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F25 Q6a

Final examination December 2025 fully solved

Question

Question (10 marks). Calculate every activation shape and trainable-parameter count for: Input 128 × 128 × 3. → Conv K = 3, filters = 16, stride = 1, padding = 0. → Pool K = 2, stride = 2, padding = 0. → Conv K = 3, filters = 32, stride = 1, padding = 0. → Pool K = 2, stride = 2, padding = 0. → Flatten. → Dense 240 units. → Dense 120 units. → Dense 5 units. Also convolve the input A with the kernel K shown below, valid padding and stride 2.

Given matrices
Input A
31221
21210
32101
03012
11023
Kernel K
10−1
20−2
10−1

Stride 2; P = 0 (no padding).

Answer — step by step

Given

Input 128 × 128 × 3.

Conv K = 3, filters = 16, stride = 1, padding = 0.

Pool K = 2, stride = 2, padding = 0.

Conv K = 3, filters = 32, stride = 1, padding = 0.

Pool K = 2, stride = 2, padding = 0.

Dense 240 units.

Dense 120 units.

Dense 5 units.

Also convolve the input A with the kernel K shown below, valid padding and stride 2.

To find

Find every layer shape, its parameter count and the network total. Find every convolution output cell.

Formula — which one and why

Complete CNN calculation. Carry each output shape into the next layer. Convolution changes spatial size and channel count; pooling changes spatial size; flattening converts all activations into dense-layer inputs.

H,W,C = feature-map height, width and channels; K = kernel size; S = stride; P_pad = padding. One bias is assumed for each learned filter or dense neuron unless specified otherwise.

N_{\mathrm{out}}=\left\lfloor\frac{N+2P_{\mathrm{pad}}-K}{S}\right\rfloor+1LaTeX: N_{\mathrm{out}}=\left\lfloor\frac{N+2P_{\mathrm{pad}}-K}{S}\right\rfloor+1
P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}LaTeX: P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}
n_{\mathrm{flat}}=HWCLaTeX: n_{\mathrm{flat}}=HWC
P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}LaTeX: P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}
P_{\mathrm{pool}}=P_{\mathrm{flatten}}=0LaTeX: P_{\mathrm{pool}}=P_{\mathrm{flatten}}=0

Yaad rakhne ka cue: Ek layer ka output, next layer ka input; shape aur parameters alag count.

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.

Use output size \left\lfloor\frac{N+2P-K}{S}\right\rfloor+1LaTeX: \left\lfloor\frac{N+2P-K}{S}\right\rfloor+1, Conv weights K² × Cin × Cout and biases Cout, Dense weights Nin × Nout and biases Nout. Assume a bias per learned filter/unit, full input-channel connectivity and no trainable pooling parameters. ReLU and softmax activations themselves add no trainable parameters.

Layer-by-layer calculation

Carry the complete output shape of one layer into the next. Spatial size determines the number of activations; it does not multiply the number of shared convolution parameters.

Layer 1 — convolution

Layer 1: what changes and what is learned
Input shape128 × 128 × 3height × width × channels
Convolution16 filters · 3 × 3 · S = 1 · P = 0
Output shape126 × 126 × 16height × width × channels
Weights432
Biases16
Trainable parameters448

Spatial positions share the filter weights. Output depth equals the number of filters.

Input: 128 × 128 × 3. Use 16 filters of spatial size 3 × 3, stride 1, and padding 0 on each side.

1. Calculate height and width.

N_{\mathrm{out}}=\left\lfloor\frac{128+2(0)-3}{1}\right\rfloor+1=126LaTeX: N_{\mathrm{out}}=\left\lfloor\frac{128+2(0)-3}{1}\right\rfloor+1=126

2. Determine output channels. Each filter produces one feature map. Therefore the output is 126 × 126 × 16.

3. Count trainable values. Each filter receives all 3 input channels. Its weights are reused at every spatial position.

N_{\mathrm{weights}}=3\times3\times3\times16=432LaTeX: N_{\mathrm{weights}}=3\times3\times3\times16=432
N_{\mathrm{biases}}=16LaTeX: N_{\mathrm{biases}}=16
P_1=432+16=448LaTeX: P_1=432+16=448

Layer 2 — pooling

Layer 2: what changes and what is learned
Input shape126 × 126 × 16height × width × channels
Pooling2 × 2 window · S = 2
Output shape63 × 63 × 16height × width × channels
Weights0
Biases0
Trainable parameters0

Spatial dimensions change; the channel count stays the same. Pooling learns no weights.

