratchpad and a pen from his backpack.

’Little Mathematician, passive activation!’

Instantly, that familiar feeling washed over him.

Mathematical knowledge automatically converged and organized itself in his mind, naturally forming lines of solutions.

"Let’s start with the first problem." Chen Lin picked up his pen and wrote the problem number on the scratch paper.

"For part (a) of the first problem, to verify that the function belongs to the L² space, the key is to calculate its L² norm."

He started writing on the paper as he spoke:

"Step 1: Calculate the L² norm of u.

||u||_{L²(Ω)}² = ∫_{-1}^{1} |x|² dx"

Shen Yan watched his derivation intently, nodding from time to time.

"Pay attention here," Chen Lin said, tapping the paper with the tip of his pen. "Since |x| is an even function, we can use symmetry to simplify the calculation."

"||u||_{L²(Ω)}²

= ∫_{-1}^{1} |x|² dx

= 2 ∫_{0}^{1} x² dx

= 2 * [x³/3]_{0}^{1}

= 2 * (1/3) = 2/3

"Therefore, u ∈ L²(Ω)."

Shen Yan nodded thoughtfully. "So, to verify that a function belongs to the L² space, you just need to prove that its L² norm is finite."

"Exactly." Chen Lin continued writing. "Next is finding the weak derivative, which is a bit more complicated."

He drew a simple graph on the paper showing the shape of the function |x|.

"You see, |x| is not differentiable at x=0, so we can’t use the normal concept of a derivative. This is where the definition of a weak derivative comes in."

Chen Lin began the detailed derivation, his pen flying across the paper:

"Let v(x) = sign(x) = {-1, if x0}"

"For any φ ∈ C_c^∞(Ω), we need to verify:"

"∫_{-1}^{1} |x|φ’(x) dx = -∫_{-1}^{1} v(x)φ(x) dx"

He split the integral into two parts for the calculation, writing out each step with perfect clarity.

Shen Yan watched with rapt attention, occasionally asking a question. "Why can you use integration by parts here?"

"Because φ is a smooth function with compact support," Chen Lin explained patiently. "Its value is zero at the boundary, so integration by parts doesn’t produce any boundary terms."

Time slipped by without them noticing.

Sunlight filtered through the window, casting dappled shadows on the scratch paper.

The tip of Chen Lin’s pen scratched across the paper, pausing occasionally to explain a key step to Shen Yan in detail.

"See, in the end, we get:"

"LHS = ∫_{-1}^{0} φ dx - ∫_{0}^{1} φ dx"

"The RHS gives the same result, so v is indeed the weak derivative of u."

"And since |v(x)|=1 is bounded on Ω, it’s clear that u’ ∈ L¹_loc(Ω)."

It dawned on Shen Yan. "So the essence of a weak derivative is that it’s defined through integration, which bypasses the problem of the function not being differentiable at certain points."

"A very sharp understanding." Chen Lin put on a serious face and nodded in approval. "Let’s move on to part (b)."

"This one is simple, since we already proved the key equality in part (a)."

He quickly wrote:

"As proven in (a):

∫_Ω uφ’ dx = ∫_{-1}^{0} φ dx - ∫_{0}^{1} φ dx

-∫_Ω u’φ dx = -∫_Ω vφ dx = Same result

Therefore, the equality holds."

After finishing the first problem, Chen Lin looked up at Shen Yan. "So, do you understand the basic concept of the weak derivative covered in this problem?"

Shen Yan nodded earnestly. "It’s much clearer than reading the book on my own."

"Then let’s continue with the second problem." Chen Lin turned to a new page. "This one deals with the completeness of Sobolev spaces."

He began to structure the proof on the paper:

"To prove that the W^{1,p} space is complete, the key is to use the completeness of the L^p space..."

And so, the two of them sat in the quiet study room, one explaining diligently, the other studying intently.

The sun outside gradually sank in the west, casting a warm golden glow over them.

Occasionally, Shen Yan’s face would light up with joy upon suddenly understanding a difficult point.

And after she understood, Chen Lin would calmly move on to the next problem.

This atmosphere of teaching and learning made time fly by.

Before they knew it, several pages of scratch paper had been filled with dense formulas and derivations.

"This last problem on variational principles," Chen Lin said, stretching his slightly sore wrist, "involves the Palais-Smale condition, which is a pretty abstract concept."

Shen Yan also rubbed her eyes. Staring at mathematical derivations for so long was indeed a bit tiring.

But her eyes were still bright, a clear sign that she had gained a lot.

She glanced at the time on her phone. It had been almost three hours.

Then she heard Chen Lin say, "Let’s stop here for today. I’m afraid you won’t be able to digest everything if we cover too much at once. There are still a few problems left; you can try doing them at home based on what you’ve learned. If there’s anything you don’t understand, you can ask me during our next session."

Shen Yan nodded. "Okay. Thank you so much for spending all this time tutoring me."

Chen Lin smiled. "Like I said, you’re my best customer. I have to provide top-notch service, right?"

Shen Yan froze for a second, a little embarrassed. "Oh, I almost forgot. I’ll transfer this session’s fee to you right now."

With that, she took out her phone to make a WeChat transfer.

Seeing the WeChat transfer notification, Chen Lin’s mood became even more cheerful.

Spending three hours alone with a campus belle-level beauty and earning an easy 1,500 yuan on top of that. The only thing he’d expended was brainpower, which could be easily replenished with a good night’s sleep.

’Could anything be better than this?’

So he said to Shen Yan, "For our next session, just contact me a day in advance to schedule it. I’ll definitely clear my schedule for you."

After saying that, he gave a wave, grabbed his backpack, and walked out of the study room.

Shen Yan watched his retreating back. This was the third time she had witnessed Chen Lin’s extraordinary mathematical genius.

Recalling the calm and composed expression on Chen Lin’s face as he explained the problems, and then glancing at the scratch paper he had left behind, her heart began to race for no reason. An imperceptible blush crept onto her cheeks.