Foundations — Convolutional neural networks — convolution operation, pooling

5.6.11 · D1 · Coding › Machine Learning (Aerospace Applications) › Convolutional neural networks — convolution operation, pooli

Parent note padhne se pehle, tumhe $S(i,j)={\sum }_m{\sum }_{n}I(i+m,j+n)K(m,n)+b$ jaisa symbol dekhna chahiye aur ek picture dikhni chahiye, Greek ki ek diwar nahi. Yeh page har piece ko zero se build karta hai. Upar se neeche padho — har idea sirf unhi cheezein use karta hai jo uske upar hain.

1. Ek image numbers ki ek grid hai

Figure dekho. Left side par woh hai jo tumhari aankh dekhti hai; right side par woh bilkul same cheez hai jo computer store karta hai ek grid ke roop mein.

• Picture: rows aur columns mein numbers ki ek table.
• Topic ko yeh kyun chahiye: convolution is grid par arithmetic hai. Agar tum image ko numbers-in-boxes ki tarah picture nahi kar sakte, toh formulas mein se koi bhi kuch matlab nahi rakhta.

2. Indexing: $I(i,j)$ — ek box ko naam dena

Hum rows aur columns $0$ se count karte hain (yeh coding convention hai). Toh $I(0,0)$ top-left box hai.

Topic ko yeh kyun chahiye: convolution formula poori tarah $I(i+m,j+n)$ ke terms mein likha hai. Woh sirf "box $(i,j)$ par shuru karo aur $m$ boxes neeche, $n$ boxes right step karo." Spreadsheet cell padhne se zyada mysterious kuch nahi.

3. Kernel $K$ — ek tiny pattern-stamp

Ise ek chhoti transparent stamp ki tarah socho jisme numbers printed hain. Positive numbers ka matlab hai "mujhe yahan brightness chahiye"; negative numbers ka matlab hai "mujhe yahan darkness chahiye." Jab tum stamp ko image par rakhte ho aur bright/dark pattern match karta hai, stamp "fire" karta hai.

• Picture: ek chhota $2\times 2$ stamp badi grid ke ek window ke upar hover karta hua.
• Topic ko yeh kyun chahiye: poora CNN idea ek chhota stamp har jagah reuse hota hai. Kernel wahi stamp hai. Parent ka "weight sharing" ka matlab sirf yeh hai ki same $K$ numbers har position par use hote hain.

4. Multiply-and-add: $\sum$ (sigma) ka matlab kya hai

Convolution sum se pehle, tumhe $\sum$ padhna aana chahiye.

Ek double sum ${\sum }_m{\sum }_n$ sirf do nested loops hain: har row-step $m$ ke liye, har column-step $n$ ke liye. Yeh chhote kernel ke har box par sweep karta hai.

5. Sab kuch jodna: convolution sum, dekha hua

Ab parent ka formula plain English jaisa padhta hai: $S(i,j)={\sum }_{m=0}^{k-1}{\sum }_{n=0}^{k-1}I(i+m,j+n)K(m,n)+b$

Zor se bolo: "Stamp ko slide karo taaki iska top-left corner box $(i,j)$ par baithe. Stamp mein har box $(m,n)$ ke liye, neeche wale image number, $I(i+m,j+n)$, ko stamp number $K(m,n)$ se multiply karo. Unhe sab add karo. Bias $b$ add karo. Woh total ek output number $S(i,j)$ hai."

• Picture: ek window par stamp; numbers ki har pair multiply hoti hai; arrows ek output box mein funnel hote hain.
• $S$ output feature map hai — ek nayi grid jo har position par ek match-score rakhti hai.
• Topic ko yeh kyun chahiye: yahi convolution operation hai. Baaki sab (pooling, sizing) is core step ko decorate karta hai.

6. Bias $b$

Picture: multiply-add ko $S(i,j)$ mein funnel karne ke baad, result ko ek fixed amount $b$ se nudge karo. Har stamp per ek $b$.

7. Sliding controls: stride $s$ aur padding $p$

• Picture: same grid stride-1 (dense) sweep vs stride-2 (skipping) sweep ke saath, aur ek zero border padding dikhata hua.
• Topic ko yeh kyun chahiye: yeh sizing formula $W_{out}=\lfloor (W_{in}-k+2p)/s\rfloor +1$ mein appear hote hain. Poori treatment ke liye Padding and stride dekho; yahan tumhe sirf pictures chahiye.

8. Floor brackets $\lfloor \rfloor$

Topic ko yeh kyun chahiye: tum stamp ko fractional number of steps slide nahi kar sakte, isliye positions ki count ek whole number honi chahiye — tum koi bhi leftover throw away karte ho. Yahi floor hai.

9. max aur average — pooling summaries

Average same hai lekin "unhe sab add karo aur kitne hain se divide karo" ke bajaye "sabse bada." Pooling $S$ ki har window ko ek aisi summary se replace karta hai. Koi naya stamp nahi, koi multiplying nahi — sirf dekho aur summarise karo.

Topic ko yeh kyun chahiye: pooling feature map ko shrink karta hai aur network ko ek feature ki exact position ki parwah karna band karta hai (dekho Translation equivariance vs invariance).

10. Channels $C$ aur filter count $F$

Picture: $F$ stamps ka ek deck, har ek ek output sheet produce karta hai; output $F$ sheets ki ek moti stack hai. Yeh aage Feature maps and receptive fields se connect hota hai.

Yeh foundations topic ko kaise feed karte hain

Har arrow kehta hai "right box samajhne ke liye left box chahiye." Parent topic neeche baitta hai — sirf tab reachable jab saare foundations in place hon. Agar tum seedha wahan jump karte, tumhe ek undefined symbol milta; yeh map dikhata hai exactly pehle kya build karna hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.

Connections

• Hinglish version of the parent
• Fully-connected neural networks — woh layer jiska convolution ek constrained version hai
• Padding and stride — $s$ aur $p$ ki poori story
• Feature maps and receptive fields — stacked outputs $S$ kya bante hain
• Translation equivariance vs invariance — sliding + pooling kyun matter karta hai
• Backpropagation — stamp numbers kaise seekhe jaate hain
• Image classification for aerospace inspection — yeh sab kahan use hota hai