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:

And for 2D signals:

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 is the Fourier transform of and is the Fourier transform of g, then :

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

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:

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:

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

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.