Machine LearningConvolutional Neural Networks

The Convolution Operation

LevelIntermediate

Duration90 mins

TopicConvolutional Neural Networks

4 / 5

Dilation (Dilated/Atrous Convolutions)

Expanding Context Without Losing Resolution

The fundamental trade-off in CNN design is between receptive field size and spatial resolution. Large receptive fields capture more context, essential for understanding what's in an image. But the traditional way to grow receptive fields—through strided convolutions or pooling—reduces spatial resolution, discarding fine-grained detail.

For tasks like semantic segmentation (classifying every pixel), we need both: large context to understand the scene, and high resolution to make precise pixel-level predictions. This seems contradictory.

Dilated convolutions (also called atrous convolutions from the French 'à trous', meaning 'with holes') resolve this contradiction elegantly. By inserting gaps ('holes') between kernel elements, dilated convolutions exponentially expand receptive fields while maintaining the same number of parameters—and crucially, without any downsampling.

This page provides complete coverage of dilation: its mathematics, its implementation, its receptive field effects, and the architectures that leverage it.

What You Will Learn

By the end of this page, you will understand the mathematics of dilated convolution, calculate effective kernel sizes and receptive fields with dilation, appreciate why dilation is crucial for dense prediction, and understand gridding artifacts and how to avoid them.

The Receptive Field vs. Resolution Dilemma

To appreciate dilated convolutions, we must first understand the problem they solve.

The Classification Paradigm:

In image classification, we want a single label for the entire image. The architecture progressively reduces spatial dimensions (via stride-2 convolutions or pooling) while increasing channels. This is acceptable—even desirable—because:

We only need one output, not spatially-detailed outputs
Downsampling increases receptive fields rapidly
Reduced resolution means reduced computation

The Dense Prediction Challenge:

In semantic segmentation, we need a class label for every pixel. The output should match the input resolution. But if we downsample to grow receptive fields, we lose the very resolution we need for pixel-precise predictions.

Traditional Solutions (and Their Limitations):

Approach 1: Skip downsampling entirely

Only use stride-1 convolutions
Problem: Receptive field grows only linearly (K-1 per layer)
For 224×224 input, need hundreds of layers for global context

Approach 2: Encoder-decoder architecture

Downsample (encoder), then upsample (decoder)
Skip connections transfer high-resolution features
Problem: Upsampling introduces approximation errors; fine details may be lost

Receptive Field Growth: Standard vs. Dilated Convolutions
Approach	3 Layers (3×3 kernels)	Required Layers for 200+ RF	Resolution Loss
Standard conv (s=1)	RF = 7	~100 layers	No loss
Strided conv (s=2 every 2 layers)	RF = 22	~15 layers	Significant (÷8 or more)
Dilated conv (d=1,2,4)	RF = 15	~10 layers	No loss

The Dilated Convolution Promise

Dilated convolutions provide exponential receptive field growth without any spatial downsampling. A stack of dilated convolutions with dilation rates 1, 2, 4, 8, 16 can achieve a receptive field equivalent to ~31-layer standard CNNs—with only 5 layers and no resolution reduction.

Mathematical Definition of Dilated Convolution

Standard 2D Convolution (Review):

$$(I * K)[m, n] = \sum_{i=0}^{K_h-1} \sum_{j=0}^{K_w-1} I[m + i, n + j] \cdot K[i, j]$$

Adjacent kernel elements sample adjacent input positions.

Dilated 2D Convolution:

Introduce a dilation rate d (also called dilation factor or rate):

$$(I *d K)[m, n] = \sum{i=0}^{K_h-1} \sum_{j=0}^{K_w-1} I[m + d \cdot i, n + d \cdot j] \cdot K[i, j]$$

Kernel elements are spaced d positions apart in the input. The kernel 'reaches' further without additional weights.

Visual Interpretation:

A 3×3 kernel with dilation d=1 (standard) samples a 3×3 input region.

With d=2, the same 3×3 kernel samples positions:

(0,0), (0,2), (0,4)
(2,0), (2,2), (2,4)
(4,0), (4,2), (4,4)

The kernel now covers a 5×5 input region (but still with only 9 weights).

Effective Kernel Size:

A kernel of size K with dilation rate d effectively covers:

$$K_{eff} = K + (K - 1) \cdot (d - 1) = d \cdot (K - 1) + 1$$

Examples:

K=3, d=1: Keff = 3 (standard)
K=3, d=2: Keff = 5
K=3, d=4: Keff = 9
K=3, d=8: Keff = 17

With the same 9 parameters, we can cover regions from 3×3 up to arbitrarily large by increasing d.

Output Dimension with Dilation:

The output dimension formula adjusts for effective kernel size:

$$H_{out} = \left\lfloor \frac{H + 2p - d \cdot (K - 1) - 1}{s} \right\rfloor + 1$$

Simplifying for stride 1:

$$H_{out} = H + 2p - d \cdot (K - 1)$$

To achieve 'same' output (Hout = H) with stride 1:

$$p = \frac{d \cdot (K - 1)}{2}$$

For K=3 and d=2: p = 2. For K=3 and d=4: p = 4.

dilated_conv_math.py
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import numpy as np
 
def effective_kernel_size(kernel_size: int, dilation: int) -> int:
    """
    Calculate effective kernel size with dilation.
    
