Deep Learning · TEE · section

F22 Q1a

Final examination November 2022 fully solved

Question

Question (4 marks). A normalized 5 × 5 image has every row [0.5,0.5,0.3,0.3,0.3]. Convolve it with a 3 × 3 filter whose every row is [1,0,−1], stride 2, no padding. Compute the output and identify the feature.

Answer — step by step

Given

A normalized 5 × 5 image has every row [0.5,0.5,0.3,0.3,0.3].

Convolve it with a 3 × 3 filter whose every row is [1,0,−1], stride 2, no padding.

To find

Find every convolution output cell.

Formula — which one and why

Convolution values. At each valid location, multiply corresponding window and kernel entries, add all products and add the filter bias once. Move by the stated stride and repeat.

X = input; K = kernel; S = stride; b = filter bias; u,v locate a kernel entry. Use the printed kernel without flipping under the stated deep-learning convention.

Y_{ij}=\sum_{u,v}X_{iS+u,jS+v}K_{uv}+bLaTeX: Y_{ij}=\sum_{u,v}X_{iS+u,jS+v}K_{uv}+b

Yaad rakhne ka cue: Matching cells multiply; sab products add; bias sirf ek baar.

Solution — step by step

Follow each selected input window below. Matching row colors connect its input cells, kernel coefficients and products; the outlined output cell shows where their sum belongs.

Step 1 shape: \left\lfloor\frac{5+2 \times 0-3}{2}\right\rfloor+1LaTeX: \left\lfloor\frac{5+2 \times 0-3}{2}\right\rfloor+1 = 2 along each axis. Use the deep-learning cross-correlation convention: do not flip the printed kernel.

Output position (1, 1)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (1, 1) into one output cell

Input rows 1–3, columns 1–3. The window moves 2 cells per output step.

1. Select the input window
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3

Only the outlined cells belong to this output.

2. Match the kernel
10−1
10−1
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
0.50−0.3
0.50−0.3
0.50−0.3

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 10.5 + 0 + (−0.3) = 0.2
Row 20.5 + 0 + (−0.3) = 0.2
Row 30.5 + 0 + (−0.3) = 0.2

0.2 + 0.2 + 0.2 = 0.6

5. Place it in the output
0.60
0.60

Output row 1, column 1.

Window row 1.

R_1=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2LaTeX: R_1=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2

Window row 2.

R_2=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2LaTeX: R_2=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2

Window row 3.

R_3=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2LaTeX: R_3=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2

Add the rows to obtain one output cell.

Y_{1,1}=(0.2)+(0.2)+(0.2)=0.6LaTeX: Y_{1,1}=(0.2)+(0.2)+(0.2)=0.6

Output position (1, 2)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (1, 2) into one output cell

Input rows 1–3, columns 3–5. The window moves 2 cells per output step.

1. Select the input window
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3

Only the outlined cells belong to this output.

2. Match the kernel
10−1
10−1
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
0.30−0.3
0.30−0.3
0.30−0.3

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 10.3 + 0 + (−0.3) = 0
Row 20.3 + 0 + (−0.3) = 0
Row 30.3 + 0 + (−0.3) = 0

0 + 0 + 0 = 0

5. Place it in the output
0.60
0.60

Output row 1, column 2.

Window row 1.

R_1=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0LaTeX: R_1=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0

Window row 2.

R_2=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0LaTeX: R_2=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0

Window row 3.

R_3=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0LaTeX: R_3=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0

Add the rows to obtain one output cell.

Y_{1,2}=(0)+(0)+(0)=0LaTeX: Y_{1,2}=(0)+(0)+(0)=0

Output position (2, 1)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (2, 1) into one output cell

Input rows 3–5, columns 1–3. The window moves 2 cells per output step.

1. Select the input window
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3

Only the outlined cells belong to this output.

2. Match the kernel
10−1
10−1
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
0.50−0.3
0.50−0.3
0.50−0.3

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 10.5 + 0 + (−0.3) = 0.2
Row 20.5 + 0 + (−0.3) = 0.2
Row 30.5 + 0 + (−0.3) = 0.2

0.2 + 0.2 + 0.2 = 0.6

5. Place it in the output
0.60
0.60

Output row 2, column 1.

Window row 1.

R_1=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2LaTeX: R_1=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2

Window row 2.

R_2=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2LaTeX: R_2=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2

Window row 3.

R_3=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2LaTeX: R_3=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2

Add the rows to obtain one output cell.

Y_{2,1}=(0.2)+(0.2)+(0.2)=0.6LaTeX: Y_{2,1}=(0.2)+(0.2)+(0.2)=0.6

Output position (2, 2)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (2, 2) into one output cell

Input rows 3–5, columns 3–5. The window moves 2 cells per output step.

1. Select the input window
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3
0.50.50.30.30.3

Only the outlined cells belong to this output.

2. Match the kernel
10−1
10−1
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
0.30−0.3
0.30−0.3
0.30−0.3

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 10.3 + 0 + (−0.3) = 0
Row 20.3 + 0 + (−0.3) = 0
Row 30.3 + 0 + (−0.3) = 0

0 + 0 + 0 = 0

5. Place it in the output
0.60
0.60

Output row 2, column 2.

Window row 1.

R_1=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0LaTeX: R_1=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0

Window row 2.

R_2=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0LaTeX: R_2=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0

Window row 3.

R_3=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0LaTeX: R_3=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0

Add the rows to obtain one output cell.

Y_{2,2}=(0)+(0)+(0)=0LaTeX: Y_{2,2}=(0)+(0)+(0)=0

Final output: \begin{bmatrix}0.6 & 0\\0.6 & 0\end{bmatrix}LaTeX: \begin{bmatrix}0.6 & 0\\0.6 & 0\end{bmatrix}. Each output came from its own complete window; stride changes the next window position, not the multiplication rule.

The filter compares left and right neighborhoods and responds to a vertical brightness boundary. Inputs are already normalized. No additional output-normalization rule is supplied, so the determined convolution is \begin{bmatrix}0.6 & 0\\0.6 & 0\end{bmatrix}LaTeX: \begin{bmatrix}0.6 & 0\\0.6 & 0\end{bmatrix}. If an examiner explicitly requests min-max normalization of this result, divide by 0.6 to obtain \begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}LaTeX: \begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}; that is a separate stated operation.