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# Image Reduction

[Laboratory]

Using the same principles shown on the lesson Image Enlargement , we illustrate here how to reduce the size of an image. We split the image reduction algorithm in several primitive operations to ease explanation. We also show the results of these primitive operations both in the spatial as in the frequency domains.

### Reduction without interpolation

First, lets look at the original image and its spectrum. The image was taking from a portion of a cantata screen dump. You may notice that the image has high frequency components due to its synthetic graphic nature.

a)original image; b)its spectrum
a) | b)

The easiest way to reduce the image is by just sub-sampling its pixels. So if we want to reduce its size by 4 in each dimension, we just select one pixel at every four. To fully understand how this processing can be seen in the frequency domain, we first multiply the image by an impulse image, which the impulses located in the positions of the sampling. Below is the impulse image and its spectrum. Remember that the dft of an impulse image is also an impulse image (See DFT of Simple Images ).

a)impulse; b)its spectrum
a) | b)

Multiplying the original by the impulse image in the spatial domain is equivalent of convolving their DFT's.

a)original sampled at impulses; b)its spectrum (convolution)
a) | b)

We can now apply the image expansion DFT property. See in the DFT Properties lesson. The resultant DFT is exactly the inner part of the convolved DFT. We can interpret the poor result of the reduced image due to the convolution process carried out in the frequency domain, which corrupts the inner spectrum of the final image. To avoid this corruption process, we have to limit the bandwidth of the image before re-sampling. We will see this in the next steps.

a)reduced image; b)its spectrum (the inner of dft)
a) | b)

### Reduction with interpolation

We repeat the above steps, but before, we low pass filter the original image in order to limit its bandwidth. The cutoff frequency to use depend on the amount of reduction. We use an ideal circular filter to illustrate the following steps. By using other low pass techniques, both in frequency and spatial domains, one can get a large variety of interpolation techniques. All these interpolation technique uses the same principles and concepts shown here.

a)low pass filtered; b)its spectrum
a) | b)

The following steps are exactly the same used in the previous reduction process: multiplication by the impulse image and image reduction. One may note that the convolution of the spectrum and the impulse DFT does not create any overlapping on the sampled image.

a)impulse; b)its spectrum
a) | b)

a)original sampled at impulses; b)its spectrum (convolution)
a) | b)

a)reduced image; b)its spectrum (the inner of dft)
a) | b)

### Comparison

Finally, for comparison, we show below the sub-sampled image with and without interpolation. Note that the grid in the original image is well preserved in the interpolated image.

a)sub-sampling; b)sub-sampling with interpolation
a) | b)

- Lab: Image Reduction