    Args:
        kernel_size: Original kernel size (K)
        dilation: Dilation rate (d)
    
    Returns:
        Effective kernel size (K_eff)
    """
    return kernel_size + (kernel_size - 1) * (dilation - 1)
 
 
def dilated_output_dim(
    input_dim: int, 
    kernel: int, 
    dilation: int, 
    padding: int, 
    stride: int
) -> int:
    """
    Calculate output dimension for dilated convolution.
    """
    k_eff = effective_kernel_size(kernel, dilation)
    return (input_dim + 2 * padding - k_eff) // stride + 1
 
 
def same_padding_dilated(kernel: int, dilation: int) -> int:
    """
    Calculate padding for 'same' output with dilation (stride 1).
    """
    k_eff = effective_kernel_size(kernel, dilation)
    return (k_eff - 1) // 2
 
 
def demonstrate_dilation():
    """
    Show how dilation affects effective kernel coverage.
    """
    K = 3  # 3×3 kernel
    
    print("3×3 Kernel with Different Dilation Rates:")
    print("-" * 50)
    
    for d in [1, 2, 4, 8, 16]:
        k_eff = effective_kernel_size(K, d)
        p_same = same_padding_dilated(K, d)
        print(f"d={d:2d}: Effective size = {k_eff:2d}×{k_eff:<2d}, "
              f"Padding for 'same' = {p_same}")
    
    print()
    print("Output dimensions for 64×64 input, 3×3 kernel, stride 1:")
    print("-" * 50)
    
    H = 64
    for d in [1, 2, 4, 8]:
        p = same_padding_dilated(K, d)
        H_out = dilated_output_dim(H, K, d, p, stride=1)
        print(f"d={d}, p={p}: Output = {H_out}×{H_out}")
 
 
if __name__ == "__main__":
    demonstrate_dilation()

Dilation Rate 1 = Standard Convolution

When d=1, dilated convolution reduces to standard convolution. All formulas generalize: Keff = K, and the output formula matches the standard case. Setting d=1 is how frameworks treat 'normal' convolutions as a special case of dilated.

Receptive Field Growth with Dilated Convolutions

The primary benefit of dilation is accelerated receptive field growth. Let's quantify this precisely.

Single Layer Contribution:

A dilated convolution with kernel K and dilation d contributes to receptive field growth by:

$$\Delta RF = d \cdot (K - 1)$$

Compare to standard convolution: ΔRF = K - 1. Dilation multiplies the contribution by d.

Stacked Dilated Convolutions:

Consider stacking L layers of 3×3 convolutions with dilation rates d₁, d₂, ..., dₗ (all stride 1):

$$RF = 1 + \sum_{l=1}^{L} d_l \cdot (K - 1) = 1 + 2 \sum_{l=1}^{L} d_l$$

For K=3, each layer adds 2·dₗ to the receptive field.

Exponentially Increasing Dilation:

The common pattern uses dilation rates that double: d = 1, 2, 4, 8, 16, ...

With L layers of 3×3 at these rates:

$$RF = 1 + 2 \cdot (1 + 2 + 4 + ... + 2^{L-1}) = 1 + 2 \cdot (2^L - 1) = 2^{L+1} - 1$$

The receptive field grows exponentially with depth!

Receptive Field Comparison: Standard vs. Dilated (3×3 kernels, stride 1)
Layers	Standard (all d=1)	Dilated (d=1,2,4,8,...)	Ratio
3	7	15 (d=1,2,4)	2.1×
5	11	63 (d=1,2,4,8,16)	5.7×
7	15	255 (d=1,2,...,64)	17×
10	21	2047 (d=1,2,...,512)	97×

Dense vs. Sparse Sampling:

There's an important subtlety: while dilated convolutions cover large regions, they sample sparsely within those regions.

A standard 3×3 conv samples all 9 positions in a 3×3 region.

A 3×3 conv with d=4 samples only 9 positions in a 9×9 region—gaps of 3 pixels between samples.

This sparse sampling isn't problematic when:

Features are inherently coarse-grained (large objects, smooth textures)
Mixed dilation rates are used (some dense, some sparse) to cover all scales
Deep stacks allow information to propagate across gaps

But it can cause gridding artifacts (covered later) if used naively.

The Power of Exponential Growth

Five layers with dilation rates 1, 2, 4, 8, 16 achieve a receptive field of 63—equivalent to 31 standard layers! This efficiency is why dilated convolutions are standard in semantic segmentation networks.

Implementation in Deep Learning Frameworks

All major deep learning frameworks support dilated convolutions as a parameter of their convolution layers.

PyTorch:

import torch.nn as nn

# Standard 3×3 convolution (d=1 implicitly)
conv_standard = nn.Conv2d(64, 128, kernel_size=3, padding=1, dilation=1)

# Dilated 3×3 convolution with d=2
conv_dilated = nn.Conv2d(64, 128, kernel_size=3, padding=2, dilation=2)

# Dilated 3×3 convolution with d=4
conv_dilated_4 = nn.Conv2d(64, 128, kernel_size=3, padding=4, dilation=4)

TensorFlow/Keras:

import tensorflow as tf

# Dilated convolution with d=2
conv_dilated = tf.keras.layers.Conv2D(
    128, 
    kernel_size=3, 
    padding='same', 
    dilation_rate=2
)

dilated_conv_implementation.py
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import torch
import torch.nn as nn
import torch.nn.functional as F
 
class DilatedConvBlock(nn.Module):
    """
    A block of dilated convolutions with exponentially increasing rates.
    Commonly used in semantic segmentation networks.
    """
    def __init__(
        self, 
        in_channels: int, 
        out_channels: int, 
        num_layers: int = 4,
        base_dilation: int = 1
    ):
        super().__init__()
        
        self.layers = nn.ModuleList()
        dilation = base_dilation
        
        for i in range(num_layers):
            # Calculate padding for 'same' output
            padding = dilation  # For 3×3 kernel
            
            self.layers.append(nn.Sequential(
                nn.Conv2d(
                    in_channels if i == 0 else out_channels,
                    out_channels,
                    kernel_size=3,
                    padding=padding,
                    dilation=dilation
                ),
                nn.BatchNorm2d(out_channels),
                nn.ReLU(inplace=True)
            ))
            
            # Double dilation rate for next layer
            dilation *= 2
    
    def forward(self, x):
        for layer in self.layers:
            x = layer(x)
        return x
 
 
class ASPPModule(nn.Module):
    """
    Atrous Spatial Pyramid Pooling (ASPP) from DeepLab.
    Applies parallel dilated convolutions at multiple rates.
    """
    def __init__(
        self, 
        in_channels: int, 
        out_channels: int,
        rates: list = [6, 12, 18]
    ):
        super().__init__()
        
        # 1×1 convolution
        self.conv1x1 = nn.Sequential(
            nn.Conv2d(in_channels, out_channels, 1, bias=False),
            nn.BatchNorm2d(out_channels),
            nn.ReLU(inplace=True)
        )
        
