Quantization

Suppose you are going to hike in the mountains. To prevent getting lost, you might build a three-dimensional scale clay model of the area you plan to hike in and take the model with you. However, this is likely impractical. In addition to extra weight, such models consume backpack space and do not fold well. So, you compromise and bring a contour map.

Figure 11-1  Sample of a contour map

Contour maps are flat and lightweight, providing a handy terrain representation you must interpret to be useful. Figure 11-1 is a portion of a United States Geological Survey (USGS) map for San Jose, California’s famous Mount Umunhum area. As the map indicates, Mount Umunhum is a 3,486-foot peak, nestled in an exquisite public-land hiking area replete with hidden, babbling mountain springs. To determine this scenic hiking area’s terrain requires simple map interpretation.

Suppose you had a massive earth-plane tool that shaves earth similar to the way wood planes shave away wood. First, set your earth plane to a 3,500-foot elevation and begin shaving Mount Umunhum by making a horizontal pass from above. Since Mount Umunhum’s summit is only 3,486 feet, you do not impact the earth surface, though you do contact a few buildings the map designates with dark rectangles. Lowering your earth plane 40 feet to a 3,460-foot elevation and making another pass shaves off a bit of earth. The peak is now gone and a flat area remains.

The map shows the flat area’s outline with a contour line that surrounds the former peak’s position—representing where the mountain descends from the new flat area. Lowering your earth plane 40 feet to a 3,420-foot elevation and making another pass shaves off more earth. The flattened area expands as the map indicates with another contour line that surrounds the first. Continuing this operation produces enclosing contour lines until you eventually reach sea level. Now let’s interpret the map.

To assist interpretation, the map provides contour line elevation labels every 400 feet starting at sea level. Labeled contour lines are thicker than others. Some thick lines are not labeled and represent 200-foot elevation changes from thick labeled lines. The remaining contour lines represent 40-foot elevation changes. So, if we descended Mount Umunhum’s thrilling Loma Almaden Road by bicycle, would our trip consist of wheeling over flat areas interspersed with precipitous 40-foot plunges naive map interpretation might suggest? Hardly! The map is merely an abstraction that quantizes reality, thereby introducing side effects requiring interpretative compensation.

Specifically, this map creates 40-foot elevation bins and assigns all mountain surface points to specific bins by dividing their elevations by 40 (using integer arithmetic) and then multiplying the quotient by 40. Dividing both 2,610 and 2,620 by 40 produces 65 which when multiplied by 40 produces 2,600. Consequently, the 2,600-foot bin owns all mountain points with elevations from 2,600 feet up to, but not including 2,640 feet.

This scheme reduces a three-dimensional object, the mountain surface elevations, to a two-dimensional contour map abstraction. It does this by converting an infinite number of continuously varying measurements (zero to 3,486-foot elevations) into 88 discrete values (zero feet, 40 feet, 80 feet, 120 feet,..., 3,480 feet). Like the bathroom scale, this quantization simplifies reality by discarding information detail. And, once discarded, no process can recover this detail from the quantized abstraction, though estimation (mathematical interpolation) can provide guesses.

Finally, understanding quantization is important in image processing because digitized images represent quantized abstractions of continuous phenomena. Specifically, image digitization quantizes continuously varying light intensity levels into a few discrete values (16, 32, 256, etc.). In color imaging, it is sometimes necessary to represent color scenes with an infinite number of colors using a very few colors. In both cases, quantization can introduce abrupt transitions the human visual system amplifies. This phenomena’s name is, you guessed it, contouring—where quantized data exerts its visually detectable rebellion.