Deconvolution vs Convolutions

Last Updated : 06 Dec, 2023

Deconvolution is a computation process that reverses the effect of convolution, aiming to recover the original signal or image from the convolved representation. This technique is being used widely in a variety of fields, like image and signal processing, computer vision, deep learning, remote sensing, scientific imaging, and communication systems. This article delves into the fundamental concepts, mathematical foundations, and practical steps involved in deconvolution.

Before understanding the concept of deconvolution, we need to understand convolution.

What is convolution?

Convolution is a mathematical operation that combines two functions to produce a third function. In the context of signal processing and image processing, it involves applying a filter (also known as a kernel) to an input signal or image. The convolution operation is often denoted by the symbol ‘*’, and it is used to express how one function (the filter) modifies another (the input signal or image).

1D Convolution:

In 1D convolution, the filter is a 1D array, and the convolution operation is performed by sliding the filter over the input signal and computing the sum of element-wise products at each position.

$(f\star g)[n] = \sum_{m = -\infty }^{\infty }f[m].g[n-m]$

The formula represents the mathematical formula for the discrete convolution of two sequences f and g.

f and g are discrete sequences.
$(f\star g)[n]$ denotes the convolution output at the position n.
the summation is performed over all possible values of m (from negative infinity to positive infinity).

2D Convolution:

In 2D convolution, the filter is a 2D matrix, and the convolution operation is applied to a 2D input, such as an image. The filter is moved across the image, and at each position, the element-wise product is computed and summed.

$(I\star K)[i,j] = \sum_{m} \sum_{n} I[i-m, j-n].K[m,n]$

The formula represents the mathematical forumla for the 2D discrete convolution of an image I with a kernal K.

I and K are 2D matrices representing an image and a convolution kernel, respectively.
$(I \star K)[i,j]$ denotes the convolution output at position (i,j).
The summation is performed over all possible values of m and n.

In convolutional neural networks (CNNs), convolution layers use learnable filters to extract features from input data. These filters are updated during the training process.

What is Deconvolution?

Deconvolution is a computational process that aims to invert a convolution operation, reconstructing the original signal or image from its convolved representation. Convolution involves the mathematical integration of an input signal with a filter or kernel, producing a transformed output. Deconvolution is employed to reverse this convolution process, undoing the effects of the convolution and restoring the initial signal or image.

Imagine taking a photo with a camera that has a limited resolution, causing some blurring in the captured image. This blurring effect can be considered a convolution operation, as the details in the scene get mixed together due to the limitations of the imaging system. Now, let’s say you want to enhance the details and recover a sharper version of the original image. This is where deconvolution comes into play.

In this real-life example, deconvolution would be akin to using a photo editing tool to reverse the blurring effect and bring back the crisp details in the image. By applying a deconvolution operation, you effectively “undo” the convolution that occurred during the photo capture process. The result is an enhanced image that more closely resembles the finer details of the actual scene. So, deconvolution, in this context, acts like a digital “sharpening” tool, allowing you to recover or emphasize details that might have been lost or blurred in the original image due to the limitations of the imaging system.

In the discrete domain, the 1D deconvolution is defined as:

$(f⊛g)[n]=∑ m f[n−m]⋅g[m]$

And for 2D signals:

$(I⊛K)[i,j]=∑ m ∑ n I[i+m,j+n]⋅K[m,n]$

In the context of neural networks and deep learning, deconvolution is often implemented using transposed convolution or fractionally strided convolution. It involves applying a filter to the input with a certain stride and padding to increase the spatial resolution of the output.

Fourier Transform:

The convolution operation in the spatial domain is equivalent to multiplication in the frequency domain. Mathematically, if $F$ is the Fourier transform of $f$ and $G$ is the Fourier transform of g, then :

$F(f∗g)=F⋅G$

Deconvolution, in turn, involves dividing the Fourier transform of the convolved signal by the Fourier transform of the kernel:

$F(f⊛g)= G F $

Inverse Filtering:

Deconvolution can be seen as a form of inverse filtering. If H is the Fourier transform of the kernel, then the inverse filter is given by:

$F(f⊛g)= 1 / H ⋅F$

However, inverse filtering can be sensitive to noise, and regularization techniques are often employed to stabilize the deconvolution process.

Wiener Deconvolution:

Wiener deconvolution is a common approach that combines information from the observed signal, the point spread function (PSF), and an estimate of the noise to recover the original signal. The Wiener deconvolution in the Fourier domain is given by:

$F(f⊛g)= G* / ∣G∣ 2 + N S G ∗ ⋅F$

Here, G* is the complex conjugate of G , and S/N is the signal-to-noise ratio.

Applications of Deconvolution

Deconvolution is widely used to reconstruct high-quality images from degraded or blurred versions, making it valuable in fields like medical imaging, where clear and detailed images are crucial for diagnosis.
In microscopy, deconvolution helps improve the resolution and clarity of images, allowing researchers to observe cellular structures and processes in finer detail.
Deconvolution is applied to enhance the quality of satellite images, aiding in tasks such as land cover classification, environmental monitoring, and disaster response.
In art restoration, deconvolution techniques are employed to restore and enhance digital images of paintings or cultural artifacts, bringing out finer details while minimizing noise.
Deconvolution is used in audio processing to recover the original signal from a convolved or distorted version. This is beneficial in fields like music production and speech analysis.
Deconvolution plays a role in communication systems to equalize the effects of signal distortion during transmission, ensuring the integrity of transmitted information.
Deconvolution contributes to super-resolution imaging by enhancing image resolution beyond the limitations of the original data. This is valuable in applications like surveillance and high-resolution photography.
Deconvolution is applied in various biomedical imaging techniques, such as positron emission tomography (PET) and magnetic resonance imaging (MRI), to improve image quality and enable more accurate diagnoses.
Deconvolution is employed in astrophysics to process images captured by telescopes, allowing astronomers to study celestial objects with greater detail and precision.

Frequently Asked Questions (FAQs)

1. What Libraries can be Used for Deconvolution in Python?

Popular libraries for deconvolution in Python include NumPy and SciPy for general numerical operations, and deep learning frameworks like TensorFlow and PyTorch, which provide specialized functions for deconvolution layers in neural networks.

2. What is the Purpose of Deconvolution in Image Processing?

In image processing, deconvolution is used to reverse the effects of blurring or convolution, enhancing image details, removing artifacts, and improving image quality. It is commonly applied in tasks like image restoration, deblurring, and super-resolution.

3. How can Blind Deconvolution be Achieved in Python?

Blind deconvolution is a challenging task where the convolution kernel is unknown. In Python, iterative optimization algorithms can be employed to simultaneously estimate the unknown kernel and the original signal.

4. What is Transposed Convolution, and How is it Related to Deconvolution?

Transposed convolution (or fractionally stride convolution) is often used for deconvolution in deep learning. It involves padding and sliding a filter over the convolved data to expand it back to its original size. In neural networks, the term “deconvolution” is sometimes used interchangeably with transposed convolution.

5. Can Deconvolution be Applied to Multidimensional Data?

Yes, deconvolution can be applied to multidimensional data, such as images or 3D volumes. The process involves using multidimensional filters or kernels and extending the deconvolution operation accordingly.

6. How to Choose the Deconvolution Kernel?

The deconvolution kernel is often chosen based on prior knowledge of the convolution process. In cases where the convolution kernel is unknown, blind deconvolution techniques may be employed. Additionally, regularization methods can be used to stabilize the deconvolution process.

Suggest improvement

What is Fractional Convolution?

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