        # Dilated 3×3 convolutions at multiple rates
        self.dilated_convs = nn.ModuleList()
        for rate in rates:
            self.dilated_convs.append(nn.Sequential(
                nn.Conv2d(
                    in_channels, out_channels, 3,
                    padding=rate, dilation=rate, bias=False
                ),
                nn.BatchNorm2d(out_channels),
                nn.ReLU(inplace=True)
            ))
        
        # Global average pooling branch
        self.gap = nn.Sequential(
            nn.AdaptiveAvgPool2d(1),
            nn.Conv2d(in_channels, out_channels, 1, bias=False),
            nn.BatchNorm2d(out_channels),
            nn.ReLU(inplace=True)
        )
        
        # Fusion 1×1 conv
        num_branches = 1 + len(rates) + 1  # 1×1 + dilated + GAP
        self.fusion = nn.Sequential(
            nn.Conv2d(num_branches * out_channels, out_channels, 1, bias=False),
            nn.BatchNorm2d(out_channels),
            nn.ReLU(inplace=True)
        )
    
    def forward(self, x):
        size = x.shape[2:]
        
        branches = [self.conv1x1(x)]
        
        for dilated_conv in self.dilated_convs:
            branches.append(dilated_conv(x))
        
        # Upsample GAP output to match spatial size
        gap_out = self.gap(x)
        gap_out = F.interpolate(gap_out, size=size, mode='bilinear', align_corners=True)
        branches.append(gap_out)
        
        # Concatenate all branches
        concat = torch.cat(branches, dim=1)
        
        return self.fusion(concat)
 
 
def demonstrate_dilated_conv():
    """
    Show dilated convolutions in action.
    """
    x = torch.randn(1, 64, 128, 128)
    
    print("Input shape:", x.shape)
    print()
    
    # Single dilated conv
    conv_d1 = nn.Conv2d(64, 64, 3, padding=1, dilation=1)
    conv_d2 = nn.Conv2d(64, 64, 3, padding=2, dilation=2)
    conv_d4 = nn.Conv2d(64, 64, 3, padding=4, dilation=4)
    
    print("Single layer outputs (all preserve 128×128):")
    print(f"  d=1: {conv_d1(x).shape}")
    print(f"  d=2: {conv_d2(x).shape}")
    print(f"  d=4: {conv_d4(x).shape}")
    print()
    
    # Dilated block
    block = DilatedConvBlock(64, 128, num_layers=4)
    y = block(x)
    print(f"Dilated block (4 layers, d=1,2,4,8):")
    print(f"  Output: {y.shape}")
    print(f"  Receptive field: {1 + 2*(1+2+4+8)} = 31")
    print()
    
    # ASPP module
    aspp = ASPPModule(64, 256, rates=[6, 12, 18])
    z = aspp(x)
    print(f"ASPP output: {z.shape}")
 
 
if __name__ == "__main__":
    demonstrate_dilated_conv()

Memory and Compute Considerations

Dilated convolutions have the same parameter count and theoretical FLOPs as standard convolutions (same kernel size). However, the sparse memory access pattern can be less cache-friendly on some hardware, potentially reducing practical speed. Modern frameworks and hardware have largely optimized this.

Gridding Artifacts and How to Avoid Them

Dilated convolutions introduce a subtle but important problem: gridding artifacts (also called checkerboard artifacts or the 'gridding effect').

The Problem:

When using dilated convolutions with rate d, the kernel samples positions 0, d, 2d, 3d, ... in each dimension. Intermediate positions (1, 2, ..., d-1, d+1, ...) are never directly sampled by this layer.

If multiple consecutive layers use the same dilation rate d, information propagates along 'grids' of spacing d, creating d² independent pathways that never interact. The output shows visible grid patterns.

Visual Intuition:

Consider three consecutive 3×3 convolutions with d=2:

Each layer samples every other pixel. The red grid of pixels (even positions) never mixes with the blue grid (odd positions). The network effectively processes d² separate sub-images.

Symptoms:

Blocky, grid-like output patterns
Reduced prediction smoothness
Visible seams at dilation boundaries

Solutions to Gridding Artifacts

•Hybrid Dilation Rates (HDC): Avoid using consecutive layers with the same or proportional dilation rates. Mix rates like (1, 2, 5, 1, 2, 5) instead of (2, 2, 2) or (2, 4, 8). Ensures all positions are sampled.
•Interleave with Standard Convolutions: Alternate dilated and non-dilated (d=1) layers. The d=1 layers 'fill in' the gaps from dilated layers.
•Use Non-Power-of-2 Rates: Dilation rates like (1, 2, 5, 9) with different prime factors reduce grid alignment compared to (1, 2, 4, 8).
•Multi-Scale Aggregation (ASPP): Apply multiple dilation rates in parallel and combine. Information from all rates fills in each other's gaps.
•Smoothing/Upsampling: Apply smoothing operations or bilinear upsampling after dilated layers to blend grid artifacts.

gridding_solutions.py
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import torch
import torch.nn as nn
 
class NaiveDilatedStack(nn.Module):
    """
    Problematic: Same dilation rate repeated → gridding artifacts.
    """
    def __init__(self, channels):
        super().__init__()
        # BAD: Three layers all with d=2
        self.conv1 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv2 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv3 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
    
    def forward(self, x):
        return self.conv3(self.conv2(self.conv1(x)))
 
 
class HDCDilatedStack(nn.Module):
    """
    Hybrid Dilated Convolution: varied rates avoid gridding.
    Inspired by "Understanding Convolution for Semantic Segmentation" (Yu et al.)
    """
    def __init__(self, channels):
        super().__init__()
        # GOOD: Rates (1, 2, 5) - no common factors allow full coverage
        self.conv1 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
        self.conv2 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv3 = nn.Conv2d(channels, channels, 3, padding=5, dilation=5)
    
    def forward(self, x):
        return self.conv3(self.conv2(self.conv1(x)))
 
 
class InterleavedDilatedStack(nn.Module):
    """
    Alternating dilated and standard convolutions.
    """
    def __init__(self, channels):
        super().__init__()
        # d=1 layers fill in gaps from d>1 layers
        self.conv1 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
        self.conv2 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv3 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
        self.conv4 = nn.Conv2d(channels, channels, 3, padding=4, dilation=4)
        self.conv5 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
    
    def forward(self, x):
        x = self.conv1(x)
        x = self.conv2(x)
        x = self.conv3(x)
        x = self.conv4(x)
        x = self.conv5(x)
        return x
 
 
def analyze_coverage(dilation_rates: list) -> dict:
    """
    Analyze which input positions contribute to each output position
    for a stack of 3×3 dilated convolutions.
    