Input: 126 × 126 × 16. Use a 2 × 2 window, stride 2, and zero padding.

N_{\mathrm{out}}=\left\lfloor\frac{126-2}{2}\right\rfloor+1=63LaTeX: N_{\mathrm{out}}=\left\lfloor\frac{126-2}{2}\right\rfloor+1=63

Pooling acts independently on each channel. The output is 63 × 63 × 16.

Trainable parameters: 0. Max/average pooling has no learned filter weights or biases.

Layer 3 — convolution

Layer 3: what changes and what is learned
Input shape63 × 63 × 16height × width × channels
Convolution32 filters · 3 × 3 · S = 1 · P = 0
Output shape61 × 61 × 32height × width × channels
Weights4,608
Biases32
Trainable parameters4,640

Spatial positions share the filter weights. Output depth equals the number of filters.

Input: 63 × 63 × 16. Use 32 filters of spatial size 3 × 3, stride 1, and padding 0 on each side.

1. Calculate height and width.

N_{\mathrm{out}}=\left\lfloor\frac{63+2(0)-3}{1}\right\rfloor+1=61LaTeX: N_{\mathrm{out}}=\left\lfloor\frac{63+2(0)-3}{1}\right\rfloor+1=61

2. Determine output channels. Each filter produces one feature map. Therefore the output is 61 × 61 × 32.

3. Count trainable values. Each filter receives all 16 input channels. Its weights are reused at every spatial position.

N_{\mathrm{weights}}=3\times3\times16\times32=4{,}608LaTeX: N_{\mathrm{weights}}=3\times3\times16\times32=4{,}608
N_{\mathrm{biases}}=32LaTeX: N_{\mathrm{biases}}=32
P_3=4{,}608+32=4{,}640LaTeX: P_3=4{,}608+32=4{,}640

Layer 4 — pooling

Layer 4: what changes and what is learned
Input shape61 × 61 × 32height × width × channels
Pooling2 × 2 window · S = 2
Output shape30 × 30 × 32height × width × channels
Weights0
Biases0
Trainable parameters0

Spatial dimensions change; the channel count stays the same. Pooling learns no weights.

Input: 61 × 61 × 32. Use a 2 × 2 window, stride 2, and zero padding.

N_{\mathrm{out}}=\left\lfloor\frac{61-2}{2}\right\rfloor+1=30LaTeX: N_{\mathrm{out}}=\left\lfloor\frac{61-2}{2}\right\rfloor+1=30

Pooling acts independently on each channel. The output is 30 × 30 × 32.

Trainable parameters: 0. Max/average pooling has no learned filter weights or biases.

The division gives 29.5; take its floor before adding 1. A partial window outside the unpadded input is not included.

Layer 5 — flatten

Layer 5: what changes and what is learned
Input shape30 × 30 × 32height × width × channels
FlattenRearrange activations
Output shape28,800output features
Weights0
Biases0
Trainable parameters0

The same activation values are rearranged into one vector. No parameter is learned.

Input: 30 × 30 × 32. Rearrange all activation values into one vector.

n_{\mathrm{flat}}=30\times30\times32=28{,}800LaTeX: n_{\mathrm{flat}}=30\times30\times32=28{,}800

Output: a vector with 28,800 entries. Flattening changes the arrangement, so it adds 0 trainable parameters.

