The tax is not what's unfair

This morning was not the first time I heard the sentiment - raising taxes on junk food and cigarettes discriminates or burdens the poor.
We are loathe to increase taxes, even when it makes sense. For instance, the soda tax can't seem to get on the books, even though evidence supports that sugary drinks lead to over consumption of calories > obesity > diabetes, and raising the price of SSB can reduce this caloric imbalance.  In Denmark, however - where there is  much less prevalence of obesity than in the US and UK, foods that have a certain amount of saturated fat will cost more than others - beginning today.
I am 100% for such taxes and I will tell you this - when people respond by saying that taxing food that is calorically dense and nutrient poor is putting a burden on disadvantaged groups my response is... "THAT is exactly the POINT".  Not to be unfair, but to target the problem.  Food that is related to obesity is, more often than not, the cheapest food in the store .  It is the PRICE that is discriminatory.  So fine - don't raise the cost through taxes, but how about we see a sign like the one posted below that tops shelves of apples and grapes instead of crap.