    This is a simplified analysis of potential gridding.
    """
    print(f"Dilation rates: {dilation_rates}")
    
    # Check if rates share common factors (potential gridding)
    from math import gcd
    from functools import reduce
    
    common = reduce(gcd, dilation_rates)
    if common > 1:
        print(f"  ⚠️ Common factor: {common} - potential gridding")
    
    # Check consecutive equal rates
    for i in range(len(dilation_rates) - 1):
        if dilation_rates[i] == dilation_rates[i+1]:
            print(f"  ⚠️ Consecutive equal rates at positions {i}, {i+1}")
    
    # Compute effective receptive field
    rf = 1 + 2 * sum(dilation_rates)
    print(f"  Receptive field: {rf}")
    print()
 
 
if __name__ == "__main__":
    # Analyze different dilation patterns
    analyze_coverage([2, 2, 2])  # Bad: gridding
    analyze_coverage([1, 2, 4, 8])  # Classic exponential
    analyze_coverage([1, 2, 5, 1, 2, 5])  # HDC pattern
    analyze_coverage([1, 2, 3, 4, 5])  # Smooth increase

Gridding in Practice

Gridding artifacts are most visible in dense prediction outputs (segmentation masks, depth maps). For classification with global pooling, they're less problematic since spatial structure is discarded. Always visualize intermediate activations when debugging dilated networks.

Architectures Leveraging Dilated Convolutions

Dilated convolutions are central to several landmark architectures, particularly in semantic segmentation.

DeepLab Family:

The DeepLab series (v1, v2, v3, v3+) pioneered the use of atrous (dilated) convolutions in CNNs.

DeepLab v2/v3 (ASPP):

Atrous Spatial Pyramid Pooling
Parallel dilated convolutions at rates like (6, 12, 18)
Multi-scale feature aggregation captures objects of different sizes
Combined with global average pooling for image-level context

DeepLab v3+:

Encoder-decoder with atrous separable convolutions
ASPP in the encoder, refined upsampling in decoder
State-of-the-art segmentation performance

Architectures Using Dilated Convolutions
Architecture	Dilation Strategy	Task	Key Innovation
DeepLab v1/v2	Dense extraction at d=6 or d=12	Segmentation	First use of atrous conv for dense prediction
DeepLab v3	ASPP with d=(6,12,18)	Segmentation	Multi-scale atrous pooling
DeepLab v3+	ASPP + decoder	Segmentation	Encoder-decoder with atrous convs
Dilated ResNet	Replace stride-2 with d=2	Various	Maintains resolution in backbone
WaveNet	d=1,2,4,8,...,512 stacked	Audio generation	Causal dilated convs for large temporal RF
PSPNet	Pyramid Pooling + dilated	Segmentation	Combines pooling and dilation

Dilated ResNet/VGG for Dense Prediction:

A common technique modifies classification backbones for dense prediction:

Take a pretrained ResNet/VGG
Replace later stride-2 convolutions with stride-1, dilation-2
Maintain resolution while preserving receptive field
Output stride becomes 8 or 16 instead of 32

For example, ResNet-101 designed for 224×224 classification (output 7×7) can be modified to output 28×28 for a 224×224 input—7× more spatial detail for segmentation.

WaveNet (Audio):

WaveNet uses causal dilated convolutions for audio generation:

1D convolutions along time
Dilation rates: 1, 2, 4, 8, 16, ..., 512 (repeated blocks)
Achieves receptive fields covering seconds of audio
Enables high-quality speech synthesis

This demonstrates that dilation benefits any domain requiring large context from sequential/spatial data.

Converting Classification Networks for Segmentation

To adapt any classification CNN (ResNet, VGG, etc.) for dense prediction: identify downsampling layers (stride 2), replace stride with dilation (starting from target output stride), adjust kernel padding for 'same' behavior. Most deep learning frameworks provide 'dilate' utilities for this transformation.

Dilation vs. Other Multi-Scale Approaches

Dilated convolutions are one of several techniques for multi-scale feature extraction. Understanding the alternatives helps in choosing the right approach.

Image Pyramids:

Process the image at multiple resolutions
Run the same network on each scale
Combine predictions

Pros: True multi-scale, mathematically explicit Cons: Computational cost multiplied by number of scales

Feature Pyramids (FPN):

Build hierarchical features from backbone stages
Top-down pathway with lateral connections
Multi-scale features with shared computation

Pros: Efficient, inherent in encoder-decoder Cons: Lower resolution features lose detail

Larger Kernels:

Use 5×5, 7×7, or larger kernels
Directly capture larger patterns

Pros: Simple, dense sampling Cons: Parameter count grows as K², expensive

Multi-Scale Approaches Comparison
Approach	Parameters	Computation	Resolution	Implementation
Standard conv	K²	H×W×K²	Preserved	Simple
Dilated conv	K²	H×W×K²	Preserved	Simple, sparse access
Larger kernel	(dK)²	H×W×(dK)²	Preserved	Simple, expensive
Image pyramid	K² (shared)	Nx(H×W×K²)	Multiple	Complex pipeline
Feature pyramid	K² per level	~2x backbone	Multiple	Architectural change
Pooling pyramid	Minimal	~1.3x	Reduced	Simple additions

When to Use Dilation:

✓ Dense prediction tasks (segmentation, depth estimation) where resolution matters ✓ Limited computation budget (same FLOPs as standard conv) ✓ Need large receptive fields without massive kernel counts ✓ Adapting classification backbones for dense output

When Alternatives May Be Better:

Object detection across extreme scales: Feature pyramids often outperform
Very fine-grained tasks: Large kernels with dense sampling may capture detail better
Small inputs: Dilation overhead may not be worth it for 32×32 images

Common Hybrid Approaches:

Modern architectures often combine techniques:

FPN backbone + dilated head
ASPP (dilation) + pyramid pooling
Encoder-decoder with dilation in encoder

The Trend Toward Attention

Vision Transformers and attention mechanisms offer another path to large receptive fields (global attention spans the entire image). Dilation remains valuable in ConvNet-based architectures and hybrid designs, but attention-based methods increasingly dominate for tasks where global context matters.

