Changing a low-resolution image into a high-resolution image is simply a matter of increasing the number of pixels per inch that are mapped in the image's meta data. This is something you can change using any good graphics app, including Photoshop, Paint.net or GIMP. Photoshop is available as a free trial from Adobe, while Paint.net and GIMP are free to use.
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Much like playing an old, low-resolution movie on a new HDTV, increasing a photo's resolution doesn't actually change what you see. However, your printer may be able to print it in higher quality, depending on the printer's capabilities. Websites that have minimum resolution requirements may also accept an image that has been edited this way. To change how an image appears on the screen, you also need to increase its height and width.
Increasing Resolution With Photoshop
Click Photoshop's Image menu and select Image Size. Make a mental note of the current width. Click the Resolution menu and change it to what you need. Photoshop changes the image size whenever you change the resolution.
Change the image width back to the original size if you only want the resolution increased. With the aspect ratio locked, Photoshop automatically resizes the height to the original size as well. Click OK.
Click the Reduce Noise slider if you intend to leave the image at its new size and not only at its new resolution. Drag the slider slowly to the right to reduce the amount of noise caused by the resampling. Watch the edges of lines and colors in the preview as you change the noise level. Drag the Reduce Noise slider to the left if the lines become blurry. Click OK after you find the best balance in noise reduction.
Note that the image is resampled by default to preserve details. If you click the Resample menu, you can change to another resampling method; however, in most cases, the default setting should work best. No resampling method is perfect and will result in noise, or pixelization, when you increase the resolution.
Increasing Resolution With Paint.net
Click the Image menu in Paint.net. Select Resize.
Click the Resolution menu and change it to what you need. Then click OK to save your changes.
Unlike Photoshop, the image dimensions don't change. If you want to increase the image size, increase the height or width or click the By Percentage option and enter a percentage to change the dimensions.
Increasing Resolution With GIMP
Click the File menu in GIMP. GIMP allows you to specify different resolutions for displaying the image on a screen and for printing. If you are only increasing the resolution for printing, select Print Size. Otherwise, select Scale Image.
Click the X Resolution and increase it as needed. Click the Y Resolution and increase it by the same amount. Click OK. While GIMP does allow you to set different vertical and horizontal resolutions, in most cases, you should keep them the same.
If you only increase the image resolution, retouching shouldn't be necessary. However, if you increased the image dimensions as well as the resolution, you may want to touch up details in the image, using a sharpening tool or by painting the image using a brush tool.
Photoshop, Paint.net and GIMP all have a in their toolboxes, which is useful for retouching images. Ctrl-click or Alt-click, in the case of Photoshop, and select a sample where the pixels look good. Then drag the Clone Stamp Tool over areas with the same color as the sample area to clean up blotchy or pixelated areas.