Layer 6 — fully connected layer

Layer 6: what changes and what is learned
Input shape28,800input features
Dense layer240 neurons
Output shape240output features
Weights6,912,000
Biases240
Trainable parameters6,912,240

Every output neuron receives every input feature, with one bias per output neuron.

Input: 28,800 features. Output: 240 neurons. Every output neuron connects to every input feature and has one bias.

N_{\mathrm{weights}}=28{,}800\times240=6{,}912{,}000LaTeX: N_{\mathrm{weights}}=28{,}800\times240=6{,}912{,}000
N_{\mathrm{biases}}=240LaTeX: N_{\mathrm{biases}}=240
P_6=6{,}912{,}000+240=6{,}912{,}240LaTeX: P_6=6{,}912{,}000+240=6{,}912{,}240

Layer 7 — fully connected layer

Layer 7: what changes and what is learned
Input shape240input features
Dense layer120 neurons
Output shape120output features
Weights28,800
Biases120
Trainable parameters28,920

Every output neuron receives every input feature, with one bias per output neuron.

Input: 240 features. Output: 120 neurons. Every output neuron connects to every input feature and has one bias.

N_{\mathrm{weights}}=240\times120=28{,}800LaTeX: N_{\mathrm{weights}}=240\times120=28{,}800
N_{\mathrm{biases}}=120LaTeX: N_{\mathrm{biases}}=120
P_7=28{,}800+120=28{,}920LaTeX: P_7=28{,}800+120=28{,}920

Layer 8 — fully connected layer

Layer 8: what changes and what is learned
Input shape120input features
Dense layer5 neurons
Output shape5output features
Weights600
Biases5
Trainable parameters605

Every output neuron receives every input feature, with one bias per output neuron.

Input: 120 features. Output: 5 neurons. Every output neuron connects to every input feature and has one bias.

N_{\mathrm{weights}}=120\times5=600LaTeX: N_{\mathrm{weights}}=120\times5=600
N_{\mathrm{biases}}=5LaTeX: N_{\mathrm{biases}}=5
P_8=600+5=605LaTeX: P_8=600+5=605

Summary of the complete network

LayerSpatial or flatten calculationOutputWeightsBiasesTotal
1 conv\left\lfloor\frac{128+2 \times 0-3}{1}\right\rfloor+1LaTeX: \left\lfloor\frac{128+2 \times 0-3}{1}\right\rfloor+1126 x 126 x 1643216448
2 pool\left\lfloor\frac{126-2}{2}\right\rfloor+1LaTeX: \left\lfloor\frac{126-2}{2}\right\rfloor+163 x 63 x 16000
3 conv\left\lfloor\frac{63+2 \times 0-3}{1}\right\rfloor+1LaTeX: \left\lfloor\frac{63+2 \times 0-3}{1}\right\rfloor+161 x 61 x 324,608324,640
4 pool\left\lfloor\frac{61-2}{2}\right\rfloor+1LaTeX: \left\lfloor\frac{61-2}{2}\right\rfloor+130 x 30 x 32000
5 flat30 × 30 × 3228800000
6 dense28800 → 2402406,912,0002406,912,240
7 dense240 → 12012028,80012028,920
8 dense120 → 556005605

Parameter substitutions, in layer order:

Layer 1: (3 x 3 x 3 + 1) x 16 = 448.

Layer 3: (3 x 3 x 16 + 1) x 32 = 4,640.

Layer 6: (28800 + 1) x 240 = 6,912,240.

Layer 7: (240 + 1) x 120 = 28,920.

Layer 8: (120 + 1) x 5 = 605.

Total: 448 + 4,640 + 6,912,240 + 28,920 + 605 = 6,946,853 trainable parameters. Pooling changes activation size; it does not itself add learned weights. Always floor noninteger spatial dimensions.

Separate convolution calculation:

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
31221
21210
32101
03012
11023

Only the outlined cells belong to this output.

2. Match the kernel
10−1
20−2
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
30−2
40−4
30−1

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 13 + 0 + (−2) = 1
Row 24 + 0 + (−4) = 0
Row 33 + 0 + (−1) = 2

1 + 0 + 2 = 3

5. Place it in the output
35
3−7

Output row 1, column 1.