Summary: Dilation for Efficient Large-Context Convolution

Dilated convolutions provide an elegant solution to the receptive field vs. resolution trade-off that defines dense prediction tasks. Let's consolidate the key insights:

Key Takeaways

•Sparse Kernel Sampling: Dilation inserts gaps between kernel elements, expanding coverage without adding parameters. Effective kernel size: Keff = d(K-1)+1.
•Exponential Receptive Field Growth: Stacking layers with d=1,2,4,8,... grows receptive field as 2^(L+1)-1, far faster than standard convolutions.
•Resolution Preservation: Unlike stride>1, dilation maintains output spatial dimensions (with appropriate padding).
•Gridding Artifacts: Repeated dilation rates cause sparse, non-interacting grids. Use HDC patterns or interleaved standard convs to avoid.
•Key Architectures: DeepLab (ASPP), WaveNet, dilated ResNets all leverage dilation for their domains.
•Practical Guidelines: Mix dilation rates, avoid consecutive equal rates, ensure all positions are sampled across layers.

What's Next:

The next page covers implementation details—the practical engineering aspects of convolutional operations. We'll examine different algorithmic approaches (im2col, FFT, Winograd), memory layouts (NCHW vs. NHWC), hardware acceleration considerations, and the tradeoffs that determine real-world performance.

Dilation Mastered

You now understand dilated convolutions from theory through practice. This knowledge is essential for dense prediction tasks where understanding global context while preserving spatial detail is paramount. Whether adapting classification backbones or designing custom segmentation networks, dilation is a fundamental tool in your CNN arsenal.

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Machine LearningConvolutional Neural Networks

The Convolution Operation

LevelIntermediate

Duration90 mins

TopicConvolutional Neural Networks

4 / 5

Dilation (Dilated/Atrous Convolutions)

Expanding Context Without Losing Resolution

This page provides complete coverage of dilation: its mathematics, its implementation, its receptive field effects, and the architectures that leverage it.

What You Will Learn

The Receptive Field vs. Resolution Dilemma

To appreciate dilated convolutions, we must first understand the problem they solve.

The Classification Paradigm:

We only need one output, not spatially-detailed outputs
Downsampling increases receptive fields rapidly
Reduced resolution means reduced computation

The Dense Prediction Challenge:

Traditional Solutions (and Their Limitations):

Approach 1: Skip downsampling entirely

Only use stride-1 convolutions
Problem: Receptive field grows only linearly (K-1 per layer)
For 224×224 input, need hundreds of layers for global context

Approach 2: Encoder-decoder architecture

Downsample (encoder), then upsample (decoder)
Skip connections transfer high-resolution features
Problem: Upsampling introduces approximation errors; fine details may be lost

Receptive Field Growth: Standard vs. Dilated Convolutions
Approach	3 Layers (3×3 kernels)	Required Layers for 200+ RF	Resolution Loss
Standard conv (s=1)	RF = 7	~100 layers	No loss
Strided conv (s=2 every 2 layers)	RF = 22	~15 layers	Significant (÷8 or more)
Dilated conv (d=1,2,4)	RF = 15	~10 layers	No loss

The Dilated Convolution Promise

Mathematical Definition of Dilated Convolution

Standard 2D Convolution (Review):

$$(I * K)[m, n] = \sum_{i=0}^{K_h-1} \sum_{j=0}^{K_w-1} I[m + i, n + j] \cdot K[i, j]$$

Adjacent kernel elements sample adjacent input positions.

Dilated 2D Convolution:

Introduce a dilation rate d (also called dilation factor or rate):

$$(I *d K)[m, n] = \sum{i=0}^{K_h-1} \sum_{j=0}^{K_w-1} I[m + d \cdot i, n + d \cdot j] \cdot K[i, j]$$

Kernel elements are spaced d positions apart in the input. The kernel 'reaches' further without additional weights.

Visual Interpretation:

A 3×3 kernel with dilation d=1 (standard) samples a 3×3 input region.

With d=2, the same 3×3 kernel samples positions:

(0,0), (0,2), (0,4)
(2,0), (2,2), (2,4)
(4,0), (4,2), (4,4)

The kernel now covers a 5×5 input region (but still with only 9 weights).

Effective Kernel Size:

A kernel of size K with dilation rate d effectively covers:

$$K_{eff} = K + (K - 1) \cdot (d - 1) = d \cdot (K - 1) + 1$$

Examples:

K=3, d=1: Keff = 3 (standard)
K=3, d=2: Keff = 5
K=3, d=4: Keff = 9
K=3, d=8: Keff = 17

With the same 9 parameters, we can cover regions from 3×3 up to arbitrarily large by increasing d.

Output Dimension with Dilation:

The output dimension formula adjusts for effective kernel size:

$$H_{out} = \left\lfloor \frac{H + 2p - d \cdot (K - 1) - 1}{s} \right\rfloor + 1$$

Simplifying for stride 1:

$$H_{out} = H + 2p - d \cdot (K - 1)$$

To achieve 'same' output (Hout = H) with stride 1:

$$p = \frac{d \cdot (K - 1)}{2}$$

For K=3 and d=2: p = 2. For K=3 and d=4: p = 4.

dilated_conv_math.py
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import numpy as np
 
def effective_kernel_size(kernel_size: int, dilation: int) -> int:
    """
    Calculate effective kernel size with dilation.
    
