Image Expression Null Value Handling

• The entire expression is one single expression resulting in one single value.

• Everything is an expression and produces one single value. So, even an if-then-else is just one expression resulting in one single value. Therefore, you can sum the results of two separate if-then-else's like (if the else) + (if then else). Since even the test clause of an if-then-else is an expression, you can also replace that with another if-then-else. Example: if (if i1 > 0 then 1 else i2) then i2 else 1 + i3

• The if-then-else test assumes that 0 is false and any non-zero value is true.

• Every time the expression being evaluated results in null, the entire expression becomes null. So, i1 + null will always be null since something + null is always null.

• The only operators that can contain the destructive power of a null value are isNull( expression ) or expression1 ? expression2:
  1. isNull( expression ) evaluates the expression and results 1 if the expression evaluates to null, and 0 otherwise.
  2. expression1 ? expression2 evaluates the expression1 and, if the result is non-null, returns that value. Otherwise, it evaluates expression2 and returns the resulting value as its own result.

• Null can come from an image, but it can also be produced by some other events, such as dividing by zero, failing to find a key in a lookup table etc.

• If the if-then-else test evaluates to null, the entire if-then-else evaluates to null. Example:

• (if (i1/3 > 9) then i2 else 1 + i3) + 10:
  1. If i1 is null, i1/3 is also null, and i1/3 > 9 also becomes null. Therefore, the if-then-else test is null and the entire if-then-else is null.
  2. Now, null is being added to 10, and that also becomes null.

• ( (if (i1/3 > 9) then i2 else 1 + i3) + 10 ) ? 20 – Note that we are using the operator ? now:
  1. If i1 is null, i1/3 is also null, and i1/3 > 9 also becomes null. Therefore, the if-then-else test is null and the entire if-then-else is null.
  2. Then, null is being added to 10, and that also becomes null.
  3. Now, null is being caught by the expression null ? 20, since the if then else + 10 is null, and the result is 20.

The null evaluation mechanism allows us to write short expressions like i1 + 10 to add 10 to every valid cell of the i1 map, or i1 * i2 to multiply two maps. If one of the map cells is null, the result is null, so in the last example, we are basically just multiplying the corresponding cells where the two values are valid: everything else will be null.

If we want to treat all null values as values 1s, in the previous expression, we can write (i1 ? 1) * (i2 ? 1). We can also use the more complicated form using the if-then-else to achieve the same result such as (if isNull(i1) then 1 else i1) * (if isNull(i2) then 1 else i2), or even

if isNull(i1) and isNull(i2) then 1
else if not isNull(i1) and not isNull(i2) then i1*i2
else if not isNull(i1) and isNull(i2) then i1
else i2

If you need to retrieve the number corresponding to an image null value, you can use the null( iX ) expression. For instance, null(i1) returns the number corresponding to the null value of image i1. So if, so some reason, you need to retrieve the image value or its corresponding numeric null value, you can use i1 ? null(i1) or if isnull(i1) then null(i1) then i1. Note that null(i1) is very different from expression null, since null(i1) always returns a valid numeric value and null returns a special construct that, if not contained, invalidates the expression evaluation producing null as a result.

Lets assume that we want to replace all occurrences of a given valid value in a image with some other valid valid: if i1 = 2 then 10 else i1. Since we are not explicitly guarding the if-then-else test against null, the expression is internally equivalent to if isNull(i1) then null else if i1 = 2 then 10 else i1. So, simply using the most compact if-then-else definition works as expected.

Now, lets assume the we want do combine two images, a landscape map and road map, as a single map, where roads in the road map are represent by the value 1 and everything else is represented using the null value. Assuming the i1 is the landscape map and i2 is the road map, we might be tempted to write an expression as if i2 = 1 then 1 else i1. Unfortunately that is not going to work, since the expression is internally equivalent to if isNull(i2) then null else if i2 = 1 then 1 else i1 (we not guarding against null values). As you can see, all occurrences of null values on the road map will be replicated in the output map.

So, the proper way of doing the calculation is using a isNull( iX ) to protect the if-then-else test against the null: if not isNull(i2) and i2 = 1 then 1 else i2. Now we only test for the value in the road map if we are sure that value is not going to contaminate the whole if-then-else, turning the result into null.