Window row 1.

R_1=(3)\times(1)+(1)\times(0)+(2)\times(-1)=1LaTeX: R_1=(3)\times(1)+(1)\times(0)+(2)\times(-1)=1

Window row 2.

R_2=(2)\times(2)+(1)\times(0)+(2)\times(-2)=0LaTeX: R_2=(2)\times(2)+(1)\times(0)+(2)\times(-2)=0

Window row 3.

R_3=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2LaTeX: R_3=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2

Add the rows to obtain one output cell.

Y_{1,1}=(1)+(0)+(2)=3LaTeX: Y_{1,1}=(1)+(0)+(2)=3

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
31221
21210
32101
03012
11023

Only the outlined cells belong to this output.

2. Match the kernel
10−1
20−2
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
20−1
400
10−1

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 12 + 0 + (−1) = 1
Row 24 + 0 + 0 = 4
Row 31 + 0 + (−1) = 0

1 + 4 + 0 = 5

5. Place it in the output
35
3−7

Output row 1, column 2.

Window row 1.

R_1=(2)\times(1)+(2)\times(0)+(1)\times(-1)=1LaTeX: R_1=(2)\times(1)+(2)\times(0)+(1)\times(-1)=1

Window row 2.

R_2=(2)\times(2)+(1)\times(0)+(0)\times(-2)=4LaTeX: R_2=(2)\times(2)+(1)\times(0)+(0)\times(-2)=4

Window row 3.

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

Add the rows to obtain one output cell.

Y_{1,2}=(1)+(4)+(0)=5LaTeX: Y_{1,2}=(1)+(4)+(0)=5

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
31221
21210
32101
03012
11023

Only the outlined cells belong to this output.

2. Match the kernel
10−1
20−2
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
30−1
000
100

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 13 + 0 + (−1) = 2
Row 20 + 0 + 0 = 0
Row 31 + 0 + 0 = 1

2 + 0 + 1 = 3

5. Place it in the output
35
3−7

Output row 2, column 1.

Window row 1.

R_1=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2LaTeX: R_1=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2

Window row 2.

R_2=(0)\times(2)+(3)\times(0)+(0)\times(-2)=0LaTeX: R_2=(0)\times(2)+(3)\times(0)+(0)\times(-2)=0

Window row 3.

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

Add the rows to obtain one output cell.

Y_{2,1}=(2)+(0)+(1)=3LaTeX: Y_{2,1}=(2)+(0)+(1)=3

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
31221
21210
32101
03012
11023

Only the outlined cells belong to this output.

2. Match the kernel
10−1
20−2
10−1

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell
10−1
00−4
00−3

Each cell is input × matching kernel coefficient.

4. Add the products row by row
Row 11 + 0 + (−1) = 0
Row 20 + 0 + (−4) = −4
Row 30 + 0 + (−3) = −3

0 + (−4) + (−3) = −7

5. Place it in the output
35
3−7

Output row 2, column 2.

Window row 1.

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

Window row 2.

R_2=(0)\times(2)+(1)\times(0)+(2)\times(-2)=-4LaTeX: R_2=(0)\times(2)+(1)\times(0)+(2)\times(-2)=-4

Window row 3.

R_3=(0)\times(1)+(2)\times(0)+(3)\times(-1)=-3LaTeX: R_3=(0)\times(1)+(2)\times(0)+(3)\times(-1)=-3

Add the rows to obtain one output cell.

Y_{2,2}=(0)+(-4)+(-3)=-7LaTeX: Y_{2,2}=(0)+(-4)+(-3)=-7

Final output: \begin{bmatrix}3 & 5\\3 & -7\end{bmatrix}LaTeX: \begin{bmatrix}3 & 5\\3 & -7\end{bmatrix}. Each output came from its own complete window; stride changes the next window position, not the multiplication rule.