    Args:
        kernel_size: Original kernel size (K)
        dilation: Dilation rate (d)
    
    Returns:
        Effective kernel size (K_eff)
    """
    return kernel_size + (kernel_size - 1) * (dilation - 1)
 
 
def dilated_output_dim(
    input_dim: int, 
    kernel: int, 
    dilation: int, 
    padding: int, 
    stride: int
) -> int:
    """
    Calculate output dimension for dilated convolution.
    """
    k_eff = effective_kernel_size(kernel, dilation)
    return (input_dim + 2 * padding - k_eff) // stride + 1
 
 
def same_padding_dilated(kernel: int, dilation: int) -> int:
    """
    Calculate padding for 'same' output with dilation (stride 1).
    """
    k_eff = effective_kernel_size(kernel, dilation)
    return (k_eff - 1) // 2
 
 
def demonstrate_dilation():
    """
    Show how dilation affects effective kernel coverage.
    """
    K = 3  # 3×3 kernel
    
    print("3×3 Kernel with Different Dilation Rates:")
    print("-" * 50)
    
    for d in [1, 2, 4, 8, 16]:
        k_eff = effective_kernel_size(K, d)
        p_same = same_padding_dilated(K, d)
        print(f"d={d:2d}: Effective size = {k_eff:2d}×{k_eff:<2d}, "
              f"Padding for 'same' = {p_same}")
    
    print()
    print("Output dimensions for 64×64 input, 3×3 kernel, stride 1:")
    print("-" * 50)
    
    H = 64
    for d in [1, 2, 4, 8]:
        p = same_padding_dilated(K, d)
        H_out = dilated_output_dim(H, K, d, p, stride=1)
        print(f"d={d}, p={p}: Output = {H_out}×{H_out}")
 
 
if __name__ == "__main__":
    demonstrate_dilation()

Dilation Rate 1 = Standard Convolution

Receptive Field Growth with Dilated Convolutions

The primary benefit of dilation is accelerated receptive field growth. Let's quantify this precisely.

Single Layer Contribution:

A dilated convolution with kernel K and dilation d contributes to receptive field growth by:

$$\Delta RF = d \cdot (K - 1)$$

Compare to standard convolution: ΔRF = K - 1. Dilation multiplies the contribution by d.

Stacked Dilated Convolutions:

Consider stacking L layers of 3×3 convolutions with dilation rates d₁, d₂, ..., dₗ (all stride 1):

$$RF = 1 + \sum_{l=1}^{L} d_l \cdot (K - 1) = 1 + 2 \sum_{l=1}^{L} d_l$$

For K=3, each layer adds 2·dₗ to the receptive field.

Exponentially Increasing Dilation:

The common pattern uses dilation rates that double: d = 1, 2, 4, 8, 16, ...

With L layers of 3×3 at these rates:

$$RF = 1 + 2 \cdot (1 + 2 + 4 + ... + 2^{L-1}) = 1 + 2 \cdot (2^L - 1) = 2^{L+1} - 1$$

The receptive field grows exponentially with depth!

Receptive Field Comparison: Standard vs. Dilated (3×3 kernels, stride 1)
Layers	Standard (all d=1)	Dilated (d=1,2,4,8,...)	Ratio
3	7	15 (d=1,2,4)	2.1×
5	11	63 (d=1,2,4,8,16)	5.7×
7	15	255 (d=1,2,...,64)	17×
10	21	2047 (d=1,2,...,512)	97×

Dense vs. Sparse Sampling:

There's an important subtlety: while dilated convolutions cover large regions, they sample sparsely within those regions.

A standard 3×3 conv samples all 9 positions in a 3×3 region.

A 3×3 conv with d=4 samples only 9 positions in a 9×9 region—gaps of 3 pixels between samples.

This sparse sampling isn't problematic when:

Features are inherently coarse-grained (large objects, smooth textures)
Mixed dilation rates are used (some dense, some sparse) to cover all scales
Deep stacks allow information to propagate across gaps

But it can cause gridding artifacts (covered later) if used naively.

The Power of Exponential Growth

Implementation in Deep Learning Frameworks

All major deep learning frameworks support dilated convolutions as a parameter of their convolution layers.

PyTorch:

import torch.nn as nn

# Standard 3×3 convolution (d=1 implicitly)
conv_standard = nn.Conv2d(64, 128, kernel_size=3, padding=1, dilation=1)

# Dilated 3×3 convolution with d=2
conv_dilated = nn.Conv2d(64, 128, kernel_size=3, padding=2, dilation=2)

# Dilated 3×3 convolution with d=4
conv_dilated_4 = nn.Conv2d(64, 128, kernel_size=3, padding=4, dilation=4)

TensorFlow/Keras:

import tensorflow as tf

# Dilated convolution with d=2
conv_dilated = tf.keras.layers.Conv2D(
    128, 
    kernel_size=3, 
    padding='same', 
    dilation_rate=2
)

dilated_conv_implementation.py
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import torch
import torch.nn as nn
import torch.nn.functional as F
 
class DilatedConvBlock(nn.Module):
    """
    A block of dilated convolutions with exponentially increasing rates.
    Commonly used in semantic segmentation networks.
    """
    def __init__(
        self, 
        in_channels: int, 
        out_channels: int, 
        num_layers: int = 4,
        base_dilation: int = 1
    ):
        super().__init__()
        
        self.layers = nn.ModuleList()
        dilation = base_dilation
        
        for i in range(num_layers):
            # Calculate padding for 'same' output
            padding = dilation  # For 3×3 kernel
            
            self.layers.append(nn.Sequential(
                nn.Conv2d(
                    in_channels if i == 0 else out_channels,
                    out_channels,
                    kernel_size=3,
                    padding=padding,
                    dilation=dilation
                ),
                nn.BatchNorm2d(out_channels),
                nn.ReLU(inplace=True)
            ))
            
            # Double dilation rate for next layer
            dilation *= 2
    
    def forward(self, x):
        for layer in self.layers:
            x = layer(x)
        return x
 
 
class ASPPModule(nn.Module):
    """
    Atrous Spatial Pyramid Pooling (ASPP) from DeepLab.
    Applies parallel dilated convolutions at multiple rates.
    """
    def __init__(
        self, 
        in_channels: int, 
        out_channels: int,
        rates: list = [6, 12, 18]
    ):
        super().__init__()
        
        # 1×1 convolution
        self.conv1x1 = nn.Sequential(
            nn.Conv2d(in_channels, out_channels, 1, bias=False),
            nn.BatchNorm2d(out_channels),
            nn.ReLU(inplace=True)
        )
        
