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# Image Enlargement

[Laboratory]

Interpolation is necessary when the image is transformed geometrically and there are pixels which must have their values predicted.

Interpolation can be seen as a low pass filtering. Depending on the kind of filtering, one can get many different interpolation methods.

What follows is a case study showing the expansion by 4 in both dimensions and four interpolation methods: nearest-neighbor, linear, ideal and butterworth. In all cases the spectrum is calculated to give a better understanding of its frequency components.

Given the original image and its spectrum:

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

Making a geometrical expansion by 4, using the equation g(4x,4y) = f(x,y), where f is the original image and g is the expanded. The other pixels of g are left as zero.

Below is the expanded image and its spectrum:

a)expanded by 4; b)its spectrum
a) | b)

We can note that the DFT of the expanded image was replicated by 4. This can be explained by recalling the DFT scaling property ( see DFT Properties ) and remembering that the DFT is periodic.

The interpolation process must filter higher frequencies of the expanded image in order to preserve the original ones. If on the original image the minimum period possible was 2 pixels (by the Nyquist criteria), in the 4 times expanded image we should have the minimum period as 8 pixels. It is necessary to filter the image with a cutoff period of 8.

We can use four different low pass filters:

- Average
- Bilinear
- Ideal
- Butterworth

### Nearest-Neighbor interpolation: average 4x4 filter:

Also called pixel replication and zero order interpolation.

a)expanded by 4; b)average kernel
a) | b)

a)interpolated; b)its spectrum
a) | b)

### Linear interpolation

For two dimension (image) it is called bilinear interpolation or first order interpolation.

a)expanded by 4; b)bilinear kernel(multiplied by 16)
a) | b)

a)interpolated; b)its spectrum
a) | b)

### Ideal interpolation

Filtering in the frequency domain using an ideal filter:

a)expanded by 4; b)ideal low pass filter
a) | b)

a)filtered spectrum; b)interpolated
a) | b)

### Butterworth interpolation

Filtering in the frequency domain using the butterworth filter:

a)expanded by 4; b)low pass butterworth
a) | b)

a)filtered spectrum; b)interpolated
a) | b)

### Comparing the four together:

a)Nearest Neighbor; b)Bilinear
a) | b)

a)Ideal; b)Butterworth
a) | b)

- Lab: Image Enlargement