        # Dilated 3×3 convolutions at multiple rates
        self.dilated_convs = nn.ModuleList()
        for rate in rates:
            self.dilated_convs.append(nn.Sequential(
                nn.Conv2d(
                    in_channels, out_channels, 3,
                    padding=rate, dilation=rate, bias=False
                ),
                nn.BatchNorm2d(out_channels),
                nn.ReLU(inplace=True)
            ))
        
        # Global average pooling branch
        self.gap = nn.Sequential(
            nn.AdaptiveAvgPool2d(1),
            nn.Conv2d(in_channels, out_channels, 1, bias=False),
            nn.BatchNorm2d(out_channels),
            nn.ReLU(inplace=True)
        )
        
        # Fusion 1×1 conv
        num_branches = 1 + len(rates) + 1  # 1×1 + dilated + GAP
        self.fusion = nn.Sequential(
            nn.Conv2d(num_branches * out_channels, out_channels, 1, bias=False),
            nn.BatchNorm2d(out_channels),
            nn.ReLU(inplace=True)
        )
    
    def forward(self, x):
        size = x.shape[2:]
        
        branches = [self.conv1x1(x)]
        
        for dilated_conv in self.dilated_convs:
            branches.append(dilated_conv(x))
        
        # Upsample GAP output to match spatial size
        gap_out = self.gap(x)
        gap_out = F.interpolate(gap_out, size=size, mode='bilinear', align_corners=True)
        branches.append(gap_out)
        
        # Concatenate all branches
        concat = torch.cat(branches, dim=1)
        
        return self.fusion(concat)
 
 
def demonstrate_dilated_conv():
    """
    Show dilated convolutions in action.
    """
    x = torch.randn(1, 64, 128, 128)
    
    print("Input shape:", x.shape)
    print()
    
    # Single dilated conv
    conv_d1 = nn.Conv2d(64, 64, 3, padding=1, dilation=1)
    conv_d2 = nn.Conv2d(64, 64, 3, padding=2, dilation=2)
    conv_d4 = nn.Conv2d(64, 64, 3, padding=4, dilation=4)
    
    print("Single layer outputs (all preserve 128×128):")
    print(f"  d=1: {conv_d1(x).shape}")
    print(f"  d=2: {conv_d2(x).shape}")
    print(f"  d=4: {conv_d4(x).shape}")
    print()
    
    # Dilated block
    block = DilatedConvBlock(64, 128, num_layers=4)
    y = block(x)
    print(f"Dilated block (4 layers, d=1,2,4,8):")
    print(f"  Output: {y.shape}")
    print(f"  Receptive field: {1 + 2*(1+2+4+8)} = 31")
    print()
    
    # ASPP module
    aspp = ASPPModule(64, 256, rates=[6, 12, 18])
    z = aspp(x)
    print(f"ASPP output: {z.shape}")
 
 
if __name__ == "__main__":
    demonstrate_dilated_conv()

Memory and Compute Considerations

Gridding Artifacts and How to Avoid Them

Dilated convolutions introduce a subtle but important problem: gridding artifacts (also called checkerboard artifacts or the 'gridding effect').

The Problem:

Visual Intuition:

Consider three consecutive 3×3 convolutions with d=2:

Each layer samples every other pixel. The red grid of pixels (even positions) never mixes with the blue grid (odd positions). The network effectively processes d² separate sub-images.

Symptoms:

Blocky, grid-like output patterns
Reduced prediction smoothness
Visible seams at dilation boundaries

Solutions to Gridding Artifacts

•Hybrid Dilation Rates (HDC): Avoid using consecutive layers with the same or proportional dilation rates. Mix rates like (1, 2, 5, 1, 2, 5) instead of (2, 2, 2) or (2, 4, 8). Ensures all positions are sampled.
•Interleave with Standard Convolutions: Alternate dilated and non-dilated (d=1) layers. The d=1 layers 'fill in' the gaps from dilated layers.
•Use Non-Power-of-2 Rates: Dilation rates like (1, 2, 5, 9) with different prime factors reduce grid alignment compared to (1, 2, 4, 8).
•Multi-Scale Aggregation (ASPP): Apply multiple dilation rates in parallel and combine. Information from all rates fills in each other's gaps.
•Smoothing/Upsampling: Apply smoothing operations or bilinear upsampling after dilated layers to blend grid artifacts.

gridding_solutions.py
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import torch
import torch.nn as nn
 
class NaiveDilatedStack(nn.Module):
    """
    Problematic: Same dilation rate repeated → gridding artifacts.
    """
    def __init__(self, channels):
        super().__init__()
        # BAD: Three layers all with d=2
        self.conv1 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv2 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv3 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
    
    def forward(self, x):
        return self.conv3(self.conv2(self.conv1(x)))
 
 
class HDCDilatedStack(nn.Module):
    """
    Hybrid Dilated Convolution: varied rates avoid gridding.
    Inspired by "Understanding Convolution for Semantic Segmentation" (Yu et al.)
    """
    def __init__(self, channels):
        super().__init__()
        # GOOD: Rates (1, 2, 5) - no common factors allow full coverage
        self.conv1 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
        self.conv2 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv3 = nn.Conv2d(channels, channels, 3, padding=5, dilation=5)
    
    def forward(self, x):
        return self.conv3(self.conv2(self.conv1(x)))
 
 
class InterleavedDilatedStack(nn.Module):
    """
    Alternating dilated and standard convolutions.
    """
    def __init__(self, channels):
        super().__init__()
        # d=1 layers fill in gaps from d>1 layers
        self.conv1 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
        self.conv2 = nn.Conv2d(channels, channels, 3, padding=2, dilation=2)
        self.conv3 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
        self.conv4 = nn.Conv2d(channels, channels, 3, padding=4, dilation=4)
        self.conv5 = nn.Conv2d(channels, channels, 3, padding=1, dilation=1)
    
    def forward(self, x):
        x = self.conv1(x)
        x = self.conv2(x)
        x = self.conv3(x)
        x = self.conv4(x)
        x = self.conv5(x)
        return x
 
 
def analyze_coverage(dilation_rates: list) -> dict:
    """
    Analyze which input positions contribute to each output position
    for a stack of 3×3 dilated convolutions.
    
    This is a simplified analysis of potential gridding.
    """
    print(f"Dilation rates: {dilation_rates}")
    
    # Check if rates share common factors (potential gridding)
    from math import gcd
    from functools import reduce
    
    common = reduce(gcd, dilation_rates)
    if common > 1:
        print(f"  ⚠️ Common factor: {common} - potential gridding")
    
    # Check consecutive equal rates
    for i in range(len(dilation_rates) - 1):
        if dilation_rates[i] == dilation_rates[i+1]:
            print(f"  ⚠️ Consecutive equal rates at positions {i}, {i+1}")
    
    # Compute effective receptive field
    rf = 1 + 2 * sum(dilation_rates)
    print(f"  Receptive field: {rf}")
    print()
 
 
if __name__ == "__main__":
    # Analyze different dilation patterns
    analyze_coverage([2, 2, 2])  # Bad: gridding
    analyze_coverage([1, 2, 4, 8])  # Classic exponential
    analyze_coverage([1, 2, 5, 1, 2, 5])  # HDC pattern
    analyze_coverage([1, 2, 3, 4, 5])  # Smooth increase

Gridding in Practice

Architectures Leveraging Dilated Convolutions

Dilated convolutions are central to several landmark architectures, particularly in semantic segmentation.

DeepLab Family:

The DeepLab series (v1, v2, v3, v3+) pioneered the use of atrous (dilated) convolutions in CNNs.

DeepLab v2/v3 (ASPP):

Atrous Spatial Pyramid Pooling
Parallel dilated convolutions at rates like (6, 12, 18)
Multi-scale feature aggregation captures objects of different sizes
Combined with global average pooling for image-level context

DeepLab v3+:

Encoder-decoder with atrous separable convolutions
ASPP in the encoder, refined upsampling in decoder
State-of-the-art segmentation performance

Architectures Using Dilated Convolutions
Architecture	Dilation Strategy	Task	Key Innovation
DeepLab v1/v2	Dense extraction at d=6 or d=12	Segmentation	First use of atrous conv for dense prediction
DeepLab v3	ASPP with d=(6,12,18)	Segmentation	Multi-scale atrous pooling
DeepLab v3+	ASPP + decoder	Segmentation	Encoder-decoder with atrous convs
Dilated ResNet	Replace stride-2 with d=2	Various	Maintains resolution in backbone
WaveNet	d=1,2,4,8,...,512 stacked	Audio generation	Causal dilated convs for large temporal RF
PSPNet	Pyramid Pooling + dilated	Segmentation	Combines pooling and dilation

Dilated ResNet/VGG for Dense Prediction:

A common technique modifies classification backbones for dense prediction:

Take a pretrained ResNet/VGG
Replace later stride-2 convolutions with stride-1, dilation-2
Maintain resolution while preserving receptive field
Output stride becomes 8 or 16 instead of 32

For example, ResNet-101 designed for 224×224 classification (output 7×7) can be modified to output 28×28 for a 224×224 input—7× more spatial detail for segmentation.

WaveNet (Audio):

WaveNet uses causal dilated convolutions for audio generation:

1D convolutions along time
Dilation rates: 1, 2, 4, 8, 16, ..., 512 (repeated blocks)
Achieves receptive fields covering seconds of audio
Enables high-quality speech synthesis

This demonstrates that dilation benefits any domain requiring large context from sequential/spatial data.

Converting Classification Networks for Segmentation

Dilation vs. Other Multi-Scale Approaches

Dilated convolutions are one of several techniques for multi-scale feature extraction. Understanding the alternatives helps in choosing the right approach.

Image Pyramids:

Process the image at multiple resolutions
Run the same network on each scale
Combine predictions

Pros: True multi-scale, mathematically explicit Cons: Computational cost multiplied by number of scales

Feature Pyramids (FPN):

Build hierarchical features from backbone stages
Top-down pathway with lateral connections
Multi-scale features with shared computation

Pros: Efficient, inherent in encoder-decoder Cons: Lower resolution features lose detail

Larger Kernels:

Use 5×5, 7×7, or larger kernels
Directly capture larger patterns

Pros: Simple, dense sampling Cons: Parameter count grows as K², expensive

Multi-Scale Approaches Comparison
Approach	Parameters	Computation	Resolution	Implementation
Standard conv	K²	H×W×K²	Preserved	Simple
Dilated conv	K²	H×W×K²	Preserved	Simple, sparse access
Larger kernel	(dK)²	H×W×(dK)²	Preserved	Simple, expensive
Image pyramid	K² (shared)	Nx(H×W×K²)	Multiple	Complex pipeline
Feature pyramid	K² per level	~2x backbone	Multiple	Architectural change
Pooling pyramid	Minimal	~1.3x	Reduced	Simple additions

When to Use Dilation:

When Alternatives May Be Better:

Object detection across extreme scales: Feature pyramids often outperform
Very fine-grained tasks: Large kernels with dense sampling may capture detail better
Small inputs: Dilation overhead may not be worth it for 32×32 images

Common Hybrid Approaches:

Modern architectures often combine techniques:

FPN backbone + dilated head
ASPP (dilation) + pyramid pooling
Encoder-decoder with dilation in encoder

The Trend Toward Attention

Summary: Dilation for Efficient Large-Context Convolution

Dilated convolutions provide an elegant solution to the receptive field vs. resolution trade-off that defines dense prediction tasks. Let's consolidate the key insights:

Key Takeaways

•Sparse Kernel Sampling: Dilation inserts gaps between kernel elements, expanding coverage without adding parameters. Effective kernel size: Keff = d(K-1)+1.
•Exponential Receptive Field Growth: Stacking layers with d=1,2,4,8,... grows receptive field as 2^(L+1)-1, far faster than standard convolutions.
•Resolution Preservation: Unlike stride>1, dilation maintains output spatial dimensions (with appropriate padding).
•Gridding Artifacts: Repeated dilation rates cause sparse, non-interacting grids. Use HDC patterns or interleaved standard convs to avoid.
•Key Architectures: DeepLab (ASPP), WaveNet, dilated ResNets all leverage dilation for their domains.
•Practical Guidelines: Mix dilation rates, avoid consecutive equal rates, ensure all positions are sampled across layers.

What's Next:

Dilation Mastered

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